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E<Wjt, I ll«r^b,H5
►
HARVARD
COLLEGE
LIBRARY
THE GIFT OF
Miss Ellen Lang Wentworth
of Exeter, New Hampshire
5ooooooooooooooooooooooooooooooooooooooooooooo£
c
3 2044 097 012 348
* THE
Elements of Algebra.
BY
G. A. WENTWORTH, A.M.,
AUTHOR OF A SERIES OF TEXT-BOOKS IN MATHEMATICS.
TEACHERS' EDITION.
BOSTON, U.S.A.:
PUBLISHED BY GINN & COMPANY.
1895.
HARVARD COLLEGE LIBRARY
GIFT OF
MISS ELLEN L*WENTWORTH
HAY * 1W3
Entered according to Act of Congress, in the year 1882, by
G. A. WENTWORTH,
in the Office of the Librarian of Congress, at Washington.
Typography by J. 8. Cvshino & Co., Boston, U.S.A.
PRESSWOBK BY OlNN & Co., BOSTON, U.S.A.
PREFACE.
THIS edition is intended for teachers, and for them only. The
publishers will make every effort to keep the book from
pupils; and teachers are urged to exercise the utmost care not to
lose their copies, or to leave them where pupils can have access
to them.
It is hoped that young teachers will derive great advantage
from studying the systematic arrangement of the algebraic work,
for such attention has been paid to this as the limitation of the
page would allow.
It is also expected that many teachers, who are pressed for
time, will find great relief by not being obliged to work out
every problem in the Algebra.
G. A. WENTWORTH.
Exetee, N. H.,
December, 1882.
ALGEBEA.
Exercise I.
Whena=l, & = 2, c = 3, d=4, e=5, /=0:
1. 9a + 26 + 3c-2/ a *
= 9 + 4 + 9-0 &«
= 22. 4s
= F5
2. 4e-3a-36 + 5c
= 20-3-6 + 15
= 26.
10.
= 2.
e« + 6«
3. 8afc — 6crf + 9cde — def c* — 6«
= 48-24 + 540-0 ^ 58 + 2
= 564. 3* - 2s
125 + 2
a ^ac 86c bed D g__g
& d e =127
= 6 + 12-12
= 6' 11. 6" + *
5. 7e + 6crf-^-^ 2* + 4s
2ckj = -
= 35 + 24-20 22o+4l78
= 39. 8 + 64
4 + 16-8
G. aSc' + ic^-de^+Z8 =6.
= 18 + 96-20 + 0
= 94. ^ e«_*
7. e* + 6e'&* + J*-4e»&-4e&3 c +e4 + ls
= 625 + 600 + 16-1000-160 = ° ~* m
= 81 5* + 20 + 4s
125-64
8 8aa + 3&* 4c2 + 65' c* + d* 25 + 20 + 16
8 + 12 36 + 24 9+16^
4 5 25 13. 100 + 80 -+-4
= 5 + 12-1 =100 + 20
- 16. = 120.
2
ALGEBRA.
14.
75 - 25 x 2
20.
3a; + 7y-*-7+aXj/
- 75 - 50
= 9 + 35 + 7 + 2x5
= 25.
= 9 + 5 + 10
= 24.
15.
25 + 5x4-10-5-5
= 25 + 20-2
21.
6b-8y + 2yxb-2b
= 60-40-5-100-20
= 43.
= 60-f-20
16.
24-5x4-*- 10 + 3
= 39$.
= 24-20-5-10 + 3
22.
(66-8t/)-s-2yx6 + 26
= 24-2 + 3
= (60-40)-*- 10x10 + 20
= 25.
= 20-5-100 + 20
= 201.
17.
(24-5)x(4 + 10 + 3)
= 19x($ + 3)
= 19x4*
= 64|.
23.
(66-8y)-r(2yX6)+26
= (60 - 40) -5- (10 x 10) + 20
= 20-5-100 + 20
= 20*.
24.
Qb-(8y + 2y)xb-2b
18.
zy + 4ax2
= 15 + 16
= 60 -(40 + 10) X 10-20
= 60-40-20
= 31.
= 0.
19.
xy — 156-5-5
25.
6&-*-(&-y)-3a; + fory-5-10a
= 60 -5-(10-5)-9 + 150+20
= 15-1^
= 15-30
= 12-9 + 7*
= -15.
= 10*.
26. Express the sum of a and b.
a + 6.
27. Express the double of x.
2x.
28. By how much is a greater than 5?
a -5.
29. If x be a whole number, what is the next number above it?
z + 1.
30. Write five numbers in order of magnitude, so that x shall
be the middle number.
x— 2, x — lt x, a? + l, x + 2.
31. What is the sum of x + x + x + written a times?
ax.
32. If the product be xy and the multiplier z, what is the
multiplicand? xy + x=y.
teachers' edition.
33. A man who has a dollars spends b dollars; how many
dollars has he left? *
a — o.
34. A regiment of men can be drawn up in a ranks of b men
each, and there are c men over: of how many men does the reg-
iment consist? # ab + c
35. Write, the sum of x and y divided by c is equal to the
product of a, 6, and m, diminished by six times c, and increased
by the quotient of a divided by the sum of x and y.
c % + y
36. Write, six times the square of n, divided by m minus a,
increased by five b into the expression c plus d minus a.
6n* +5b(c + d-a\
m — a
37. Write, four times the fourth power of a, diminished by
five times the square of a into the square of &, and increased
by three times the fourth power of b.
4o4-5a»&2 + 36*.
Exercise IT.
3. The greater of two numbers is six times the smaller, and
their sum is 35. Required the numbers.
Let x = smaller number.
Then 6 a? = larger number,
and 6 x + x = sum of numbers.
But 35 = sum of numbers.
Therefore, 6a; + x = 35, x = 5, and Qx = 30.
4. Thomas had 75 cents. After spending a part of his money,
he found he had twice as much left as he had spent. How much
had he spent ?
Let x = number of cents spent.
Then 75 — x = number of cents left.
But 2x = number of cents left.
Therefore, 75 - x = 2x, - 3a; = - 75, and x = 25.
ALGEBRA.
5. A tree 75 feet high was broken, so that the part broken off
was four times the length of the part left standing. Required
the length of each part.
Let x = number of feet left standing.
Then 4x = number of feet broken off,
and x + 4x = number of feet in height.
But 75 = number of feet in height.
Therefore, 5 x = 75, x = 15, and 4x = 60.
6. Four times the smaller of two numbers is three times the
greater, and their sum is 63. Required }he numbers.
Let x = smaller number.
Then 63 — x = larger number,
and 4 x = 4 times smaller ;
also, 3 (63 — x) = 3 times greater.
.Ax =3 (63 -a:), 4a?=189-3z, 7x=189, x = 27, and 63 - x = 36.
7. A farmer sold a sheep, a cow, and a horse, for $216. He
sold the cow for seven times as much as the sheep, and the
horse for four times as much as the cow. How much did he
get for each ?
Let x = number of dollars received for sheep.
Then 7x = number of dollars received for cow,
and 28 x = number of dollars received for horse,
and x + Ix + 28 a = number of dollars received for all.
But 216 = number of dollars received for all.
.\a? + 7a: + 28a;=216, 36a = 216, x = 6, Ix = 42, and 28a =168.
8. George bought some apples, pears, and oranges, for 91
cents. He paid twice as much for the pears as for the apples,
and twice as much for the oranges as for the pears. How much
money did he spend for each ?
Let x = number of cents paid for apples.
Then 2x = number of cents paid for pears,
and 4» = number of cents paid for oranges,
and x + 2x + 4x*= number of cents paid for all.
But 91 = number of cents paid for all.
.\a: + 2a> + 4a: = 91, 7a=91, a?=13, 2x=26, and 4z=52.
9. A man bought a horse, wagon, and harness, for $350. He
paid for the horse four times as much as for the harness, and for
the wagon one-half as much as for the horse. What did he pay
for each ?
TEACHERS EDITION.
Let x = number of dollars paid for harness.
Then 4 a; = number of dollars paid for horse,
and 2x = number of dollars paid for wagon,
and x + 4a; + 2a; = number of dollars paid for all.
But 350 = number of dollars paid for all.
.\a; + 4a; + 2a; = 350l 7 a; = 350, x = 50, 4a; = 200, and 2a; =100.
10. Distribute $3 among Thomas, Richard, and Henry, so
that Thomas and Richard shall each have twice as much as
Henry.
Let x = number of dollars Henry has.
Then 2x = number of dollars Thomas has,
and 2 a; = number of dollars Richard has,
and a; + 2a; + 2a; = number of dollars all have.
But 3 = number of dollars all have.
Therefore, a; + 2x + 2a; = 3, 5a; = 3, x = $, and 2a; = 1J.
11. Three men, A, B, and C, pay $ 1000 taxes. B pays four
times as much as A, and C an amount equal to the sum of what
the other two pay. How much does each pay ?
Let x = number of dollars A pays.
Then 4 a; = number of dollars B pays,
and 5 a: = number of dollars C pays,
and a; + 4a; + 5a; = number of dollars all pay.
But 1000 = number of dollars all pay.
.\a; + 4a; + 5a; = 1000, 10a; = 1000, a;=100, 4 x = 400, and 5 x = 500.
Exercise III.
1. +16 + (-11) 4. -7 + (+4)
= 16-11 =-7 + 4
= 5. =-3.
2. -15 + (-25) 5. +33 +(+18)
==_15_25 =33 + 18
= -40. =51.
3. +68 + (-79) 6. +378 + (+709) + (-592)
= 68-79 = 378 + 709 - 592
= -11. =495.
7. A man has $5242 and owes $2758. How much is he
worth ?
f 5242 + (- $2758) = f 5242 - $2758 = $2484.
8. The First Punic War began B.C. 264, and lasted 23 years.
When did it end ?
- 264 + (+ 23) = - 264 + 23 = - 241 ; i.e. 241 b.c.
6 ALGEBRA.
9. Augustus Caesar was born B.C. 63, and lived 77 years.
When did he die ?
- 63 + (+ 77) = + 14 ; i.e. 14 a.d.
10. A man goes 65 steps forwards, then 37 steps backwards,
then again 48 steps forwards. How many steps did he take in
all ? How many steps is he from where he started ?
65 + 37 + 48 = 150. 65 - 37 + 48 = 76.
Exercise IV.
1. 5ab + (-bab) 7. 120my + (-95ray)
= 5 ab — 5 ab = 120my — 95 my
= 0. = 25my.
2. Smx + (-2mx) 8. -33o62 + (lla62)
= 8mx- 2mx = - 33a&2 + Ha6*
= 6mz. = -22a&2.
3. — 13 mng + (- 7 mng) 9. - 75<cy + (+ 20zy)
= — \$mng — *lmng — ~7oxy + 20xy
= — 20mng. = — 55xy.
4. -5s2 + (+8a?) 10. +15a*s2 + (-a2.T2)
= - 5ar» + 8a2 = 15a2ar» - a2x2
= 3r». = 14a2s2.
5. 25my2 + (-18my2) 11. -62ms + (+ 762m8)
= 25my2 - 18my2 = - 62ms + 7 b2m*
= 7my* *662m8.
6. 7a6 + (-5ai) " 12. 5a + (-36)+(+4a) + (-7&)
= 7ab — 5db =5a — Sb + ia — 7b
= 2a6. = 9a -105.
13. 4a2c + (- 10zyz) + (+ 6a2c) + (-9zyz) + (- lla2c) + (+ 20xyz)
= 4a2c — lOxyz + 6a2c — 9xyz — 11 a?c + 20xyz
= — a*c + xyz.
14. 3rc2y + (-4a6)+(-2mn) + (+5rc2y) + (-ariy) + (-4a;2y)
= 3a?y — 4a6 — 2mn + 52?y — x*y — 4a:2y
= 3a^y — 4a6 — 2mn.
Exercise V.
1. 5a + 36+ c 2. 7a-46+ c
3a + 36 + 3c 6a + 36-5c
a + 36 + 5c —12a + 4c
9a + 96 + 9c a- 6
TEACHERS EDITION.
3. o+ b — c
— a+ b + c
a— b + c
q + b — c
2a + 2b
4. a + 26 + 3c
2a — b-2c
— a + b — c
— a— b + c
a+ b + c
5. a-26 + 3c-4a*
-2a + 36-4c + 5a*
+ 3a-46 + 5c-6a*
— 4a + 56 — 4c + 7a*
-2a + 26
+ 2d
Xs- 4ar2 + 5a?- 3
2a* -14a?- 14a: + 5
- ar*+ 9x* + g?+ 8
2a*- 9^- 8a; + 10
11. 3a?2 — xy + xz —
— 5a?2 — xy — xz
6a?2
7. «*-2a» + 3a«
x* + x2 + x
4a;4 + 5a:8
+ 2a?2 + 3a?-4
-3s*-2a?-5
5z4 + 4ar» + 3ar2 + 2a?-9
8. a8 + 3ai2-3<z2&- 6s
2a8-6a&2 + 6a*& -76s
a8- a&2 +26*
4a8-4a&2+2a*& + ft8 -76*
9. 2a6 + 2a2a?- Sax9
12ab-6a*x + 10ax*
— 8a6 — 5a*a? + ax*
6ab — 9a2a? + 7oa?2 + aa?8
10. c4-3c8 + 2c2-4<? + 7
2c* + 3^ + 2^ + 5c + 6
-4c* -4c2 -5
- c*
3y2+ 4yz- z2
+ 5yz
+ c + 8
-4a?2
-2zy
4yz
- 5yz + 3z2
+ y2 + 3yz + 3z*
■6.J/-6Z
- 2y2 + llyz + 5z2 - 6y - 6z
12. m
»5 — 3m4n — 6m8n2
5m*n + ra8n2 + ra2n8
+ 7ra8n2 + 4m2n8 — 3mn*
— 2m2n8 — 3mn4 + 4n5
3m6 + 2mn4 + 2n6
2m5 + 7m4n — nft
6m5— m4n + 2m8n2 + 3m2n8 — 4ran4 + 5n6
Exercise VI.
1. +25-(+16)
= 25-16
= 9.
2. _50-(-25)
= - 50 + 25
= -25.
3. _3i_(+58)
= -31-58
4. +107 -(-93)
= 107 + 93
= 200.
8 ALGEBRA.
5. Borne was ruled by emperors from B.C. 30 to its fall, a.d.
476. How long did the empire last?
476 - (- 30) = 476 + 30 = 506 ; i. c. 506 years.
6. The continent of Europe lies between 36° and 71° north
latitude, and between 12° west and 63° east longitude (from
Paris). How many degrees does it extend in latitude, and how
many in longitude?
71-(+36) = 71-36 = 35.
12-(-63)=12 + 63 = 75.
Exercise VII.
1. 5x — (—4x) 8. — 4ixy~ (— 5xy)
= 5<c-f4a; = — ixy + dxy
= 9.x. = xy.
2. -3ab-(\-5ab) 9. Sax-(-3ay)
-3ab — 5ab = 8 ax + 3 ay.
= -Sab.
3a&2 - (+ lOafc2) = 2al2y - aby.
10. 2ab*y-(+<iby)
-S^lOaV n 9a, + (5^_(+8rf)
--(a0- = 9s2-f5z2-8s2
4. 15m2*2 -(-7m2*2) = 6z*.
= 22m2x2. =5W + 18x2/-10^y ^
5. -7at,-(-3ay) =13^
= -7ay + 3ay 13. 17aa* -(-a*8) -(+ 24a*8)
= -4ay. =17aaJ, + ar,-24aa!8
6. 17axs-(-24arl) = -6aa*
= 17ax8 + 24a;c8 14. -3ab + (+ 2mx) -(-4mx)
= 41 ax*. = — 3ab + 2mx + 4:mx
= — 3a6 + 6ma;.
7. 5a2»-(-3a2ic)
= 5a2x + 3a2a; 15. 3a-(+ 2&)-(-4c)
= 8a2a\ = 3a -26+ 4c.
Exercise VIII.
6a-2&~ c 2. 3a-26 + 3c
2q-26-3c 2a-86- c
4o +2c a + 66+4c
TEACHERS EDITION.
3. 7x»-8aj-l 6. x*-3xy- y*+yz-2z*
5x*-6x + 3 x2 + 2xy-Sy2 -2z* + 5xz
2x2-2x-4t -5zy + 2y2 + yz -bxz
4. 4** -33* -2a?- 7x+ 9 7. o8-3a2& + 30&2- 6»
x*-2x*-2x* + Ix- 9 -a8-+ 3a26-3a62 + 6»
3a?*- a* -14x+18 2a8-6o26 + 6aP-2P
5. 2ics-2aa; + 3aa 8. x*-bxy+ xz-y2 + 7yz + 2z-
x2 — ax+ a2 x2— xy— xz -f2yzf3z*
Xs— ax + 2a2 -4a?y + 2x2-y* + 5y2- 2*
9. 2ax2 + 3afcc-4&2a; + 12&8
clx2 — 4abx — 5b2 x + bx2 — x*
ax2 + labx + 62a: + 12 63 — bx2 + x3
10. ear1^^ + 4zy2-2y8- 5 a? + xy- 4y2 + 2
8a^-7aay+ ay2- y8 + 9s2- xy + 6y2-4
-2ar» + 3ay2- y8- 14a? + 2zy- 10y2 + 6
11. a4- ft*
-f4q86-6a*&* + 4ay
a*_ 6*-4a86 + 6a26i-4a6s
2q*-2&*-4a86+6q2&2 + 4ay
- a* + 6* -Satf1
12. a?y2-3x2y8 + 4sy*- y5
— 4sy* — 4.^ — a? + 2ar*y
ar»y2 - Sx2^ + 8*y + 3y* + a? - 2a*y
a?y2 — SaPy* + 4sy* — y5
— 4zy* — 4y* — a^ + 2ar*y
x*y2-3x2y* -Sy6- ar> + 2z*y
a?y2 — SarV + 8sy* + 3y5 + xb—2x*y
-Sxy*-8y*-2x* + lx*y
13. a252- o26c-8a62c-a2c2+ a&^-G&V2
+ 2a2 be - 5ab2c + 2 ah? - 562c2
a^-Stf&c-Sa^c-aV- a&c2- &2<?
14. 12a + 36-5c-2rf=69 16. 2a?-y2-2a# + z2
10a- 6 + 4c-3rf = 45 a?-y2 + 2sy-- 22
2<z+46-9c + d = 24 a? -4ay+222
15. b-a.
2a8-6o26 + 6a62-26s 17. 12ac+8ca*-9
a*-*Ja2b -3&8 -lac- 9cd+ 8
as+ a2b + §at?+ 6s 19ac + 17cd-17
10 ALGEBRA.
18. - 6a2 + 2a6-3c2 22. 3a;2 + 2&y- y2
4a24-6a6-4c2 - s»-3ay + 3,y2
-10a2-4a6 + c2 4s2 + 5a)y-4y2
3s2 + 4sy — 5y2
19. 9sy-4a;-3y + 7 ** + ^ + y2
sy-2s-6y + l + ex2 - <fr2
o^ — ty2 — car2 + dy1
20. — a2 6c— a62c-f ab&— abc
a2bc + ab2c- abc2+ abc 24. as + bx + by + cy
-2a2bc-2ab2c + 2abc2-2abc F~ bx~ hV + ^
2&c + 26y
21. 7a2-2a; + 4 26. 5s2 + 4a-4y + 3y2
2^-f 3s--l 5s2-3s+3y + y2
5a2 — 5a? + 5 Ix — 7y + 2y2
26. a262 + 12a6c-9ac*
4a62 — 6acx + 3 a2 x
a2b2 + 12a6c - 9ax*- 4a62 + 6acx - 3a2a;
27. a2-2a6+c2-362 29. 2z*-2y2- z2
2a2-2a6 + 362 2x2 + 3y2- 22
-5y2
-x2-2y2 + Zz2
aJ_3y2-3z2
-a2
^-^-B^2
a2 +
6» +
c2*
<p
62 +
c* +
d2
a2 +
62-
c2-
d2
a2-
62 +
(* +
d2
3a2 + 262 + 2c2 + 2d2
62 + c2* cP
S- 8; J; I 30. -2a8 + «Jc + 2«*
a2c — ac2
— a9 -\- ac2
-a2 + b2+ <? + ri2
a2 + 262 + 2c2 + 2d2
3 a2 + 262 + 2c2 + 2d2 a8 - 2a2c + Sac2
q2 + 2y + 2c2 + 2d2 - a8 + ac2
2a2 2a*-2a2c + 2ac2
Exercise IX.
1. (a + &) + (& + c)-(a + c) 2. (2a--6-c)-(a-26 + c)
=a+b +b+c—a—c = 2o— 6 — c — a+ 26— c
«26. -a + 6- 2c.
teachers' edition. 11
3. (2*-y)-(2y-2)-(2*-a?) 4. (a-x-y)-(b-x+y)+(c+2y)
= 2x — y — 2y + z — 2z + x =a— x— y— 6 + a— y + c + 2y
= 3a? — 3y — 2. = a — 6 + c.
5. (2<e-y + 32) + (-<e-y-42)-(3a>-2y-z)
= 2x — y + 32 — a; — y — 42 — 3a;+ 2y + 2
= -2a\
6. (3a - 6 + 7c) - (2a + 36) - (56 - 4c) + (3c - a)
= 3a-6 + 7c - 2a - 36 - 56 4- 4c + 3c -a
= -96 + 14c.
7. l-(l-a) + (l-a + aJ)-(l-a + a»-a»)
= 1 — 1 + a + 1 — a+a* — 1 + a — a%+ a*
= a + a*.
8. a -{26 -(3c +26) -a} 10. 3a-{6 + (2a-6)-(a-6)}
= a -{26 -3c -26 -a} =*3a -{6 + 2a- 6-a + 6}
= a-26 + 3c+ 26 + a = 3a — 6 — 2a + 6 + a-6
= 2a + 3c. = 2a-6.
11. 7a-[3a-{4a-(5a-2a)}]
__ t, v_ -26)} = 7a-[3a-{4a-5a + 2a}
= 2a-{6 — a+26} = 7a — [3a-4a + 5a- 2a
= 2a — 6 + a — 26 = 7a — 3a + 4a — 5a + 2a
= 3a-36. =5a.
12. 2x + (y - 32) - [(3x - 2y) + z] + 5x - (4y - 32)
= 2x+y — 32 — [3z — 2y + 2] +5x — 4y + 32
r — 32 — 3 a; + 2y — 2 + 5aj — 4y + 32
13.
= 2x + y-
= 4a? — y — 2.
. {(3a-26) + (4c-a)}-{a-(26-3a)-c} + {a-(6-5c-a)}
= J3a — 26 + 4c — a} — {a -26 + 3a — c} + {a — 6 + 5c + a}
= 3a — 26 + 4c — a — a + 26 — 3a + c + a — 6 + 5c + a
= -6 + 10c.
14. a-[2a + (3a-4a)]-5a-{6a-[(7a + 8a)-9a]}
= a- [2a + 3a -4a] - 5a - {6a- [7a + 8a -9a]}
= a — 2a — 3a + 4a — 5a — {6a — 7a — 8a + 9a}
= a-2a — 3a + 4a — 5a — 6a + 7a + 8a — 9a
= -5a.
15. 2a-(36 + 2c)-[5_6-(6c-66)+_5c-{2a-(c + 26)}]
.36_2c-[56-6c + 66+5c-{2a-c-26}
-36-2c-[56-6c + 66 + 5c-2a+.c + 26]
-36-2c-56 + 6c-66-5c + 2a-c-26
*2a-36-2c-[56-6c + 66+5c-{2a-c-26}J
= 2a- "' ~
= 2a-
-4a-166-2ct
12
ALGEBRA.
16. a-[26+{3c-3a-(a + 6)} + {2a-(6 + c)}]
= a-[26 + f3c-3a-a-6} + {2a-6-c{]
= a — [26 + 3c — 3o — a — 6 + 2a — 6 — c]
= a — 26 — 3c+3a + a + 6 — 2a + 6 + c
= 3a-2c.
17. 16 -a; -[7a; -{8a; -(9a; -3a; -6a;)}]
= 16 - a; -[7a; -{8a; -(9a;- 3a; + 6a;)}]
= 16-a;- lx -{8a; - 9a; + 3s -6a;}]
= 16-a;- lx - 8a; 4- 9a;- 3a; + 6a;]
= 16 - x - 7x + 8a; - 9a; + 3a; - 6a;
= 16 -12a;.
18. 2a-[36 + (26-c)-4c + {2a-(36-c~^~23)}l
= 2a-[36 + 26-c-4c + {2a-(36-c + 26)}]
= 2a- 36 + 26-c-4c + {2a-36 + c-26}]
= 2a - [36 4- 26 - c- 4c + 2a- 36 + c- 26]
= 2a-36-26 4-c4-4c-2a4-36-«4-26
= 4c.
19. a - [2 6 + {3 c - 3 a - (a + 6)} + 2 a - (6 + 3 c)]
[26 + {3c-3a-a-6 4-2a-6-3c]
[26 + 3c-3a-a-6 + 2a-6-3c]
26 -3c 4- 3a 4- a 4- 6 -2a 4- 6 + 3c
*3a.
a — |^o -
= a-[2
= a-[2
= a-2fl
20. a-[56-{a-(3c-36)4-2c-(a-26-c)}]
""" 't-3c + 36+2c-d+26+c}]
i4-3c-36-2c4-a-26-c]
-3c + 36 + 2c-a+26 + c
— iou-ia — y
•- a — [5 6 — {a -
= a — [56 — o-f
*a — 56 + a —
Exercise X.
1. 2a-36-4c4-a*4-3e-2/ 2. a- 2a; + 4y-3z-26 + c
= (2a-36)-(4c-rf)+(3e-2/) =(a_2a;)4-(4y-3z)-(26-c)
=(2a-3' • - - — - - - " - --
-36-4c)4-(d4-3e-2/).
a-^) + (4y-
>-2a;4-4y)-
(38 + 26-c)
3. a5 + 3a*-2a»-4a2 + a-l
= (a5 4- 3a4) -(2as + 4a») 4- (a- 1)
= (a6 4- 3 a* - 2 a») - (4 a» - a 4- 1 ).
4. -3a-26 + 2c-5a*-e-2/
= -(3a + 26) + (2c-5rf)-(c + 2/)
= - (3a + 26 - 2c) - (5c? + e 4- 2/).
5. aa; — by — cz — 6x + cy 4- az
= (ax — by) — (cz + bx) + (cy + az)
*= (ax - 6y — cz) — (6a; — cy — az).
TEACHERS* EDITION. 13
6. 2x*-3xiy + ±x*y*-bx2y* + xyi-2y!>
= (2x* - 3x*y) + (4ary -SaPy*) + (*y* - 2y*)
= (2a-5 - 3a*y + 4ary) - (5^^- :ry* + 2jf).
(1.) 2a-36-4c+a' + 3e-2/=(2a-36-4c) + (d+[3c-2/]).
(2.) a-2a; + 4y-3z-26 + c = (a-2a? + 4y)-(3z + [26-c]).
(3.) a5 + 3a*~2a8-4a2 + a-l=(a5 + 3o4-2a8)-(4a2-[a-l]).
(4.) -3a-26 + 2c-5d-e-2/= — (3a + 26-2c)-(5d + [e+2/])
(5.) ax — by — cz — 6o? -f cy -f az = (ax — by — cz) — (bx — [cy + az])
(6.) 2x*-3xty + 4xfy* — 5xty + xy*-2y6 = (2x*-Zx*y + 4xiyi)
-(5a?ys-[xy*-2y«])
8. 2ax — 6ay + 46z — 46a; — 2cx — 3cy
= (2a - 46 - 2c)x - (6a + 3c)y + 4 6z.
9. ax— bx + 2ay + 3y + 4az — 36z — 2z
= (a - 6)a? + (2a + 3)y + (4a - 3 b - 2)z.
10. ax — 2by + 5cz — 46a; — 3cy + az — 2cx — ay + 46z
= (a _ 46 - 2c)a; - (a + 26 + 3c)y + (a + 46 + 5c)*.
11. 12aa; + 12ay + 46y-126z-15cx-f 6cy + 3cz
= (12a - 15c)a; + (12a + 46 + 6c)y - (126 - 3c)z.
12. 2ax — Zby — jcz — 2bx + 2cx -f 8cz — 2cx — cy — cz
= (2a - 26 + 2c - 2c)a - (36 + c)y - (7 c - 8c + c)z
= (2a- 26)x- (36 + c)y.
Exercise XI.
1. -17x8 = -136. 4. -18x-5=+90.
2. -12.8x25
= -12.8xl00 + 4 = -320. 5. 43x-6 = -258.
3 3 29
5;49 6. 457x100 = 45700.
2961
1316 7. (_358-417)X-79
16*5 L-775X-79
- 18.0621 = 61225.
8. (7.512 - {- 2.894}) X (- 6.037 + {13.963})
- (7.512 + 2.894) x (- 6.037 + 13.963)
- 10.406 x 7.926
= 82.477956.
14 ALGEBRA.
9. 13 x 8 x -7 11. -20.9 12. - 78.3 x -0.57 = 44.631 ;
= 104x-7 -1.1 1.38 X -27.9 =-38.502;
= - 728. "209 44.631 x -38.502 = -1718.382762.
209
10. - 38 5^ 13. -2.906 x -2.076= 6.032856 ;
9 lA' yo 6.032586 X -1.49 = -8.98895544 ;
~~^ ^ -8.98895544 x 0.89 =-8.0001703416.
" 6™ 183.92
2052
Exercise XII.
9. 6ax-2a«-12tf. 21. lm*x
3mx*
10. 5mnx 9m = 45m2n. ^ — r^
11. 'SaxX— 4by — — 12 abxy. — 2mq
12. -$cmxdn = -8cdmn. -±2m*qa*
13. -7a&x2ac = -14a*&c. ^ Q _
22. — 3pcf
14. 5ra23X 3™^ = 15 m*a?. 6p*q
15. 5a*x-2a» = -10a'»+» "^iS
16. 3a2a2x7a8aj4 = 21a5s6. - 144p«g*
17. 7ax-46 = -28a6;
-28a6x -8c = 224a6c 23. 2a2m8j*x3amV=6a8m8a*;
18. 8a&2x3ac = 24a262c; 6 a»m8^x 4 a»m^= 24 a«m^.
24 a2b2c x - 4c2 = - 96 aW.
M , 24. 6A28X-9ajJy222=-54aV25;
27 ^ -54^26 X -3atyz=l62afy*.
19.
-39mp
243
81
- 1053 abmp
18 ap
25. 3aix 2amx— 4ma;X&a
= -24a262m2a2.
26. 7am2x362w2
= 21 ab2m2n*;
8424
1063 21 a&2m2n2 X - 4a&
- 18954 a2 bmp* = - 84 a2 6s m2 n2 ;
on a m .3 -84a2&8m2w2Xa26n
20. 6a&y = -84a4Mm2n8;
26.r -84a*Mm2n8X-262n
I2ab*tf =168a466w2n4;
-5a2y 168a*66m2n4X-?nn2
-60a8#y = -168a*66m8n«
teachers' edition. 15
Exercise XIII.
5. (4a2-36)x3o& 8. (a*x-5a*x* + ax* + 2x*)xax*y
= l2a*b — 9abi. = a4x*y-5a*x*y + a*x*y + 2ax*y.
6. (8a2-9a6)x3a2
= 24 a* -27 a8 6.
7. (3x* - 4y2 + 5z*) X 2x2y 9. (-9a6+3a8&2-4aJ68-65)X -3a6*
= 6^^-80^^ + \Qx*yz*. = 27aW - 9a466 + 12a867 + 3a&».
10. (3a^-2a^y-7ajy2 + y8)x-5x,y
= -15a^y + lOarty + 35 a1/ - Sa^y*.
11. (-4w2+5x*y + 8r,)x-3xay 12. (-3 + 2ab + a262)x -a*
- 12aY - 15x*y2 - 24a*y. - + 3a* - 2a6b - a662.
13. (-z-2xz* + 5x*yz2-6x*yi + 3x*y*z)x-3x*yz
= Sx*yz* + 6x*yz*- lbx*tfz* + l$x*ifz - 9x*y*z\
1. ^-4
a? + 5
+ 5x2-20
:e* + a^-20
2.
y-6
y + 13
y2- 6y
+ 13y-78
y2+ 7y-78
3.
a* + aix* + ar*
a6 + a*a^ + a2a^
— a* x* — a2 x* — x*
a6 -a*
4.
** + xy + y2
x -y
■ aJ + afy + sy2
— any — xy2 — y3
Exercise XIV.
5.
2x-y
x + 2y
2a? — xy
+ 4xy — 2y2
2x2 + 3xy-2y2
6.
2x8 + 4x2 + 8x + 16
3a; -6
6x* + 12x*+24x2 + 48x
-123»-24a»-48x-
-96
6x*
-96
7.
a^ + x2 + x — 1
a; -1
X* + X3 + X2 — X
— X8 — X2— X + 1
tf* -2x + l
8.
x2 — 3ax
x + 3a
ar'-Sax2
+ 3ax2-9a2x
-9a2x
16
ALGEBRA
9. 262 + 3a6-a2 11.
-56 + 7a
a2 + ab + 62
a —b
-1068-15a&2 + baH
+ 14a62 + 21a26-7a3
a* + a2b -f-aJ2
-a^-a^-fc3
-106s-
- abi + 26aib-7a*
a3 -ft3
10. 2a + b
a+26
12.
a2 - a& + b*
a +b
2a2 + ab
+ 4a5 + 262
a* — a*b + ab2
+ a2b — ab2 + J3
2a2 + 5a6 + 262
a8 +#»
13.
2a&-562
3a2-4a6
6a86-15a262
- 8a262-f20a&8
6a8&-23a262 + 20a&8
14.
-a8 + 2a26-63
4a2 + 8a&
-4a5 + 8a*6-4a268
-8a* 6 + 16a362-8a&*
- 4a5 - 4a'2 ft3 + 16a3 62 - 8a&*
15.
a2 + ab + b*
a* — ab + b*
a* + a36 + a2&2
-as6-a262-a68
+ a2&2 + a&8 + #
t
a* + a262 +6*
16. as-3a26 + 3a62-&3
a2-2a& + b2
a5-3a*6 + 3a3 b2 - a2b*
-2a46 + 6a3 i2- 6a263 + 2a64
+ a9b2- 3a263 + 3a&*-&*
17.
a5 - 5 a* 6 + 10a362 - 10a2 63 + 5a6* -
z-f 2y — 3z
x — 2y + 3z
-&»
x2 4- 2a^ — 3a?z
— 2xy — 4y2 + 6yz
+ 3a;z + 6yz-9z2
a2 -4y2 + 12yz-9z2
TEACHERS EDITION. 17
18. 2s2 + 3:ry-f 4y2
Sx2 — 4sy + yz
6tf4 + 9z3y + 12a2y2
-SaPy-WaPtf-lGxy*
+ 2a*yz + Say's + 4y»g
6a^ + afy - 16zy8 + 2x*yz + 3sy2z + 4y«2
19. x* + xy + y%
a? + xz + z2
ic* + ic*y + a?y2
+ x*z + x2yz + xy*z
+ a^z2 + xyz2 + y2zl
x* -f s?y 4- a^y2 + ar*z + x*yz -f xy2z + a£z2 4- xyz1 4 y2^
33. a2 + ab+.b*
a2
-a& + 62
-afc8
-a&84-
a*
+ a86
-a8&
4-a262
-a262-
+ a2b2-\
ft*
a*
a*
-a2£*
4a262
4 6*
4
6*
a8
+ a«62
-a«#»
4a4M
-a*6*-o26«
<
a8 +a464 +$8
34. 4a8-4a2&4-a#
4a2 + 3a6 4 b*
16a6 -16a*6 + 4a8 62
+ 12a*6 -12aW + 3a*P
4- 4a8&2-4a2684-a&4
16a5 - 4a*6 - 4a8&2- a?P + ab*
2a2ft + y
32a76- 8a662- 8a6 6s- 2d* 6* 4- 2 a8 6*
+ 16a568-4a«6*-4a86s-q2&64a&7
32a76- 8<*ft»- + 8a563-6a*6*-2a866-a266+a67
18 ALGEBRA.
)
25. ^-5^ + 13^- a? -»
a?-2x -2
3^-53* + 13s5- z4- a*
-2a* + 10^-260*4- 2a3 + 2x2
- 2s6 + 103? -26s3 + 23? + 23?
a-7 _ 736 + 21 rr5 - 17a? - 25a? + 4a? + 2a:
26. 3?-2a? + 33;-4
43? + 33? + 2s + l
4a?-8a? + 123?-163?
+ 3a?- 6a? + 9a?-12a?
+ 2a?4- 43?+ 63^-83?
+ a-3- 23? + 33?-4
" 19. ^500 — 0 0*
3a2 -4a&
6a36-15a262
- 8a262 + 20a63
6a8&-23a262 + 20a&8
14. -a8 + 2a26-&8
4a2 + 8a&
_4a5 + 8a*6-4a2&8
-Sa*b + 16a8&2-8a64
ri^ _4a2&3 + 16a8&2-8a&4
15. a2 + a& + 62
a^-afc + ft2
at + tfb + an*
-os6-a262-a&8
+ a262 + a&3 + &4
a* + a262 + Z?
16. a8-3a2& + 3a&2-&8
a*-2a6 + 62
a*-3a46 + 3a3 ft2- a268
-2a4& + 6a862- 6a2&3 + 2a&4
+ a362- 3a268 + 3a64-&»
a5 - 5a46 + 10a3 62 - 10a2 b* + 5a64 - P
17. x + 2y-3z
x-2y + 3z
a? + 23ry — 3zz
-2zy -42/2+ 6yz
+ 3 3:2 + 6yz — 9z2
& -4y* + l2yz-9z*
teachers' edition. 19
31. x-3 32.
.T-1
x2 - a? + 1
iC2 -+ JC + 1
x*-3x
- x +3
X*-4:X +3
X +1
a4-
H
x4-
a8-
-ar' + ic2
har1 — a2 + x
+ x* — x +1
+ a* +1
ar»-4s2 + 3a;
+ a2 — 4z + 3
ar1— 3a2 — a; + 3
x +3
0^-3^- a^ + 3a;
+ 33*- 9a?-3a? + 9
-z2 + l
1-a^ + ar4
- a6 — X* — X2
+ ar4 + x2 + 1
+ ** +1
x< -10a2 +9
33. tf + ab+.V*
a2 — ab + b2
a* + a86 + a262
-a8& -a262-a&8
+ a262 + a&8 +
6*
a* + a262 +
6*
a8 + a662+a46*
-a*V-a*b*-a2b*
+ a*M + a266 + &8
a8 +a4J4 + $8
34. 4a8-4a26+a6»
4a2 + 3a6 + b2
16a6 -16a4 6 + 4as62
+ 12a46 -12a862 + 3a268
+ 4a8y-4a2y + a6*
16a5 - 4a46 - 4as62- tfP + ab*
2q*& + y
32a76~ 8a662- 8a6&8- 2a4M + 2a*P
+ 16g568-4a464-4a8y-a2&6 + a&7
32a76- 8a<62+ 8a6^-6a46*-2a865-a266 + ab1
20 ALGEBRA.
35. x + a
x+2
a
+
ax
2 ax + 2a%
x* +
3ox + 2a* •
X —
3a
x* +
Sax'1 + 2a2x
Sax2 - 9a2x -6a8
X8
- 7a2x -6a8
x —
4a
X4-
7a2x2 — 6a*x
+ 28a8x -4ax8+ 24a*
x4-
7a2x2 + 22a3x -4ax3+ 24a*
X +
oa
X*-
7a2x3 + 22a8x2-4ax4-f 24a4x
20a2x8 - SSa8^ + 5ax* + 110a4x + 120a6
x5-27a2x3 - 13a3 x2 + ax*+ 134a4x + 120a5
or x*+ ax4-27a2x8-13a8x2 + 134a4x + 120a5
36. 81a4-9a262 + 64 37. y2-y2-2z2
9a2 + ft2 y* + yz-2z*
729a6 - 81 a*b2 + 9a2b* y*-ySz- 2y*z*
+ 81a462-9a264 + 66 + rz~ y%zi-2yzi
729 a6 + 66 -2yV + 2.VZ8 + 4z*
27 a8 + J8 y y
27 a8 -ft8 y*-2yz-z*
729a6 + 27a868 y2 + 2yz-z2
- 27a863-&6 y*-2y*z- ylz%
729a6 -66 2y32-4y222-2y23
729a6 + 66 - y*z* + 2yz* + z4
531441a12 -729 a6 66 y4 -ey2*2 + z4
+729a666-612 £ -by*z* +4z*
531441a12 -ft12 - y2*8 -3Z4
38. 3a2 -a&-3&2
a26-2&2
3a46-a862-3a263
-6a2&2 + 2a&8 + 664
3a4&-a8&2-3a263-6a262 + 2a&8 + 6&* .
-2aM-66»
3a46- a862 - 3a268- 6a262 + 2a&3 - 2a64 + 664 - 66*
teachers' edition. 21
39. a + h-c
a — h + c
a2+ab—ac
—ab —b2+ be
+ac + bc—c2
a2 -tf+zbe-c*
-a+b+c
-a*+ab2— 2abc+ac2
+a2b-b* +2b2c- be2
+a2c- b2c+2bc2-c*
-ai+ab2-2abc+ac2+a2b-b3+a2c+ b2c+ bc*-c*
a+b+c
a*+ a2l?—2a2bc+ a2c2+a*b-abs+a?c+ ab2c+ abc2-ac*
+ a2b*+ a2bc -a*b+aV -2ab2c+ abc2 -b*+Pc+ 6V-&C8
+ a2be+ aV -a*c+ aVtc-2abc2+aci -&c + b2c*+bc*-c*
a*+2a2b2 +2a2c2 -b* +2W -c4
40.
a + b
b +c
ab + b*
+ ac + bc
c + d
a + d
ac + ad
+ cd+d2
ac + ad + cd + d2
a + c
b-d
ab + be
— ad— cd
ab + b2 + ac + be
ab + be — ad — cd
(ab + b2 + ac + be) — (ac + ad + cd + d2) — (ab + bc — ad— cd)
= ab + b2 + ac + be — ac — ad — cd — d2 — ab — be + ad + cd
= b* - d\
41. a + b + c + d
a + b + c + d
a2 + ab + ac+ ad
+ ab +b2+ bc+ bd
+ ae + be +c2 + cd
+ ad + bd + cd+d2
a2 + 2ab + 2ac + 2ad + b2 + 2bc + 2bd + c2 + 2cd + d2
a—b—e + d
a—b—c+d
a2 — ab— ac+ ad
— ab +b2 + be— bd
— ac + be + c2 — cd
+ ad - bd - cd + d2
a* _ 2ab -2ac + 2ad +b2 + 2bc- 2bd + c2 - 2cd + d2
22 ALGEBRA.
a—b+c—d
a — b + c — d
a* — ab + ac— ad
— ab +b2— bc+ bd
+ ac —be + c2
— ad + bd
- cd
- cd+d*
a2 - 2ab + 2ac- 2ad + 62 - 26c + 2bd + c2
a + 6 — c — a*
a + 6 — c — d
a2 + aft — ac — ad
+ a6 + 6*- be- bd
— ac —be +
— ad — -bd
-2cd + d*
c* + cd
+ cd +
d2
ai + 2ab-2ac-2ad+ b*-2bc-2bd +
a2 + 2a6 + 2ac + 2ad + 62 + 26c + 26a* +
a2 - 2a6 - 2ac + 2ad + 62 + 26c - 26a* +
a2-2a6 + 2ac-2ad + 62-26c + 26d +
c* + 2cd +
c* + 2cd +
c*-2cd +
<?-2cd +
a*2
rf2
d2
d2
4a2 + 462 +4C2 + 4d2
42. (a + 6 + c)2 — a (6 + c — a) — 6 (a + c - 6) — c (a + 6 — c)
= a2+62+c2+2a6+2ac+26c— a6— ac+a2— ab— 6c+62— ac— 6C+C2
= 2a2+262+2c2.
(a-6)a;-(6-c)a-{(6-a;)(6-a)-(6-c)(6 + cft
= ax — bx — ab + ac — {(ft2 — bx + ax — ab) — (62 — c2)}
= ax — 6a; — a6 + ac — Jo2 — 6a; + ax — ab — 62 -f c2}
= ax — bx — ab + ac — o* + bx — ax + ab + 6s — c2
= ac — c2.
44. (m + n) m — {(m — n)2 — (n — in) n\
»» m2 + win — {m2 — 2mn + n2 — n* + mn]
= m2 + ron — ra2 + 2mn — n2 + n2 — mn
=- 2mn.
45. (a - 6 + cy — {a (c — a — 6) — [6 (a + 6 + c) — c (a — 6 — c)]}
~ ' ab)
(ab + 62 + 6c) - (ac -be- c2)]}
:lac — a2 — a6)
= a2 — 2a6 + 2ac — 26c + 62 + c2 — {(ac- a2 — a6)
= a2 - 2a6 + 2ac - 26c + 62 + c2
— [a6 + 6'2 + 6c — ac + 6c + c2]}
=» a2 - 2a6 + 2ac ~ 26c + 62 + c2
— {ac — a2 — a6 — a6 — 62 — 26c + ac — c2}
=» a2 — 2a6 + 2 ac — 26c + 62 + c2 — ac + a2
+ a6 + a6 + 62 + 26c — ac + c2
=■ 2a2 + 26* + 2c2.
teachers' edition. 23
46. {p2 + q2) r-(p + q)(p{r -q}-q{r -p})
= p2r + q*r-(p + q)({pr -pq} - {qr -pq})
=p2r + q2r — (p +q)(pr—pq — qr +pq)
= tfr + q2r-(p+q)(pr- qr)
=p2r -f q2r — (p2r — q1r)
=p2r + g^r — p2r + q*r
= 2q*r.
= yx*yA-L6x*y*+W-i2xy—Zy*ittx9y+6xy°-
= 9a?y2-13a?yH4>-{9aty-13a?yH4yf
= 92*y2-l$x*y*+fy-9a*y*+lSx*if-4y
= 0.
48. a2-{2o6-[-(a + {6-c}Xa-{6-c}) + 2a6]-46c}-(6 + c)«
= a2-{2a6-[-(a + 6-c)(a-6 + c) + 2a6]-46c}-(6 + c)2
= a2-{2a6-[-a24^,-2Wc2 + 2a6]-46c}-(6+c)1
= a2-{2a6 + a2-62 + 26c-<?-2a6-46c}-(6 + c)2
= a2 -2a6-a2 + 6*-26c + ^ + 206 + 46c- bi-2bc-<*
= 0.
49. {ac-(a-6)(6 + c)}-6{6-(a-c)}
= {ac - (a6 — 6* + ac - be)} - 6 (6 — a + c)
= ac — ab + & — ac + be — b2 + ab — be
= 0.
60. 5{(a-b)x-cy}-2{a(x-y)-bx}-{3ax-(5c-2a)y]
= 5 {aa; — bx — cy} — 2 {ax — ay — bx\ — {3 ax — 5 cy + 2 ay]
= 5ax — 56a; — ocy — 2 ax + 2ay + 26x — 3aa? + 5cy — 2ay
= -3bx.
51. (a;-l)(a;-2}-3a!(a; + 3) + 2{(a; + 2)(a; + l)-3}
= x2 - Sx + 2 - Zx2 - 9a? + 2 {x2 + 3a; + 2 - 3}
= a?- 3a; + 2- 3a?- 9a; + 2a? + 6a; + 4-6
= -6a;.
52. {(2a + 6)2 + (a-26)2}{(3a-26)2-(2a-36)2}
= {(4a2 + 4a6 + 62) + (a2-4a6 + 462)}{(9a2-12a6 + 462)
-(4a2-12a6 + 962)}
= {4a2 + 4a6 + 62 + a2-4a6 + 462}{9a2- 12a6 + 46s
-4a2 + 12a6-962}
= {5a2 + 56n{5a2-562}
53. 4(a-36)(a + 3 6)-2(a-66)2-2(a2 + 662)
= 4(a2-962)-2(a2-12a6 + 3662)-2a2-1262
- 4a2 - 366» - 2a2 + 24a6 - 72 62 - 2a2 - 1262
= 24a6-12062.
24 ALGEBRA.
54. x2 (x2 + y2)2 - 2x*y2 (x + y) (x - y) - (a? - y3)2
= x2 (x4* + 2a?2y2 + y*) - 2x2y2 (a* - y2) - (a» - y3)2
= a* + 2**y» + a*y* - 2x*y2 + 2x2y4 - x6 + 2a?y3 - y«
= 3a?y4 + 2ic33/3-y6.
55. 16(a* + 62)(a2 - b2) - (2a - 3) (2 a + 3)(4a* + 9)
+ (2&-3)(2& + 3)(4&2 + l/j
= 16(a* - ft*) - (4a2 - 9) (4 a* + 9) + (4 J2 - 9)(46» + 9)
= 16a*-16M-16a4 + 81 + 1664-81
= 0.
Exercise XV.
7. (s + y)2 19. (Sabc-bcdf
= a* + 2a?y+y2. -9as-Pc>-6aftV<Z + Pc><*'.
8. {y-zf 20. (4ar>-a!y2)2
= y2 - 2yz + z2. = 160* - 8x*y2 + x*y*.
9. (2x+l)2 21. (x + y)(x-y)
= 4ar* + 4a; + l. = x2-y2.
10. (2a + 56)2 22. (2a + 6)(2a-6)
= 4a2 + 20afc + 2562. = 4a2 - b2.
11. (1-z2)2 23. (3 -a?) (3+*)
-l-23» + aJ*. = 9-a*.
12. (3aa?-4^)2 24. (3a6 + 26*)(3a& - 2&2)
= 9a2z2- 24aars + 16a?*. = 9a262 -464.
13. (l-7a)2 25. (4a?2-3y2)(4a;2H-3y2)
= i - 14a + 49a2. = 16ar* - 9y4.
14. (5sy + 2)2 26. (a*x2-by*)(a*x2 + by*)
= 25x2y* + 20 ay + 4. = a6 or4 — ^y8.
15. (ab + cd)2 27. (6<ry-5y2)(6sy + 5y2)
= a2 b2 + 2a6ca- + <*d\ = 36a*y2 - 25y*.
16. (3mn-4)2 28. (4a*- l)(4af> + 1)
= 9m2w2 - 24mn + 16. = 16x10 - 1.
17. (12+ 5s)2 29. (l+3a63)(l-3a&3)
= 144 + 120s + 25a* = 1 - 9a26«.
18. (4ajy2-yz2)2 30. (ax + by)(ax-by)(a2x2 + b2y2)
= WaPy* - Sxfz2 + y2z*. = a*xA - b*y*.
Exercise XVI.
1. (x + y + z)2 = x2 + y2 +z2 + 2xy + 2xz + 2yz.
2. (a; - y + z)2 = x2 + y2 + z2 - 2a:y + 2arz - 2yz.
teachers' edition. 26
3# (m+n_p_<j)2 = ra2+n2+pa+2*+2 mn— 2 rap-2 mq-2 np-2 nq+2pq.
4. (a? + 2x -3)2 = a? + 4a?-2a? -12* + 9.
5. (a? - 6a; + 7)2 = a? - 12a? + 50a? - 84a; + 49.
6. (2a?2 - 7x + 9)2 = 4a? - 28a? + 85a? - 126a; + 81.
7. (a? + y* _ z*)* = a? + y4 + z4 + 2 a?y2 - 2a?z2 - 2y2z2.
8. (a?-4a?y2+y4)2 = a?+18a?y4 + y8-8a?y2-8a?y«.
9. (a8 + 6s + c3)2 = a6 + 66 + c6 + 2a8 68 + 2a8 c8 + 26s c8.
10. (a*-tf-Jf = x* + tf + z*-2x*y*-2x'* + 2yi2*.
11. (a; + 2y - 3z)2« a? + 4y2 + 9Z2 + 4 scy - 6a;z - 12yz.
12. (a? - 2y2 + 5z2)2 = a? + 4y* + 25 z4 - 4a?y2 + 10a? z2 - 20y2z«.
13. (a? + 2x-2)2 = a? + 4a?-8a; + 4.
14. (a? - 5a; + 7)2 = a? + 39a? - 10a? - 70a; + 49.
15. (2a? - 3a; - 4)2 = 4a* - 12a? - 7a? + 24a; + 16.
16. (a; + 2y + 3z)2 = a? + 4y2 + 9Z2 + 4an/ + 6a;z + 12yz.
Exercise XVII.
1. (x + 2)(a?+3) = a? + 5a; + 6. 12. (x-4y)(x + y) = xi-3xy-4yi.
2. (a; + l)(x + 5) = a? + 6a; + 5. 13. (a-26)(a-56) = a2-7a6+1062.
4. (a;-8)(a;-l) = a?-9x>8. ar-i-o*-y-t- y
5. (*-8)(* + l)-**-7*-8. La?- 2a?y - 3z2y2.
6. (a;-2)(a; + 5) = a? + 3a;-10. 16 (aa._9)(oa; + 6)
7. (x-3)(a; + 7) = a? + 4a;-21. = a2 a? - 3 ax- 54.
8. (.-,(._4)-*-e. + a. it. ^y)faZ^m_A
10. (a:-2a)(a;+3a)=a?+aa;-6a2.
19. (a; + 12)(a;-ll) = a?+a;-132.
11. (x-c)(x-d)
= a?_(c + cZ)a; + cd. 20. (a;-10)(a;-5) = a?-15a; + 50.
Exercise XVIII.
1. (x + a)s = a? + 3a?a + 3a;a2+a8. 3. (a; + l)8 = a? + 3a? + 3a; + 1.
2. (a;-a)8=a?-3a;2a + 3ara2-a8. 4. (x- l)8 = a?- 3a? + 3a;- 1.
5. (a; + a)4 = a? + 4a?a + 6a;2 a2 + 4<ra8 + a4.
6. (z-a)4 = a?-4a?a + 6a?a2-4a?a8 + a4.
26 ALGEBRA.
7. (x + l)* = z* + 438*6s8 + 4a; + l.
8. (s-l)* = a?*-4aj8 + 6x2-4a; + l.
9. (x + y)5 = i5 + 5z*y + 10ar»y» + lO^y8 + 5sy* + y6.
10. (a;-y)5 = a^-5a^y + 10rBsy,-10x2y8 + 5a!y4-y5.
11. (x + l)6 = x6 + 5s* + 1038 + 10a* + 5a; + l.
12. (x-l)5 = a*-5a?* + 10arJ-10(r8 + 5a;-l.
4.
11.
Exercise XIX.
+ 264 Aft 6.
+ 4 "66* 3
-1.23
9.
-0.1123
45)-424.35
-61)6.8503
345
61
-|648-581.
-8
793
75
690
61
1035
140
+ 3840 = 128
1035
122 ,
-30
183
7.
-11
183
-2568 = 214 +
-24)- 264
24^
+ 12
24
24
-21.7
-49.) 1063.3 8.
+ «A
10.
-0.022*
98
-85)- 3670
+■ 324)- 7.1560
83
340
648
49
270
676
343
255
648
343
tt-
■A
AWt
0.31831
+ 12.
0.0101321+
-314159.)-100000.0
314159)-
-3183.10
942477
314159
575230
415100
314159
314159
2610710
1009410
2513272
942477
974380
669330
942477
628318
319030
410120
314159
314159
4871 95961
teachers' edition. 27
Exercise XX.
5. «~-3m. 18. 30a^ = _6£
2x — bx*y x
6 12g4= ia3 19 ^tfwtx* _4ma^
7.
— 3a 5a?msx 5as
10a*_6« * 20. M^^„6^
26c c 7ayV
8 _a^___l 21. -S^*8^6- 3Wd"
9.
-x5 a?" -a462crf» a2
12am n 22 * ^ am5 n* ^ ^* = ^ am3 n
— 2m " 47»2ns|?*j5 j>g»
in 35 a6c^ ,. 23. (4a'&z»xl0a'ft»c)-i-5as'Psl
""l^T = 40a464«4-*-5a36222
= 8a62*2.
U* ^t = ^~; **• (21a^2«-t-3sy2z)(-2a*y2Z)
12. -?^ = -9a4. — M^A
-3a3 25. 104a6V-*-(91aWB7-*-7a464a:)
oA ... = 104a6sa*-*-13a62a*
13. -36ma?== _£jg!. = 86ar>.
4aa^ 4oa;
26. (24aW* + 3a'P)
14 a62c8_,2 + (35a862a* + -5a86a;)
^67 ~(8a86a0 + (-7a»6a0
=» a* 6a;.
15. ^£j?L = m4a:2. 27. 85a**+i-i- 5a<— »
mF^ =17a*"»+i-(4«-«)
1<L-5i«hy =17a3-
36ay y 28 84a""4
nnc , A ' 12a2
,7. 225 m2y _9m = 7a»-4-2
25 my2 y = 7a«"«.
Exercise XXI.
3. (ISamy - 27bny + 36cpy) -^ - 9y = - 2am + 3 bn - ±cp.
4. (21aa;-186a; + 15ca;)-*--3a; = -7a + 66-5c.
5. (12a* - 8ar> + 4a;)-*- 4a; = 33*- 2x* + 1.
28
ALGEBRA.
6. (3a? - 63? + 9z7 -12a?)-*- 3a? = a; -2a? + 33? -4a*.
7. (35rasy + 28m2y2 - 14my8) -+■ - 7 my = - 5m2 - 4my + 2y2.
8. (4a*6 - 6as&* + 12a* 6s) -*- 2a2 6 = 2a2 - 3ab + 662.
9. (12arV- 15a?y2 - 24a?y) -*■ - 3a?y = - 4sy2 + 5a?y + 8a*.
10. (12a?y*-24a?y2 + 363?y8-123?y2)^12a?y2 = a?y2-2a? + 3zy- I
11. (3a*-2a56-a662)H-a*=3-2a6-a262.
12. (3a?yz2 + Gatyz* - ldx^y2^ + 18zVz) -5- - 3a*yz
= — z — 2arz2 + 5a?yz2 — 6a?y2.
13. (- 16a8 52<? + 8a*&V- 12a5 6V) ■*- - 4a2 6V
= 4ac8-2a2c2 + 3a36c.
Exercise XXII.
6.
3?-7a; + 12
a?-3a;
X-
-3
2a;-3
11.
12.
13.
7a?-
7a?-
a?-
a?-
-24a?+58a;-21|7a;-3
3J-4
- 3a;2 |a?-3a;+7
7.
8.
-4a; + 12
-4a; + 12
a? + x-72
a? + 9a;
-8a; -72
-8a;-72
2a?- a? + 2
3J + 9
3J-8
ia;-9
-21
-21
-1
-a?
a?+58a;-21
*?+ 9s;
49x-21
49a;-21
x-1
2a?-33?
3? + s; + 3
2a;+6
a? + a? + a? + a? + 3; + l
9.
2a? + 3a;-9
2a?-33;
6a;-9
6a;-9
6a?+143?- 4a;+24
6a?+18a?
a8-
a8-
a?
a?
-1
-3?
3?-l
3?-3?
3?-l
a? — a?
33?-2a;+4
a?-l
- 4a?- 4a;+24
- 43?-12a;
r-1
a? — a;
x-1
10.
83J+5
$x+'<
9a?+33?+ 3J-1
9a?-33?
24
>A
3.
-2aP + P
-a26
x-l
a — b
3a?+2$+l
a* + ab- b2
6a?+ x
6 a?-2 x
3x
3^
-1
-1
a2
a2
b-2al
b- al
— al
-al
2 + 68
2 + 68
2 + 68
TEACH HRS EDITION.
29
14. x*-8ly*
a;-3.y 15. xA-
-y5
-x*y
*-.v
x*~ 3afy a?+3a?y+9ay2+27iya a?-
xt+aPy +x2i/2 -fay1 +y4
3x*y — Sly* x*y
3afy-9a;V a^
9a?y2-81y*
9^^-27^
27*^-8 ly4
27a^-8b/
-y4
— afy2
X*y* - y5
arty2 — xhp
a;2^3 — xy*
xy'-f
xy^-y1
16. a5 + 3265
a + 2b
a5 + 2a*6
a*-2a36 + 4a262-8a63 + 166*
-2a*b
-2a*b
+ 326*
-4a362
4a3&2 + 32#>
4a862 + 8a2&3
-8a2&3 + 32&*
-8a263-16aM
16a64+3265
16a&* + 326*
17. 2a* + 27a&3-8164|a-f 36
2a* + 6a*b |2a3-6a26 + 18a&2-2768
-6a36+27a63-8U*
-6a36-18a262
-Sib*
18a262 + 27a&3-
18a262 + 54a&3
- 27a63-8U*
-27a&3-816*
18. a^-5ar>+ 11a;2- 12& + 6
x2-3x + S
a^-3a* + 3»2
a? -2a; + 2
-2ar»+ 8a;2-12a;
-2ar*-f 6a*- 6a;
2a;2- 6a? + 6
2a?- 6a; + 6
19. x* + x*- 9a;2-16x-4la;24-4a; + 4
a^ + 4ar> + 4ar* ja^8 — 3x — 1
-3ar»-13a^-16a;-4
-33*- 12a;2 -12a;
— a?— 4a; — 4
-a?- 4a?-4
30
ALGEBRA.
^+5^jf6^J x2 - 5 x 4- 6
-5^-19^ + 36
6a^ + 30a; + 36
6a? 4- 30x + 36
21. x4 + 64 |g24-4g + 8
x44-4x84- 8g8|a2-4a;-f 8
-4ar>- 8.jr2 + 64
-4x8-16x2-32a;
8x2 + 32x + 64
8x* 4- 32x4- 64
22. x*4- x3-243r-35x4-57|x24-2x-3
x44-2x8- 3a? |a£-&-19
- x8-21x2-35x
- x8- 2x24- 3x
-19x2-38x4-57
-19x2-38x4-57
23. 1- x-3x2-x5|l4-2x4-x2
1 + 2x4- x2 |l-3x4-2x2-x8
-3X-4X2-X5
-Sx-ea^-Sx3
2x24-3x8-x5
2x24-4x34-2x4
- xs-2x*-xB
- x8-2x4-x5
24.
x«-2x84-l
s^-^-fx4
x2-2x4-l
2x5-x4
2x6-4x44-2x8
x44-2x34-3x24-2x4-l
2xs4-l
26. 4x*- ar»4-4x
4x54-6x44-4x8
3x4-4x34-l
3xi-6x84-3x8
2x3-3x2+l
2x3-4xa4-2x
x*-2x+l
x2-2x+l
25. a4+2a2624-964
a4-2a3&4-3a262
a*-2a54-35*
a24-2a&4-362
2a»6- a262+964
2a36-4a2624-6a68
3a*V-6dP+9&
3a262-6a&8+9&4
2x24-3x4-2
2x8-3x24-2x
-6x4— 5x34-4x
4x84-6x24-4x
4x84-6x24-4x
27. a5-243
a6-3a4
a-3
a44-3as4-9a24-27a4-Sl
3a*-243
3a4-9a8 ,
9a3-243
9a8-27a2
27a2-243
27a2-81a
81a-243
81a-243
teachers' edition.
3i
28
18a;* - 45 ar» + 82a;2 - 67a; + 40 . 3s2 - 4 a; + 5
18ar* - 24ar* + 30a;2 Ua;2 - 7a? + 8
-21ar* + 52x2-67a;
-21arJ + 28a;2-35a;
24a;2 -32a; + 40
24a?-32a; + 40
29.
3ii-9xt-6xy-y2\xa + 3x-{-y
xt + SoP + xty |a;2-3a;-v
- 3ar» - 9a;2 - x2y - 6xy - y2
-3a? -9a;2 -3a;y
-x*y-Sxy-y*
— x2y — 3a?y — y2
30.
a*-6a*y + 9ary~4y4
a^-Sar'y + 2a;2y2
-3ar»y + 7a;2y2-4y4
-3ar,y + 9ar!y2-6a^
-2a?y2 + 6a^
— 2x2y2 + 6a;y
x2-3xy-\-2y2
•
x2 — '6xy-2y2
3
8-4y4
1-4.V*
31.
x4 + a?2y2 + y* 1 a;2 — xy + y2
a;4 - a?y + x*y2 J a? + xy + y2
afy + y4
a^y — a?y2 -faty3
»2.V* - *y3 + y4
a?y2 — ajy5 ■+ y*
32.
a? + ar* + oty-aPy2 - 2xy2 + y*\x* + x-y
a£ + a? — a*y | a?2 + an/ — i/2
a^y + a:2y — a^y2 — 2xy2 + y3
a**y + a;2y — xy2
— x*y2 — xy2 + y3
— sc'y8 — xy2 + y3
33.
2a? + xy—xz — Sy2 — 4yz — z2
2a;2 + 3ary + a;z
- 2arcy — 2a;z - 3y2 - 4yz - z2
~2xy -Sy2- Vz
-2xz -Syz-z2
-2xz -Zyz-z2
2a; + 3y + z
x-y-z
32
ALGEBRA.
34. 12-38a;-f 82o;2-112j3+ 106^-70^ 1 3-5s+7a*
12-20a?+28a? 1 4-6a;+8z2-10arJ
-18a;+54aj2-112ar>
-lSx+Wx2- 42ar»
24a?-
24a?-
70ar*-fl06;c*
40ar>+ 56a^
- 30ar*+ 50^-70.^
- 30^+ 50**-7Qg»
36.
3*
a5 — x*y + x*y* -
+ 3Z5
a^y8 + xy*
x* _ gJy + j?y2 — ayy3 -f y*
s + y
a*y — a^y2 -t- a;2^3 — xy* + y3
a^y — a^V2 + a^y8 — %y* + y5
36. 2z4-7a;8y + 232y2-2zy8-y4
2a^ — x*y + a;2y2
— 6ar*y -f a?y2 — 2a*y8
-6ar}y + 3ac2y2-3.Ty3
2a;2 — ay-f y2
sP — Sxy—y2
■l&y1^
-2ar»y2-f
sy8-
a;;/8-
37. 16ar* + 4x2y2 + y*
16g*-8ar*y + 4a:2y2
4iX2 — 2xy -f y2
4a?2 + 2zy + y!
8ar*y + y4
8ar,y-4a?2y2 + 2a;y8
4a?y2 — 2#y8 + y*
4s2y2 — 2ay8 + y4
38. 32a56-56a462+ 8 a8&8-4 a264-a65 1 -4 a26+6 a&2+68
32a55-48a462- 8a868
- 8a4&2+16a368-4a8&4
- 8tt462+12a868+2a264
4a363-6a264-a66
4a868-6a264-a65
-8a3+2a2&-a&2
39.
l + 5ar»-6a^
l-a; + 3a^
l-a? + 3a:2
l+a-2a>2
x-3x2 + 5x*-6x*
x— x2 + 3S8
- 2a? + 2ar»- 6a;4
-2a^ + 2ar^-6a4
teachers' edition.
33
40. l-51a3&3-52g464|-l+3a6 + 4a262
l-3a6-4a262 \- I -Sab- lSa2b2
Sab + 4«262 - 51 a3 b3 - 52a464
Sab-9a?b*-12a?b3
13a2 b2-
13a262-
-39a363-52a464
-39a363-52a464
41. x7y — xy7
x1y-2x6y2 + 2x&y3- x*y*
x9y-2x2y2 + 2xyi-y*
x4 + 2^2/ + 2a?y + ay1
2x*y2 — 2xby* + a^y4 — an/7
2aV-43*y3 + 4aV-23*,v*
2a£y3-3z4y4 + 2z8y5-a*/7
2s5y3— 4E4y4 + 4g3ys-2a2,y6
a^y4-2ar,y5 + 2a;2i/a-an/7
g4y4-2ariys-f 2a?2y6-a;y7
42. z*-6 afy +15 ay~20 afy8 +15 a?y-6 an/5 +y6
3^-3^+ 3a?V- a^y3
" -3a^y+12a?4y2-19a^y3+15a;y
-3a*y+ 9a^,y2- 9ar»y3+ 3afy4
3a;y-10ar,y3+12arJy4-6an/6
3**y»- 9a;y+ 9x2y*-Sxy*
-x*tf + 3ay-3ay+y«
-arV + SxfySxyt+y*
a*-3a;2y+3an/2-y3
3^-3^+3 a^2-^3
43.
ar _ 2a6& - 2a4 J3 + 2a3 64 - 6a2 6s - 3a66
a3-2a2b~ab2
a7-2a66-a5&2
a562 - 2a463 + 2a3 A4 - 6a2b* ■
a*b2-2a*V- a364
a4 + a262 + 364
-3a&«
3a364-6a265-
3a364-6a265-
-3 aft6
-3a6«
44.
Slafy-54ay -183y+18ay-18ay-9y7
81a*y +27aV +27 *y
-27^
-27a.y
-9ay
-9a?y
3a*+ay+y4
~54ay-27ay-18ay - 9*y-18an/6-9y7
-54sy ' -18gy -18ay
27aj2y-18an/2-9y3
-9y7
-9£
34
ALGEBRA.
45. a4 + 2a*b + Sa2b2 + Sab* + lG64[a2 + 462
a* + 4a262
a2 + 2a& + 462
2a36 + 4a262 + 8a63
2a36 -+8a63
4a262 + 166*
4a262 +166*
46. -a* + 2la*tf-24xy* + 8y«
— aP + SaPy — xKy2
-3x*y + tf*y« + 21a*y»
-So^y + dx^y2- Sx^y3
-x2 + 3xy — y2
x* + 3x*y + 8x2y2-$y*
24zy5 + 8y6
- $x*y2 + 24arY - 24a*/5 + 83^
- 8x*y2 + 24s3 y3 - 8 x2y*
8a;2 3/4- 24ay» + 8/
8sV-24gn/b + 8y6
47. 16a* + 8a262 + 96* |4a2
16a4-16a36 + l2a2&2
-4a6 + 362
4a2 + 4a6 + 362
16a36- 4a262 + 964
16a36-16a262 + 12a&3
12a262-
12a262-
12a634-964
12a63 + 964
48. a3
as + a?b + a2 c
— Sabc + ¥ + c*\a 4- b + c
— a2 6 — a2 c — 3 a6c + 63 + c3
— a2 6 — a62 — a&c
-2abc + & + <*
a& — ac 4- 62 — 6c + c2
— a?c + ab'1
-a2c
— abc — ac2
+ ab'2 — abc + ac2 + b* + c*
+ ab2 + 63 + b2c
— abc + ac2
— abc
- b2c + c3
- 62c - 6c2
ac2
ac2
+ 6c2 + c3
+ be2 + (?
49. a3
-6 abc+8 ¥+<* . a2-2a6-ac+462-26c+c2
a3-2 a26-a2c-f 4 a&2-2 abc+ac2
a+2b+c
+2a26+a2c-4a62-4a6c-ac2+863+c3
+2a26 -4a62-2a6c +863-462c+26c2
+a2c
+a2c
—2 abc—ac2
—2 abc—ac2
+462c-26c2+c3
+4&2c-26c2+c3
teachers' edition.
35
50. o3 + 'Sa2b+3ab2 + b* + <?
a + b + c
a3+ a2b + a2c
a2 + 2ab + b'l-ac-bc + c2
2a2b + 3ab2-a2c + b* + c*
2a*b + 2ab2 + 2abc
ab2-a2c-2abc + b* + c*
ab2 +b* + b*c
— a2c-2abc-b2c + c*
— a?c— abc — ac2
-
abc — b2 c + ac2 -\- c9
abc — b2c —be2
ac2 -{-be2 + c*
af + bct + c*
Exercise XXIII.
a2 (b + c) + b2(a-c) + c2(a-b)+ abc\ a + Hc
a2(b + c) + b2(a ) + c2(a ) + 2abc\ a(b + c)-bc
— abc
— abc
+ b2(
2. a^ — (g + 6 + c)a:2 -f (ab + ac + 6c)s- abc\ x2- (a + b)x + ab
x*—(a + b )x2 + (ab \x___\x-c
-ex2
—ex2
+ (ac-\-bc)x—abc
+ (ac + bc)x—abc
3. x*-2axi+(a2 + ab-
x9— ax2 -f bx2
■b2)x—a2b -f ab2\x — a + b
-(a + b)x2 + (a2 + ab-b2)x-a2b + ab2
-(g + 6)rt + (a2 -b2)x
Ix2— (a + b)x + ab
+ abx— a2b + ab2
+ afoc — a26 + a&2
4. a?4 —(a2 — 6 — c) a2 — (6 — c) as + ftcls2 — ax-\-c
x4 + ( + c)a? — aar» | a? + asc + 6
ax*-
aa*-
— 6 ) x2 — (b — c) oa; + be
}jt? + cox
-{-bx2
+ bx*
— box + 6c
— bax + be
36 ALGEBRA.
5. y3 — (ra + n + p) y2 + (ran + mp -f r?p) y — ronp y —p
Vs ~ ( + P) .V2 I y2 - (w + w)y + ran
— (m + n ) ,y2 + (ran + mp + rip) y
— (m + n ) y2 -f ( + mp + np) y
mny — winp
mriy — mnp
6. x* + (5 + a) x8 - (4 - 5a + b) x2 - (4 a + 56) x + 4 b
x* + (5 ) r» - (4 \x*_
a* + 5x-4
x2 + ax — b
or* -( — 5a + b)x* -(4a + 5b)x
ax* + ( +5a )g*-(4q )x
-6x2 — 5&X + 46
-bx* -56* + 46
1 x2— (a+c)x-t-ac
7. a»-(&+a>+W
a?*— (a+J+c+^Ja^+fai+ac+aa'+ic+ftrf+crf)^— (abc+abd+acd+bcd)x+abc<l
x*— (a +c )as+( +oc Jx*_
id )xt—(abc + abd+acd+ bed \x
i.d)x*—(abc +acd )x
bdx2—^ -\-abd \-bcd)x+abcd
bdx*-( +abd ~\ bcd)x+abcd
— ( +6 -\-d)x* \-(ab +ad+bc-tbd+cd)x*—(abe+abd+acd+bcd
— ( +6 +a*)s8+(a6 H-ad-i-frc +cd)xt—(abc +acd
8.
x*-(m— c)a^+(n— cm+d)xi+(r+m-dm)x2+(cr+dn)x+dr\xsj-mx2-{-nx+r
x*-(m )xi+(n )x*+(r )a? |x2+cx+d
car*+r — cra-f o*)ar,+( -fen— dra^+^r+anb?
cx*+( —era )x*+( +cn )x2+(cr )x
dx* —dmx2 -\-dnx-\-dr
dx* —dmx2 +dnx+dr
9.
a^-ma^ + rtx8 — nx* + mx — 1 x — 1
x*-x*
x4 +(1 — m)x*+(l -ra + n)x* +(1— m)x+\
(1 — m)x* + nx* — nx2 + ma; — 1
(1 — m) x* — (1 — m) x3
(1 — m + n) x8 — nx2 -f mx — 1
(1 — m + n)x3 — (1 -ra + n)x2
(1 — ra) a;2 + raa; — 1
(1 — m)x2 — (1 — ra)x
x-1
x-1
10. (x + y)» + 3 (x + y)2« + 3 (x + y)z2 -f 2s 1 (x + yf + 2(x + y)z + z*
(x + yY + 2(x + yfz+ (x+y)z2 \x + y + z
(x + y)22 + 2(x+y)22 + 2s
(x + y)22 + 2(x + y)22 + 28
TEACHERS EDITION.
Exercise XXIV.
1. (y» _ 1) + (y - 1) 7. (l-8a»)+(l-2*)
= y*+y + 1. =1 +2x + ±x*.
2. (i3- 125) -*-(&- 5) 8. (ar*-3265) -*-(*- 26)
= b* + 56 + 25. = a*+2aM44aW+8*y+l(W.
3. (a* -216) -(a -6) 9. (8a3.**8- 1) + (2az- 1)
= a2 + 6a + 36. = 4a2ar2 + 2aa; + 1.
4. (*»- 343) -*-(*- 7) 10. (l-27ary) + (l-3a!y)
= x2 + 7* + 49. = 1 + 3sy + 9s*y2.
5- (**-y*) + (*-y) U- (64oW-27a£)-i-(4aft-3ar)
= x* + ar»y + a;2y2 + xtf + y4. = 16a2 b* + 12ate + 9 a*.
6. (o*-l) + (o-l) 12. (243a5-l) + (3a-l)
= a4 + a3 + a2 + a + 1. = 81a4 + 27a3 + 9a2 + 3a + 1.
13. (32a5-24365)-*-(2a-3&)
= 16a4 + 24a3& + 36a262 + 54a&3+81&4.
Exercise XXV.
1. (s» + y») + (* + y) 7. (a* + 3265)-5-(« + 26)
= x*-xy + y2. = a4-2a36+4a262-8a&3+1664.
2. (a^ + y^ + fc+y} 8. (512 a*y» + *») + (8asy + e)
= x4 — ar'y -fa^y2 — xy3 + y4. = 64 a^y2 — 8 xyz + ?.
3. (l+8a3)-*-(l + 2a) 9. (729a3 + 21663) + (9a + 66)
= 1 - 2a + 4a2. = 81 a2 - 54a6 + 3662.
4. (27a3 + ft3)-*- (3 a + 6) 10. (64 a3 + 1000 J3) + (4 a + 10 h)
= 9a2 - 3a6 + 62. = 16a2 - 40a& + 1006*.
5. (8a3ar» + l)-i-(2az + l) ' 11. (64 a363 + 27^)^(4 ab + 3a)
= 4a2a? - 2az + 1. = 16a2 6» - 12a&c + 9a?2.
6. (a» + 27ys) + (a + 3y) 12. (x» + 343) + (* + 7)
= a* - 3zy + 9y2. = x2 - 7x + 49.
13. (27arY + 828)-J-(3ary + 22)
= 9ar2y2~6a;y2 + 422-
14. (1024 a6 + 243 fr5) -*- (4 a + 3 6)
= 256a4 - 192a36 + 144 a2 62 -
108a63+8164.
38 ' ALGEBRA.
Exercise XXVI.
1. (x*-y*) + (X-y) 6. (*»-81iV«) + (* l-3y)
= x* + a^y + ay* + y3. = re3 - 3 a?y + 9 xy2 — 2T?/3.
2. (aj*-y*) + (a? + y) 7. (16**- l) + (2ar-l)
= x*-x2y + xyi-yi. = 8a^ + 4ar* + 2a: + 1.
3. (a6 -a6)-*- (a -a?) 8. (16a* - 1) -s-(2a: + 1)
= ab+a*x+a?x2+a2a?+atfi+x&. = 8 r1 — 4 a? + 2a; — 1 .
4. (a«-a*)-«-(a + a;) 9. (81 a*x*- 1)-*- (3az - 1)
= cP-ah+aPtf-aW+ax'-x5. = 27a3 ar, + 9a2a* + 3aa;+l.
5. (z*-81y*) + (x-3y) 10. (81a**4- l) + (3a* + 1)
= x* + 3x2y + 9xy2 + 27^. = 27a3ar» - 9a2a? + 3ax - 1.
11. (64a6-6«)-s-(2a-&)
- 32a5 + 16a*6 + 8a362 + 4a2fc* + 2a&* + 6».
12. (64a6 -ft6)-*- (2a + 6)
-32a5-16a*& + 8a362-4a2&3 + 2a&*-&5.
13. (*«-729y«)^(a;-3y)
= a-5 + 3a^y + 9ar»y2 + 27a?Y + 81 xy* + 243y*.
14. (x«-729y8)-*-(x + 3y)
= x5 - 3x*y + 9x3y2 - 27x2y3 + 81 xy* - 243^.
15. (81 a* - 16c*) -h (3a -2c)
- 27a3 + 18a2c + 12ac* + 8c*
1G. (81 a* - 16c*) h- (3a + 2c) 18. (256a* - 10,000) + (4a + 10)
= 27a3-18a2c+12ac2-8c3. = 64 a3- 160 a2 +400 a -1000
17. (256a*- 10,000) -*-(4tf- 10) 19. (625x*- 1)-?- (5x- 1)
= tiia*+ 160a2+400a + 1000. = 125X3 + 25a? + 5a; + 1.
Exercise XXVII.
1. (aB + ys) + («« + yI) 6. (*12 + 1) -*- (x* + 1)
« x* — x2y2 + y*. = xB — x* + l.
2. (a6 + 1) -*- (a2 + 1) 7. (64a* + y8) -5- (4a;2 + y2)
= a*-a2 + l. =16a^-4a?y2+y*.
3. (a10 + y10) -f- (a2 + y2) 8. (64 + a6) -*- (4 + a2)
= a*-a*y2 + ay-dY + if. - 16 - 4a2 + a*.
4. (6,0 + l) + (6» + l) 9. (729rf + ft*) + (9a* + M)
= ft«_Ji + J4_ ji+ i, = 81a*-9a262 + fc*.
5. (a12 + ft12) -s- (a* + &*) 10. (729c« + l)-?-(9c2 + l)
= a8 - a*V + 68. --= 81 c* - 9 c2 + 1.
TEACHERS EDITION.
39
Exercise X;XVIII.
5x- 1 = 19,
5z = 19 + 1,
51 = 20,
16a: -11 = 7x + 70,
16a; -7a =70 + 11,
9a; = 81,
x = 9.
3a; + 6 = 12,
3rc = 12-6,
3x = 6,
a: = 2.
9. 24a: -49 = 19a: -14,
24a; -19a; = 49 -14,
5a: = 35,
a: = 7.
24a: = 7a; + 34,
24 a: -7a? = 34,
17a; = 34,
a; = 2.
10. 3a: + 23 =78 -2a:,
3a: + 2a: = 78 -23,
bx = 55,
a; =11.
4. 8rr-29 = 26-3a:,
8a; + 3a; = 26 + 29,
11a; = 55,
a; = 5.
11. 26-8a: = 80-14x,
14a; -8a: =80 -26,
6a; = 54,
x = 9.
12 -5a; = 19 -12a:,
- 5a; + 12a; =19 -12,
7a; »= 7,
a;=l.
12. 13 - 3a; = 5a; -3,
-5a;-3a; = -3-13,
-8a; = -16,
x = 2.
6. 3a; + 6- 2a; = 7a;,
3a; -2a; -7a; = -6,
-6a; = -6,
*-l.
13. 3a; - 22 = 7a; + 6,
3a:-7a; = 6 + 22,
-4a; = 28,
x = -7.
7. 5a; + 50 =4x + 56,
5a; -4a; = 56 -50,
a; = 6.
14. 8 + 4a; = 12a; -10,
4a:-12x = -16-8,
-8a; = -24,
x = 3.
15. 5a: - (3a; - 7) = 4a; - (6a; - 35),
5a; - 3a: + 7 = 4a; - 6a: + 35,
- 4a; + 5a; - 3a; + 6a; = 35 - 7,
4a; = 28,
a; = 7.
16. 6a; - 2(9 - 4x) + 3(5a? - 7) = 10z - (4 + 16a; + 35),
6a; - 18 + 8a; + 15a; - 21 = 10a: - 4 - 16a; - 35,
6a; + 8a; + 15a; - 10a; + 16a; = 18 + 21 - 4 - 36,
35 a; = 0,
a: = 0.
40 ALGEBRA.
17. 9a; -3 (5a; -6) + 30-0,
9 a? -15a; + 18 + 30 = 0,
9a? -15a; = -18 -30,
-6a? = -48,
x = 8.
18. a? - 7 (4 as - 11) = 14 (a? - 5) - 19 (8 - x) - 61,
x - 28 x + 77 = 14 x - 70 - 152 + 19 x - 61,
a: - 28a: - 14a? - 19a; = - 70 - 152 - 61 - 77,
-60x = -360,
x = 6.
19. (a? + 7) (a? -3) = (a; -5) (a; -15),
a?2 + 4z - 21 = x2 - 20a; + 75,
4a; + 20 a; = 75 + 21,
24 a; = 96,
as-4.
20. (a; - 8) (a; + 12) = (a? + 1) (a? - 6),
0^ + 43? -96 = a?2 -5a; -6,
4a? + 5a? = -6 + 96,
9a; = 90,
x = 10.
21. (»-
-2)(7-a?)+(a;-5)(x + 3)-2(a;-l) + 12 = 0,
9a; - 14 - x2 + x2 - 2a; - 15 - 2a; + 2 + 12 = 0,
9a; - 2a? - 2a? = 14 + 15 - 2 - 12,
5a? =15,
a; = 3.
22. (2a;- 7) (a; + 5) - (9 - 2a?) (4 -a?) + 229,
2a?8 + 3a; - 35 = 36 - 17a; + 2a?2 + 229,
20 a; = 300,
x =15.
23. 14-a?-5(a?-3)(a; + 2) + (5-a;)(4-5a?) = 45a;-76J
14 - a; - 5a?2 + 5a? + 30 + 5a?2 - 29a; + 20 = 45a; - 76,
5 x - 29 a? - 45 a? - a? = - 7 6 - 20 - 30 - 1 4 ,
-70a? = -140,
a? = 2.
24. (a; + 5)2- (4 -a?)2 = 21 a?,
(a? + 10a? + 25) - (16 - 8a? + a?2) = 21a?,
a?2 + 10a; + 25 - 16 + 8a; - a?2 = 21 a?,
10a; + 8a? - 21 a? = - 25 + 16,
_3a? = -9,
a; = 3.
25. 5(a?-2)2 + 7(a?-3)2=(3a?-7)(4a?-19)+42,
5(a?2-4a?+4) + 7(a?2-6a; + 9)=12a?2-85a; + 133+42,
50^-200? + 20 + 7a?2-42a? + 63 = 12a?-85a? + 133 + 42,
23 a? = 92,
a? = 4.
teachers' EDITION. ' 41
Exercise XXIX.
6. To the double of a certain number I add 14, and obtain as
a result 154. What is the number ?
Let x = the number.
Then 2a? = its double,
and 2 a; + 14 = its double increased by 14.
But 154 = its double increased by 14.
Therefore, 2 a; + 14 = 154, 2 a; = 140, x = 70.
7. To four times a certain number I add 16, and obtain as a
result 188. What is the number ?
Let x = the number.
Then 4 a; = 4 times the number,
and 4a; + 16 = 4 times the number increased by 16.
But 188 = 4 times the number increased by 16.
Therefore, 4a; + 16 = 188, 4a; = 172, a; = 43.
8. By adding 46 to a certain number, I obtain as a result a
number three times as large as the original number. Find the
original number.
Let x = the original number.
Then 3 x = 3 times the original number.
But x + 46 = 3 times the original number.
Therefore, 3 a; = a: + 46, 2 a; = 46, a; = 23.
9. One number is three times as large as another. If 1 take
the smaller from 16 and the greater from 30, the remainders are
equal. What are the numbers ?
Let x = the smaller number.
Then 3 x = the larger number,
and 16 — x = 16 diminished by the smaller number ;
also, 30 — 3 a; = 30 diminished by the larger number.
Therefore, 16 - x = 30- 3a;, 2a; =14, x =7, 3a; = 21.
10. Divide the number 92 into four parts, such that the first
exceeds the second by 10, the third by 18, and the fourth by 24.
Let x = the first part.
Then x — 10 = the second part,
x — 18 = the third part,
x — 24 = the fourth part,
and 4 x — 52 = the whole number.
But 92 = the whole number.
.\4a;-52=92, 4a;=144, a;=36, a;-10 = 26, 3-18 = 18, a;-24 = 12.
42 ALGEBRA.
11. The sum of two numbers is 20; and, if three times the
smaller number is added to five times the greater, the sum is 84.
What are the numbers ?
Let x = the greater number.
Then 20 — x = the smaller number,
5x = 5 times the greater number,
3 (20 — x) = 3 times the smaller number,
5 a: -f 3(20 — a;) = 5 times the greater -f 3 times th© smaller.
But 84 = 5 times the greater + 3 times the smaller.
/.5a; + 3(20-a;) = 84, 5a; + 60-3a; = 84, 2z = 24, a?=12, 20-x=S.
12. The joint ages of a father and son are 80 years. If the
age of the son were doubled, he would be 10 years older than his
father. What is the age of each ?
Let x = number of years of father's age.
Then 80 — x = number of years of son's age,
2(80 — x) = number of years of father's age + 10,
x + 10 = number of years of father's age + 10.
.\2(80-a:) = a; + 10, 160-2x = a;+10, -3a? = -150, a;=50, 80-a; = 30.
13. A man has 6 sons, each 4 years older than the next younger.
The eldest is three times as old as the youngest. What is the
age of each?
Let x = number of years of age of youngest.
Then x + 4 = number of years of age of second,
x + 8 = number of years of age of third,
x + 12 = number of years of age of fourth,
x + 16 = number of years of age of fifth,
x + 20 = number of years of age of sixth.
3 x = 3 times age of youngest.
/. 3« = a: + 20, 2a; = 20, a; = 10, x + 4=14, x + 8 = 18,
x + 12 = 22, a; + 16 = 26, x + 20 = 30.
14. Add #24 to a certain sum and the amount will be as much
above $80 as the sum is below $80. What is the sum?
Let x = number of dollars in sum.
Then x + 24 — 80 = number of dollars above 80,
and 80 — x = number of dollars below 80.
.-.a; + 24 -80 = 80 -a;, 2a; = 136, a: = 68.
15. Thirty yards of cloth and 40 yards of silk together cost
$330 ; and the silk twice as much a yard as the cloth. How much
did each cost a yard ?
Let x = number of dollars one yard of cloth cost.
Then 2 a; ^= number of dollars one yard of silk cost.
30 a; + 80 a; = number of dollars all cost.
But 330 — number of dollars all cost.
/.30a; + 80a; = 330l 110a; = 330, x = 3, 2a; -6.
teachers' edition. 43
16. Find the number whose double increased by 24 exceeds
80 by as much as the number itself is less than 100.
Let x = the number.
Then 2x + 24 = its double increased by 24,
2 x + 24 — 80 = excess over 80,
100 — x == difference between the number and 100.
.\2x + 24-80 = 100-*, 3a; = 156, x = 52.
17. The sum of $500 is divided among A, B, C, and D. A
and B have together $280, A and C $260, and A and D $220.
How much does each receive?
Let x = number of dollars A has.
Then 280 — x = number of dollars B has,
260 — x = number of dollars G has,
220 — x = number of dollars D has,
760 — 2x = number of dollars all have.
But 500 = number of dollars all have.
.-.760 -2x = 500, -2x = -260, x = 130,
280-* = 150, 260-* = 130, 220-*= 90.
18. In a company of 266 persons composed of men, women,
and children, there are twice as many men as women, and twice
as many women as children. How many arc there of each?
Let x = number of children.
Then 2x = number of women,
and 4* = number of men,
7* = whole number.
But 266 = whole number.
.-.7* =266, 3 = 38, 2* =76, 4* = 152.
19. Find two numbers differing by 8, such that four times the
less may exceed twice the greater by 10.
Let x = greater number.
Then * — 8 = smaller number.
4(3 _ 8) -2* =10.
.\4*-32-2*=10, 2* = 42, * = 21, *-8 = 13.
20. A is 58 years older than B, and A's age is as much above
60 as B's age is below 50. Find the age of each.
Let x = number of years of B's age.
Then x + 58 = number of years of A's age,
(* + 58) — 60 = number of years of A's age above 60,
50 — x = number of years of B's age below 50.
.•.(* + 58) ^0 = 50-s, 2* = 52, x = 26, a; + 58 = 84.
44 ALGEBRA.
21. A man leaves his property, amounting to $7500, to be
divided among his wife, his two sons, and three daughters,
as follows : a son is to have twice as much as a daughter, and
the wife $ 500 more than all the children together. How much
was the share of each?
Let x = number of dollars in a daughter's share.
Then 2x = number of dollars in a son's share,
and 3 a? = number of dollars given to all the daughters .
also, 4 a; = number of dollars given to all the sons.
7 a; = number of dollars given to all sons and
daughters,
7 a; + 500= number of dollars given to wife,
7 a; + 7 a; + 500= number of dollars in whole estate.
But 7500 = number of dollars in whole estate.
/. 7 a; + 7a; + 500 =7500, 14a; =7000, x = 500,
2a;=1000, 7a; + 500 = 4000.
22. A vessel containing some water was filled by pouring in
42 gallons, and there was then in the vessel seven times as much
as at first. How much did the vessel hold?
Let x = number of gallons the vessel holds.
Then a?— 42 = number of gallons there were in the vessel,
7 (x — 42) = 7 times number of gallons there were at first.
:.x =7 (x -42), a;=7a;-294, -6a; = -294, a; = 49. '
23. A has $ 72 and B has $ 52. B gives A a certain sum ; then
A has three times as much as B. How much did A receive
from B?
Let x = number of dollars A receives from B.
Then 52 — x = number of dollars B has left,
and 72 + x = number of dollars A has.
-.72 + x = 3(52 -a;), 72 + x = 156 - 3a;, 4x = 84, a; = 21.
24. Divide 90 into two such parts that four times one part
may be equal to five times the other.
Let x = larger number.
Then 90— x = smaller number.
/.4a; = 5(90-a?)J 4a; = 450-5a;, 9a; = 450, a; = 50, 90-a; = 40.
25. Divide 60 into two such parts that one part exceeds the
other by 24.
Let x = lesser part.
Then x + 24 = greater part.
,\a; + a; + 24 = 60, 2a; = 36, .? = 18, a; + 2i««42.
teachers' edition. 45
26. Divide 84 into two such parts that one part may be less
than the other by 36.
Let x = lesser part.
Then x + 36 = greater part.
.\a; + a; + 36=84, 2a; = 48, x =24, 84 -a; = 60.
27. A is twice as old as B, and 22 years ago he was three
times as old as B. What is A's age?
Let x = number of years of B's age.
Then 2a; = number of years of A's age ;
also, x — 22 = number of years of B's age 22 years ago,
and 2 a; — 22 = number of years of A's age 22 years ago.
.\3(a;-22) = 2a;-22, 3a:- 66 = 2a;- 22, a; = 44, 2a; = 88.
28. A father is 30 and his son 6 years old. In how many
years will the father be just twice as old as the sou?
Let x = number of years.
Then x + 30 = number of years of father's age x years hence,
and x + 6 = number of years of son's age x years hence.
.\30 + a: = 2(a; + 6), 30 + x = 2a; + 12, x = 18.
29. A is twice as old as B, and 20 years since he was three
times as old. What is B's age?
Let x = B's age.
Then 2x = A's age ;
also, x — 20 = B's age 20 years since,
and 2 a; — 20 = A's age 20 years since.
.\2a;-20 = 3(a;-20), 2a;- 20 = 3a;- 60, a: = 40.
30. A is three times as old as B, and 19 years hence he will
be only twice as old as B. What is the age of each?
Let x = number of years of B's age.
Then 3a; = number of years of A's age ;
also, x + 19 = number of years of B's age 19 years hence,
and 3 a; -f 19 = number of years of A's age 19 years hence.
.\3a: + 19 = 2(a; + 19), 3a; + 19 = 2a; + 33, x =19, 3a; = 57.
31. A man has three nephews ; his age is 50, and the joint
ages of the nephews is 42. How long will it be before the joint
ages of the nephews will be equal to that of the uncle?
Let x = the number of years.
Then 50 + x = number of years of uncle's age x years hence.
3 a; + 42 = number of years of nephews' age x years hence
.-.3a; +42 = 50 + *, 2a; = 8, a; = 4.
46 ALGEBRA.
32. A sum of money consists of dollars xand twenty-flve-cent
pieces, and amounts to #20. The number of coins is 50. How
many are there of each sort?
Let x = number of dollars.
Then 50— a; = number of quarters,
and x + — — = sum in dollars.
4
But 20 = sum in dollars.
...x + 5^11? = 20, 4a;-f50-a; = 80, 3» = 30, as- 10, 50-3 = 40.
4
33. A person bought 30 pounds of sugar of two different
kinds, and paid for the whole $2.94. The better kind cost 10
cents a pound, and the poorer kind 7 cents a pound. How many
pounds were there of each kind?
Let x = number of pounds of the better kind.
Then 30 —x = number of pounds of the poorer kind,
and 10a: + 7(30— x) = number of cents he paid tor all.
But 294 = number of cents he paid for all.
.'. 10* + 7(30 -x) = 294, 10s + 210 - Ix - 294,
3a; = 84, a; =28, 30 -a; = 2.
34. A workman was hired for 40 days, at $ 1 for every day
he worked, but with the condition that for every day he did not
work he was to pay 45 cents for his board. At the end of the
time he received #22.60. How many days did he work?
Let x = number of days he was idle.
Then 40— x = number of days he worked,
and 45a; = number of cents he paid for board ;
also, 4000 — 100 a; = number of cents he received for work,
(4000— 100a;)— 45a; = number of cents cleared.
But 2260 = number of cents cleared.
.-. 4000 -100a;- 45a; = 2260, - 145a; = -1740, x = 12, 40 - x = 28.
35. A wine merchant has two kinds of wine ; one worth 50
cents a quart, and the other 75 cents a quart. From these he
wishes to make a mixture of 100 gallons, worth $2.40 a gallon,
ilow many gallons must he take of each kind?
Let x = number of gallons at f 2.
Then 100 — x = number of gallons at $ 3,
and 2 a: = number of dollars one part cost ;
also, 3(100 — x) = number of dollars the other part cost,
and 2a:+3(100 — a;) = number of dollars all cost.
But 240 «= number of dollars all cost.
• 2a; + 3(100 - x) = 240, 2 a; + 300 -3 a: = 240, a? = 60, 100-a; = 40.
TEACHERS' EDITION. 47
36. A gentleman gave some children 10 cents each, and had
a dollar left. He found that he would have required one dollar
more to enable him to give them 15 cents each. How many
children were there?
Let x = number of children.
Then 10 x = number of cents given ,
and 10 x + 200 = number of cents required to give each 15 cts.
But 15 x = number of cents required to give each 15cts.
.-.10a; + 200 =15 x, -bx = -200, a; = 40.
37. Two casks contain equal quantities of vinegar : from the
first cask 34 quarts are drawn ; from the second, 20 gallons ; the
quantity remaining in one vessel is now twice that in the other.
How much did each cask contain at first?
Let x = number of quarts each contained at first.
Then x — 34 = number of quarts first now contains,
and x— 80 = number of quarts second now contains.
2 (x — 80) = twice the No. quarts second now contains.
2(z-80) =
2 (a? -80) =
.\2(a:-80) = a;-34, 2a>-160 = a;-34, z=126.
38. A gentleman hired a man for 12 months, at the wages of
$90 and a suit of clothes. At the end of 7 mouths the man quits
his service, and receives $33.75 and the suit of clothes. What
was the price of the suit of clothes?
Let x = number of dollars the suit cost.
Then x + 90 = number of dollars he receives by the year.
and — — = number of dollars he receives by the month.
12 J
and v — l = number of dollars he receives for 7 months.
12
But x + 33.75 = number of dollars he receives for 7 months.
.?(« + 30) -33.75 7a; + 630 = 405 + 12a;, 5a; = 225, a? = 45.
12
39. A man has three times as many quarters as half-dollars,
four times as many dimes as quarters, and twice as many half
dimes as dimes. The whole sum is $7.30. How many coins
has he in all ?
Let x = number of half-dollar pieces.
Then 3 x = number of quarter-dollar pieces,
12 a; = number of dimes,
24 a; = number of half-dimes.
x , 3x , 12a? , 24a; ,i , , • j n
- + — + -zr^r + =• the whole sum in dollars.
2 4 10 20
But 7.30 = the whole sum in dollars.
^ + 3£ + 12a;+24» = ^ 1Qx + 15a. + 24a; + 24a; = 146,
2 4 10 20
73x = 146, a; = 2, 3x = 6, 12a; = 24, 24a; = 48
48 ALGEBRA.
40. A person paid a bill of $ 15.25 with quarters and half-
dollars, and gave 51 pieces of money all together. How many of
each kiud were there?
Let x = number of half-dollars.
Then 51 — * = number of quarter-dollars.
50 a: = number of cents in half-dollars,
25(51 — x) = number of cents in quarter-dollars,
50* + 25(51 — *) = number of cents in all.
But 1525 = number of cents in all.
.-. 50s + 25(51 - x) = 1525, 50* + 1275 - 25a? = 1525,
25* = 250, a =10, 51-* = 41.
41. A bill of £ 100 was paid with guineas (21 shillings) and
hall-crowns (2J shillings), and 48 more half-crowns than guineas
were used. How many of each were paid?
Let * = number of guineas.
Then * + 48 = number of half-crowns,
21* + 2j (* + 48) = number of shillings in the lot.
But 2000 = number of shillings in the lot.
.\21* + 2£(* + 48)=2000, 21* + 5x + 240 = 2000,
42* + 5* + 240 = 4000, 47* = 3760, * = 80, * + 48 = 128.
Exercise XXX.
1. 5a* -15a 6. 6 a5 fc8 - 21 a4 b2 + 27a8 b4
= 5a(a - 3). = 3a862(2a26 - 7a + 9b2).
2. 6a3 + 18a2 -12a 7. 54 x2 y6 + 108*V~ 243*6 f
= 6a (a2 + 3a- 2). = 27*2y8(2 + 4*2t/2 - 9*V).
3. 49*2-21* + 14 8. 45*V°-90*5y7-360*V
= 7(7** - 3* + 2). = 45*V(*V - 2* - 8y).
4. 4*^ -12*2y2 + 8*^ 9. 70asy4- 140 a2 y* + 210 ay6
= 4*t/(*2 - 3*y + 2y2). = 70ay* (a2 -2ay + 3y2).
10. 32a36« + 96a668-128a869
= 32a366(l + 3a862-4a56s)
y* — aw8 + by2 + cy
= y(y3-ay* + 6y + c).
Exercise XXXI.
1. *2 — a* — bx + ab 3. be + bx — ex— x2
= (* — a) (* — b). = (b — *) (c + *).
2. ab + ay — by — y2 4. mx + mn + ax + an
= (a — y)(b + y)- = (m + a) (a? + w).
teachers' edition. 49
5. cdx2 — cxy + dxy — y2 8. obey — b2dy — acdx + bd* x
= (cx + y) (dx — y). = (ae — ba ) {by — dx).
6. abx — aby +pqx —pqy 9. ax — ay — bx + by
= (ab +pq) (x - y). = (a - 6) (x-y).
7. cdx1 + adxy — bcxy — aby2 10. cdz2 — cyz + dyz — y2
= (ex + ay)(dx-by). = (c* + y) (cfe - y).
Exercise XXXII.
1. s2 + lla; + 24 14. a* + 5a2 + 6
= (* + 8) (x + 3). - (a2 + 3) (a1 + 2).
2. 3? + 11* + 30 15. 28 + 4zs + 3
= (z + 6)(s + 5). ~(* + 3)(«» + l).
3. y2 + lfy+60 16. a262 + 18a6 + 32
= (y + 12) (y + 5). = (aft + 16) (ab + 2).
4. 2* + 132 * 12 17. sV + 7x*y* + 12
= (z + 12) (2 + 1). = (**y2 + 4) (x*y2 + 3>
5. a2 + 21 x + 110 18. 210 + 102* + 16
= (as + 1 1) (x + 10). = (2s + 8) (2* + 2).
6. y2 + 35y + 300 19. a2 + 9a6 + 2062
= (y + 20) (y + 15). = (a + 56) (a + 46).
7. 6* + 236 + 102 20. z8 + 9x8 + 20
= (6 + 17) (6 + 6). = (*» + 5) (x9 + 4).
8. re2 + 3a + 2 21. a2*2 + 14a6x + 3362
= (x + 2) (x + 1). - (a* + 11 6) (ax + 3 6).
9. 2? + 7x + 6 22. a2c2 + 7acz + 10x2
= (a + 6) (a? + 1). = (ac+ 5x)(ac + 2x).
10. a2 + 9<x6 + 8 62 23. x*y*z* + 19zy2 + 48
= (a + 86) (a + 6). = (xyz + 16) (xy2 + 3).
11. a? + 13a* + 36a2 24. 6V+ 18a6c + 65a2
= (x + 9a) (x + 4a). = (6c + 13a) (6c + 5a).
12. y* + 19oy + 48p2 25. rV + 23 rsz + 90 22
= (y + 16p) (y + 3p). = (rs + 18 2) (rs + 52).
13. 22+29a2 + 100g2 26. m*n4 + 20m2n2pa + 51p2o2
= (2 + 25a) (2 + 4a). = (m2n2 + 17pa)(m2n2 + 3^o)
50 ALGEBRA.
Exercise XXXIII.
1. a»-7« + 10 13. a2 b2 <? - 13 abc + 22
- (x - 5) (a - 2). = (a&c - ll)'(a&c - 2).
2. a2 -29a? +190 * 14. a?2 -15a? + 50
= (a? -19) (a; -10). = (a?-*0)(a?-5).
3. a2 -23 a + 132 15. a?2- 20 a? +100
= (a - 12) (a - 11). = (a? - 10) (x - 10).
4. &2-30& + 200 16. a^ - 21 oa; + 54
- (6 - 20) {b - 10). = (ax - 18) (ax - 3).
5. 2a- 432 + 460 17. a2a*-16a&a? + 39 62
= (2 - 23) (2-20). = (ox - 13 b) (ax - 3 b).
6. a;2 -7a? + 6 18. a2 c2 - 24 ac2 + 143 2*
= (a? - 6) (x - 1). = (ac - 13z)(ac - llz).
7. a?*-4a2a?2 + 3a* 19. a?2 -20a? +91
= (x* - 3a2) (a?2 - a2). = (x - 13)(a? - 7).
8. a?2-8a? + 12 20. a?*-23a: + 120
= (a?_6)(a?-2). =(x-15)(x-S).
9. 22-57z + 56 21. 22- 532 + 360
= (2 - 56) (2 - 1). = (2- 45) (z - 8).
10. y6-7y8+12 22. a?2-^ c)a? + ac
-(y»-W-3): =(a?-a)(a;-c).
11. a?2y2-27a?y + 26 23. y222 - 28 abyz + 187a2 62
= (xy- 26) (ary - 1). = (yz - 17a6) (yz - 11 ab).
12. a*&6-lla2&8 + 30 24. c2 d2 - 30 abed + 221 a2 62
= (a2 6s - 6) (a2 6s - 5). = (cd - 17a6) (erf - 13 ab)
Exercise XXXIV.
1. r« + 6.T-7 . 5. 22 + ll2-12
= (a? + 7)(a?-l). =(2+12)(2-.l).
2. a?2 + 5a?-*84 6. z2 + 132-140
= (x + 12) (a?- 7). = (2 + 20) (2-7).
3. y2 + 7y-60 7. a2 + 13a -300
= (y + 12) (y -5). = (a + 25) (a - 12).
4. y2 + 12y-45 8. a2 + 25a- 150
= (y + 15) (y - 3). = (a + 30) (a - 5).
TEACHERS* EDITION. 51
9. &8 + 364_4 12. c2 4- 17c -390
= (6* + 4) (V - 1). - (c + 30) (c - 13).
10. 6V + 3&C-154 13. a2 + a- 132
= (be 4- 14) (6c - 11). = (a 4- 12) (a - 11).
11. c"> + 15c5 -100 14. x2y2z2 4- 9xyz- 22
= (c5 + 20) (c5 - 5). = (xyz + 11) (syz - 2).
Exercise XXXV.
1. s*-3a:-28 9. y2-5ay-50a2
= (x - 7) (x + 4). = (y - 10a) (y 4- 5a).
2. y2_7y_i8 10. a262-3a&-4
= (y-9)(y + 2). -(o6-4)(a6 + l).
3. a3_9a._36 11. a2x2-3ax-54
= (x - 12) (x + 3). = (ax - 9) (ax 4- 6).
4. z2-llz-60 " 12. c2^2- 24ca*- 180
= (z - 15) (z + 4). =s (erf - 30) (cd 4- 6).
5. 2a_l3z-14 13. a«c2-a»c-2
= (z - 14) (z + 1). = (a8c - 2) (a*c 4- 1).
6. a2 -15a -100 14. y8z*-5y*z2- 84
= (a - 20) (a 4- 5). = (y4z2 - 12) (y*z2 + 7).
7. c10-9c5-10 15. a262-16a&-36
= (c5 - 10) (c5 + 1). = (ab - 18) (a& + 2).
8. x2 — 8x-20 16. x2-(a-6)x — ab
= (x - 10) (x + 2). = (x - a) (x + 6).
Exercise XXXVI.
1. x2 4- 12x4- 36 6. z* + 14z2 + 49
= (x + 6)2. = (z2 + 7)2.
2. x2* 28x4- 196 7. x2 + 36xy 4- 324y2
= (x + 14)2. = (x4-18y)2.
3. x2 4- 34x4- 289 8. y* 4- 16VZ2 4- 64z*
= (x + 17)2. =(y2V8z2)2.
4. z24-2z4-l 9. yi + 24y8 + 144
-(z-fl)2. =(3/3 + 12)2.
5. y2 4- 200 y 4- 10,000 10. x2z2 4- 162 xz 4- 6561
= (y 4- 100)2. = (xz 4- 81)2.
52 ALGEBRA.
11. 4a2 + 12aft2 + 9ft4 13. 9z2 + \2xy + 4y2
= (2a+3ft2)2. = (3z + 2y)2.
12. 9x2y* + 30ary2z + 25z2 14. 4 a4 a?2 + 20a2ar>y + 25 z*y2
= (3zy2 + 52)2. = (2a2a; + 5a?2y)2.
Exercise XXXVII.
1. a2 -8a + 16 13. 16*«-8a^y2 + a^y4
= (a-4)2. =(4^ -ay2)2.
2. a2 -30a + 225 14. a**4- 2asftx2y4 + ft2y*
^(a-15)2. =(a8a;2-6y*)<
3. i2 -38* +361 15. 36r»y2-60zys + 25y4
= (s-19)2. =(6sy-5y2)2.
4. a2 -40a; + 400 16. l-6aft8 + 9a2ft*
= (a;-20)2. ' =(1-3 oft3)2.
5. y2-100y + 2500 17. 9m2n2-24mn + 16
= (y-50)2. . =(3mn-4)2.
6. y4-20y2+100 18. 4ft2sa- 12 bx*y + 9 x*y2
= (y2-10)2. = (2ftx-3xyj*.
7. tf-50yz + 625z* 19. 49a2- 112 aft + 64 ft2
= (y-25z)2. = (7a -8ft)2.
8. ar4-32*2y2 + 256yi 20. 64x*y«- IMsttfz + lOOz4*2
= (z2 - 16y2)2. = (8x*y* - 10 A)2.
9. z6- 342s + 289 21. 49 a2 ft2 c2- 28 aftcz + 4 a?
= (2»-17)2. = (7aftc-2a:)2.
10. 4a4y2 - 20arly8z + 25y4z2
= (2a?2y-5yV.
22. 121x4-286r,y+169y2
= (llr«-13y)2.
11. lGaJy*-8jrys22+y2z4 23. 289 afyV-^zyVo^yVd1
= (4*y2 - yz2)2. = (17zyz-3yza7.
12. 0a262c2-6a62c2a' + 62c2a'2 24. 361xsy2z2-76a6cayz+4a262c2
= (3aftc - bed)*. = (19 xyz-2abc)*.
Exercise XXXVIII.
1. a2 -ft2 , 3. 4a2 -25
- (a + ft) (a - ft). = (2 a + 5) (2 a - 5).
2. a2 -16 4. a* -ft4
- (a + 4) (a - 4). = (a2 + ft2) (a + 6) (a - 6).
TEACHERS* EDITION. 53
5. a*-l 10. 1-49*2
= (a2 + l)(a + l)(a - 1). = (1 + 7*) (I - 7*).
6- a8-*8 11. a* -25b2
= (aH6*)(a2+62)(a+6)(a-6). = (ai + db){a2 _ 56).
7' t (^-l)(a2+l)(a+l)(a-l). *• L^- 6~)-cH(a-6) +
8. 36s2 -49y2 = (a- 6-c)(a-6 + c).
= (Gx + 7y)(6x-7y).
9. 100a^y2-121a262
= (10a;y+lla6)(103y-lla6).
13. x*-(a-b)*
= {x-(a-b)}{x + (a-b)}
= (a; — a + 6) (a? + a — b).
14. (a + 6)2-(c+rf)2
= [(« + &) + (c + <*)] [(« + &)-(« + <*)]
= (a + 6 + c + d) (a + 6 - c - d).
15. (« + yf - (x - y)2
= {(* + y) + (*-y)M(z + y)-(z-y)}
= (x + y + x -y)(x + y - x + y)
= 4xy.
16. 2a6 - a2 - 6* + 1 19. a2 + 126c - 462 - 9c2
= 1 - (a2 - 2ab + b2) = a2 - (46*- 126c + 9c2)
= l-(a-6)2 =a2-(26-3c)2
= {1 + (a - 6)}{1 - (a - 6)} = {a + (2 6 - 3 c)}{a-(2 6-3 c)}
~(1 + a - 6)(1 - a + 6). = (a + 26- 3c)(a-26 + 3c).
17. s?-2y*-y*-^ 20. a2- 2ay + y"-3i-2«- 22
= a* _ (y2 + 2y2 + 22) = (aa_2 av+y2)-(ic2+2a;2+22)
= s2-(y + 2}2 ={a-y)*-(x + z?
= (a; + (y + z)} {x - (y + 2)} = {(a-t/)+(a?+2)}{(a-y)-(ir+2)}
= (x + y + 2) (x - y - 2). = (a-y+x+z)(a-y-x-z).
18. s*-2zy + y2-22 21. 2xy - a^-y2 + 22
= (z2 - 2 ay + y2) - z2 = z2 -(x2 - 2xy + y2)
'(x-yY-z2 =z2^lx_yy
= te + (x-y)\{z-{x-y)}
= (z+x-y)(z-x+y).
= (x-y + z)(z-y-z).
22. x* + y2-z2-d2-2xy-
= (x*-2xy + y2)-(d2-
■2dz
l + 2dz + z2)
= (x-y)2-(d + z)2^
= fa ~ V) ~(d + z)}{(x - y) + (d + 2)}
= (x — y - d — z) (x — y + d + 2).
23. »2-y2+22-a2-2a;2 + 2ay
= (xi~2xz + z2) - (a2 - 2ay + y2)
= (x-zf-(a-y)2
= ](x-z)-(a-y)}{(x-z)+{a-y)}
— (a; — 2 — a + y) {x — z + a — y).
54 ALGEBRA.
24. 2a6 + a24-62-c2
= (a2 + 2a& + 62)-c2
= (a + 6)2 - c2
= (a + 6 -h c) (a + 6 — c).
25. 2xy-x2-y2 + a2 + &2-2a&
= (a2 - 2a& + i2) - (x2 - 2xy + y2)
= l(a-4) + (x-y)}{(a-6)-(x-y)}
«= (a — b + x — y) (a — 6 — x + y).
26. (ax + 6y)2-l 27. l-x2-y* + 2xy
= (ax + by + l)(ax + by-l). = 1 - (x2 - 2xy + y2)
= l-(x-y)2
= (l+x-y)(l-x+y).
28. (5a-2)2-(a-4)2
= {(5a - 2) + (a - 4)}{(5a - 2)-(a - 4)}
= (5a - 2 + a - 4) (5a - 2 - a + 4)
= (6a - 6) (4a + 2) = 12 (a- 1) (2a + 1).
29. a?-2ab + V-x2 32. d2-x* + 4xy -4y*
»(a-6)»-^ = d2-(x2-4xy + 4y2)
= (a-6+x)(a-6-x). =d*-{x-2yf
= (d + x — 2y)(d—x + 2y).
30. (x + l^-ty+l)2 33. c?-b*-2bc-<?
«(s + l+y + l)(a> + l-y-l) = a2-(62 + 26c + c2)
= (x + y + 2)(x-y). =a2-(6 + c)2
= (a + 6 + c)(a — b — c).
31. (x + l)2-^-!)2 34. 4x*-9x2 + 6x-l
= (x + H-y-l)(x + l-y+l) =4s*-(9x2- 6x4-1)
= (x + y)(x-y + 2). = 4x*-(3x-l)2
= (2x2+3x-l)(2x2-3x+l).
Exercise XXXIX.
1. a3 -ft3 4. f-\2b
= (a - 6)(a2 + a& + 62). = (y - 5)(y2 + 5y + 25).
2.'s»-8 5. y»-216
= (x -2)(x2 + 2* + 4). = (y - 6)(y2 + 6y + 36).
3. x8 -343 6. 8x3-27y3
- (x - 7)(x2 + 7x + 49). = (2x-3y)(4x2 + 6xy + 9y2)
7. 64y3 - lOOOz3 = (4y - 10z)(16y2 + 40yz + 100s2).
8. 729X3 - 512y3 = (9x - 8y)(81x2 + 72xy + 64y2).
9. 27a3 - 1728 = (3a - 12)(9a2 + 36a + 144).
10. 1000a3-1331&3 = (10a-ll&)(100a2-l-110a6 + 12U2).
teachers' edition. 55
Exercise XL.
1. x* + y* 4. y:j + 64z3
= (x + y)(a*-xy + y2). = (y + 4*)(y2 - 4y* -f 16*2).
2. ar» + 8 5. 646s + 125c8
= (a; + 2) (a2 - 2 x + 4). =(46+5 c)(16 &2-20 6c+25 c2).
3. a? + 216 6. 216 a8 + 512 c8
= (x + 6) (a? - 6 a; + 36). = (6 a+8 c)(36 a2-48 OC+64C2).
7. 729a* + 1728^ = (9a; + 12y) (81 x2 - lOSxy + 144y2).
8. ar5 + y5 = (a; + y)(a?* — a^y + a^y2 — xy3 + y4).
9. a?7 + y7 = (a; + y) (a8 — ar>y + ar4 y2 — a^y8 + a^y4 — a^/6 + y6).
10. 3265 + 243c5 = (26 + 3c)(16&4- 2463c + 3662c2-546c8 + 81c4).
Exercise XLI.
1. a6 + b« - (a2 + 62) (a4 - a2 62 + ft4).
2. a10 + /;10 = (a2 + i2) (a8 - a6&2 + a464 - a266 + 6s).
3. a;12 + y12 6. a12+l
= (ar4 + y4) (a* - x*y* + y8). = (a4 + l)(a8 - a4 + 1).
4. ft6 + 64c6 7. 64a6 -h a^
= (&* + 4c2)(&4-462c2 + 16c4). " = (4a2+a?)(16a4-4a2a*+a*).
5. a*+l 8. 729 + c6
= (as" + !)(** -«■ + !). =(9 + c»)(81 -9<? + *»).
Exercise XLII.
1. a4 + a2 J2 + &4 = (a2 + ab + &2)(a2 - a& + &2).
2. 9a4 + 3z2y2 + 4t^ = (3s2 + 3a;y + 2y2)(3z2 - 3a*/ + 2y2).
3. 16a?4 - 17ar2y2 + y4 = (4ar* + 3a;y - y2)(4a;2 - 3ay - y2).
4. 81a4 + 23a2 52 + 1664 = (9a2 + lab + 462)(9a2 - 7a6 + 462).
5. 81a4- 28a2 62 + 16&4 = (9a2 + 10a& + 462)(9a2 - 10a& + 462)
6. 9a?* + 38a?y2 + 49y4 = (3a;2 + 2xy + 7y2)(3a;2 -2xy + 7y2).
7. 25a4 -9a2 62 + 16M = (5a2 + 7a6 + 468)(5a2 - 7a& + 462).
56 ALGEBRA.
8. 49m4 + 110m2«2 + 81 n* = (7m2 + 4mn + 9n2)(7m2 - 4mn + 9n2).
9. 9a4 + 21 a2c2 + 25c4 = (3a2 + 3ac + 5c2) (3a2 - 3ac + 5c2).
10. 49a4-15a262 + 121&4 = (7a2 + 13a& + 1162)(7a2-13a6 + n&2).
11. 64a?4 + 128sV + 81/ = (8a2 + 4ay + 9y2)(8o2 - 4oy + 9y2).
12. 4a4 - 37*V + 9^ = (2a2 + 5oy - 3y2) (2a2 - 5oy - 3y2).
13. 25a* - 41 x*y* + 16y4 = (5a2 + zy- 4y2) (5a2 - ay - 4y2>
14. 81a4-34a2y2+y4«=(9a2 + 4ay-y2)(9a2-4ay-y2).
Exercise XLIII.
1. 12a2 - 5a -2 13. 6a2a2 + aa-l
= (4a+l)(3a-2). = (2aa + l)(3aa- 1).
2. 12a2-7a + l 14. 6&2-76a-3a2
= (3a-l)(4a-l). = (36 +a)(26-3a).
3. 12a2 - a- 1 15. 4a2 + 8a + 3
= (4a + l)(3a - 1). = (2a + l)(2a + 3).
4. 3a2 -2a -5 16. at-ax-Gx2
= (a + l)(3a - 5). = (a + 2a)(a - 3a).
5. 3a2 + 4a-4 17. 8a2 + 14a&- 1562
= (a + 2)(3a - 2). = (2a + 5 6)(4a - 36).
6. 6a2 + 5a -4 18. 6a2 - 19ac + 10c2
= (3a + 4)(2a - 1). = (3a - 2c)(2a- 5c).
7. 4a2 + 13a + 3 19. 8a2 + 34ay + 21 y2
= (4a + l)(a + 3). = (4a + 3y)(2a + 7y).
8. 4a2-flla-3 20. 8a2- 22ay- 21y2
= (a + 3)(4a -1). = (4a + 3y)(2a - 7y).
9. 4a2 -4a -3 21. 6a2 + 19ay-7y2
= (2a + l)(2a- 3). = (2a + 7y)(3a-y).
10. a2-3aa + 2a2 22. 11a2- 23a6 + 2ft2
= (a- a)(a- 2a). = (11a- b)(a-2b).
11. 12a4 + a2a2-a4 23. 2c2 - 13ca* + 6d2
= (3a2 + a2)(4a2 - a2) = (2c- d)(c -6d).
12. 2a2 + 5ay + 2y2 24. 6y2 + 7yz-3z2
= (2a + y)(a + 2y). = (2y + 3z)(3y-z).
teachers' edition. 57
Exercise XLIV.
1. a* + Zatb+Zdt>lT& 5. «*-4a" + 6a?-4*+ 1
= (a + 6)3. ={x-lf.
2. o8+3a» + 3a+l 6. a4-4a8c + 6a2c2-4ac8 + c4
= (a + l>». -(a-c)».
3. a8-3a2 + 3a - 1 7. st? + 2xy + y2 + 2xz + 2yz+zi
= {a-lf. = (x+y + zf.
4. x4 + 4ar,y +6ic2y* + 4a^s+3/4 8. x* — 2xy + y2 — 2xz + 2yz+z*
= (* + y)4. -{x-y-zf.
9. a2 + 8" + c2 + 2a&-2ac -26c
= (a + b - c)\
Exercise XLV.
1. 2xi-5xy + 2y*-17x+13y + 21.
2 a? - 5 zy + 2 yJ = (a; - 2 y ) ( 2 a; - y ) ,
2a* -17a; +21 - (a;- 7) (2 x- 3),
2y2 + 13y + 21 = (-y-3)(- 2y -7).
a; — 2y, a; — 7, — 2y — 7;
2x-y, 2a;--3, -y-3.
(x-2y-7)(2a>-y-3).
2. 6a?-37ary + 6y2-5a;-5y-l.
6a*-5a;-l = (6a; + l)(a;-l),
6y2-5y-l = (6y + l)(y-l),
6x* - Wxy + 6y7 = (6a: - y) (z - 6y).
6a; — y, 6a; + 1, 1 — y;
a;-6y, -6y-l, *-l.
(6x-y + l)(a;-6y-l).
3. 6a?-5a#-6y2-a;-5y-l.
6 a? - 5xy - 6y2 = (2a; - 3y) (3a; + 2y\
6a£-*-l-(3as + l)(2s-l),
-6y2-5y-U(-3y-l)(2y + l). '
2a;-3y, 2a;-l, -3y-l;
3a; + 2y, 3a; + 1, 2y + 1.
(2a;-3y-l)(3a; + 2y + l).
58 ALGEBRA.
4. 5a?-8a^4-3y24-7a;-5y + 2.
5 a? - Sxy + 3y2 = (5a; - 3y) (a; - y),
5a? 4- 7a; 4- 2 = (5x4- 2) (a; +1),
3y2-5y4-2 = (3y-2)(y-l).
5a; -3y, -3y4-2, 5x4-2;
*-y» -y + j> * + i-
(5x-3y4-2)(x-y + l).
5. 2a?-ary-3y2-8aj + 7y + 6.
2x2 - 8x - 3y2 = (2x - 3y) (as 4- y),
2a? - 8x + 6 = (2x - 2) (as - 3),
-3y* + 7y + 6 = (-3y-2)(y-3),
2x-3y, 2a; -2, _3y-2;
x4-y, a?-3, y-3.
(2x-3y-2)(a; + y-3).
6. a?-25y2-10x-20y4-21.
a? - 10 a; 4- 21 = (x - 7) (a? - 3),
- 25y2 - 20y 4- 21 = (5y - 3) (- by - 7),
a?-25y2 = (x4-5y)(x-5y).
x — 7, x — 5y, — 5y — 7;
x — 3, x4- 5y, 5y — 3. '
(x-5y-7)(x + 5y-3).
7. 2a? — 5xy + 2ys — xz — yz — z2.
2 a? - 5xy + 2y2 = (2x - y) (a; - 2y),
2a:2 — xz — z2 = (2a: 4- z) (x — z),
2y2 -yz -z2 = (2y 4- z) (y - z).
2a; — y, 2a; + z, — y+z;
x — 2y, x — z, — 2y — z.
(2x-y + z){x-2y-z).
8. 6x24-xy-yi-3xz4-6yz-9z2.
6a?4-xy-y2 = (3x-y)(2x4-y),
6a?-3xz-9z2 = (3x4-3z)(2x-3z),
-y2-6yz + 9z2 = (-y + 3z)(y-3z).
3a; -y, 3x4-3z, -y + 3z;
2x4-y, 2x — 3z, y— y — 3z.
(3x4-3z-y)(2x-3z4-y).
9. 6a;2 — 7xy + y2 + 35 xz — 5yz — 6z2.
6 a:2 — 7xy + y2 = (6 a; — y) (x — y),
6a? 4- 35 xz - 6z2 = (6a; - z) (a; 4- 62),
y2 - 5yz - Gz2 = (y 4- z) (y - 6z).
6a; — y, 6a; — z, —y — z\
x — y, x 4- 6z, — y -h 6z.
(6x-y-z)(x-y 4- 62).
teachers' edition. 59
5x*-8xy + 3y2 = (5a; - 3y) (x - y),
5x2-3a;z-2z2 = (bx + 2z)(x-z)t
3y2 + yz-2z2 = (3y-2z)(y + z).
5a: -3y, 5x + 2z, - 3y + 2z ;
x — y, x — z, —y — z.
(5x — 3y + 2z) (x — y — z).
11. 2x*-xy-3y2-5yz-2z2.
2a? - 2xy - 3y2 = (2a; - 3y) (a; + y),
- 3y - byz - 2z2 = (- 3y - 2z) (y + z),
2a;* - 2z2 = (2a; - 2z) (x + z).
2x-3y, -3y-2z, 2x-2z;
x + y, y + z, x + z.
(2a; - 3y - 2z) (x + y + z).
.12. Ba^-lSxy + 6y2 + 12xz-13yz+6z2.
6X2 - 13xy + 6y2 = (3x - 2y)(2y - 3y),
6x* + 12xz + 6z2 = (3x + 3z) (2a; + 2z),
6y* - 13yz + 6za = (3y - 2z) (2y - 3z).
3x-2y, 3x + 3z, -2y + 3z;
2x-3y, 2x+2z, -3y + 2z.
(3 x - 2y + 3z) (2a; - 3y + 2z).
13. x»— 2xy + y2 + 5x — by 14. 2x2 + 5xy-3y2-4xz + 2yz
= (x2 - 2xy + f) + (5a; - 5y) = (2a?+5a?y-3y2)-(4a;z-2yz)
= (x-y)2 + 5(x-y) = (2x-y)(x+3y)-2z(2x-y)
= (a;-y)(a;-y + 5). - (xH-3y-2z)(2x-y).
Exercise XL VI.
1. 5a?— 15a;— 20 4. a2 + 2ax H-x2 + 4a + 4a;
= 5(x2-3x-4) = fa2 + 2ox + x2)+(4a + 4x)
= 5(x + l)(x — 4). = (a + x}2 + 4(a + x)
= (a + a;Ha + a; + 4V
2. 2x5-16x* + 24xs V *a« + » + *;-
= 2x*(x* -8a; + 12)
= 2x3(x-2)(x-6).
3. ZtfV-Sab-W =(a2-2a6 + 62)-c2
= 3(a262 - 3ab - 4) = (a - 6)2-c2
- 3 (aft - 4) (a& + 1). = (a - 6 + c) (a - 6 - c).
6. x2-2xy + y2-c2-{-2ca*-rf2
=» (a;2 - 2xy + y2) - (c2 - 2ca* + d2)
- {(* - y) + (« - <*)} {(* - v) -i« - <*)>
= (a; — y + c — d) (x — y — c -f a).
60 ALGEBRA.
7. 4-x2-2.is-.r4 13. a? - x* - ab - br
= 4-(x2 -f 2ar* + at) = (a2 - x2) - (a6 + 6a;)
= 4-(a; 4- ar1)2 = (a 4- a:) (a - a;) - 6 (a 4- a;)
= (2 + x + a;2) (2-x-x*). = (a + x)(a-x- 6).
8. a2-62-a-6 14. a2 -2ax + x2 + a-x
- (a2- 62) -(a + 6) = (a2 - 2a* 4- x2) + (a-x)
= (a + 6) (a - 6) - (a 4- 6) = f a - a-) (a - x) + 1 (a - .>
= (a 4- 6) (a - 6 - 1). = (a - a;) (a - x + 1).
9. a* + a2 + l 16. 3a*- 3y2- 2a; 4- 2y
r)
= (a44-2a24-l)-a
= (a2 + l)2 - a2
= (a2 + a 4- 1) (a2 - a 4- 1).
= (3a;- 3y2)- (2a; -2y)
-a^-y2)-^-*)
= 3(a;-t/)(x + y)-2(x- y)
= (a;-y)(3a;4-3y-2).
10. a;2 — y2 — xz+yz
= (x2-y2)-(xz-yz) 16. a^ + ar» 4- x2 4- a;
= (a; 4- y) (x - y) - z(x - y) = x* (x 4- 1) 4- x (x + 1)
= (x -y)(x 4- y -z). = (a? + a;) (a? + 1)
11 ^ w^t. -*(x» + l)(*+l).
11. a6 — ac — 62 + 6c
= (ab - ac) - (62 - be) 17. a4*4 - a8 a* - a2a? 4- 1
= a (6 — c) — b (6 — c) = a?.t?(ax—l)—(ax+l)(ax—l)
- (a - b) (b - c). = (ax - 1) (a^ar1 - aa;- 1).
12. 3a;2 - Sxz -xy + yz 18. 3ar»- 2a:2y- 27a#2 + 18y*
**(3x2-3xz)-(xy-yz) = a*(3a;-2y) -9y2(3a?-2y)
= Sx(x-z)~y(x-z) = (x2 - 9y2) (3a; - 2y)
= (Zx-y)(x-z). = (a;-3y)(a?4-3y)(3a;-2y).
19. 4a*-a;2 + 2a;-l
= 4a^-(a;2-2x + l)
= 4a*-(a;-l)2
= (2a?4-a;-l)(2a?-a; + l).
=(2x-l)(z + l)(2x2-x+l).
20. a;6-*/8
= (a<+y»)(a^-y»)
= (x + y) (x2 - xy + y2) (x - y) (ar» + a# 4- y2).
21. a^ + y8
= (x2+y2)(a^-a^y2+y*).
22. 729 -a*
= (27 + a*) (27 -a*)
= (3 4- a;)(9 - 3 a; + a?) (3 - a;) (9 +3a; + a?).
23. x12y + yl* 25. x2 4- 4a; -21
= y(x12 + y12) =(a; + 7)(ar-3).
= y (a* + y*)(a? - a^y4 4- y8).
24. c(a4-c*) 26. 3 a2 -21a6 4- 306*
- c (a2 + J) (a2 - c2) - 3 (a2 - 7a6 4- 10 62)
= c(a24-c2)(a \- c) (a - c). = 3 (a~26) (a- 56V
TEACHERS EDITION. 61
27. 2x*-4x*v-6x2y2 33. x2-2xy-2xz + y2 + 2yz + z2.
= 2a* (2x*- 2xy - 3y2) ^-2^ +v2 = (x-t/)(x-t/)
= 2x*(x-3y)(x + y). ^^M'-SK
28. 4a2-4a6 + 62 *+2y. +*-& + .& + .).
-(2a-w *-*• *"'» -y-«;
v ' *-y, *-*, -y-«.
29. le^-SOw + lOOy1 (»-y-*)(*-y-«).
= 4(4x2-20xy + &y2
= 4(2x-5y)». 34. ^-06-6^-40 + 126
= (a-a6-662)-4(a-36)
30. 36a*x*y*-25b*x*yi = (o-36)(a + 26)-4(a-36)
= aty»(36a1-25ftV) =»(o-36)(a+26-4).
= xV(6a + 6fec)(6a — 5bx).
«- ^ • j. ™ • «^. * 35. x* + 2xy+ y2-x-y- 6.
31. 9x2y4-30xy1* + 25z2 « ,_ 0 * J , , '
-dx^-hh x2 + 2x + y2 = (x+y)2,
-(dxy* 5z). x2-x-6t(x + 2)fx-3),
32. i6x*-x y2-y-6 = (y + 2)(y-3).
= x(16x*-l) * + y> *+2, y + 2;
= x(4x2 + l)(4x2-l) * + y. «-3, y-3.
-*(4a» + l)(2» + l)(2*-l). (x + y + 2)(x+y-3).
36. (o + 6)4-c*
= {(o + 6)s + cn«a + 6)»-c«}
=■ ((<* + &r + 07 (« + & + c)(a + & - c)
= (a2 + 2a6 + 6* + c*)(o + b + c)(a + 6 — c).
37. x2-xy-6y2-4x + 12y 38. l-x + x«-xs
= (x»-xy-6y«)-4(x-3y) - (1 -art + x»(l -x)
= (x + 2y)(s-3y)-4(x-3y) = (1 -x)(l + X2).
= (x + 2y-4)(x-3y). 39. 3x,_llxy + 6y,
= (3x-2y)(x-3y).
40. x2 + 20x + 91
= (x + 7)(x + 13).
41. (a;-y)(x2-z2)-(x-2)(x2-y«)
- (* -y)(* + *)(*- *)-(*-*) (x -y)(x + y)
-(x-y)(x-z)(z-y).
42. x*-5x-24
= (x-8)(x+3).
43. (x«-y*-za)s-4y2z2
= {(** ~ y2 - **) + SyzMx2 - y2 - z2) - 2yz}
= fo» _y« _2» + 2yz)(x2 - y2 _ z2 - 2yz)
- {«■ - (y2 - 2yz + z2)}**2 - (y2 + 2y* + *2)}
-p-(y-*-(y + z)2}
= (x + y - z) (x - y + z) (a; + y + z) (x - y - z).
62
ALGEBRA.
44.
48.
49.
jx3y2 + 5x2yz- 60 xz2
= 5x(a?y* 4- xyz— I2z2)
= 5a; {xy + 4z)(rry — 3z).
45. Sxt-xt + Sx-l
= a»(3ar-l) + (3*-l)
= (a* + l)(3a;-l).
46. x2 — 2mx + m2 — n2
= (a? — 2mi + m2) — n2
(a; — m)2 — w2
= (a; — m + n) (a; — m — ?i).
47. 4a2&2-(a2 + 62-c2)2
!2ab + (a2 + 62 - c2)}{2a& - (a2 + 62 - c2)}
2a& + a2 + 62 - c2}{2a& - a2 - J2 + c2}
(a2 + 2a6 + 62) - c*}{e* - (a2 - 2ab + 62)}
(a + 6)" -<?}{(? -(a -6)*}
a + 6 + c)(a + 6 — c)(c + a — b)(c — a + b).
a7 -ha?
= a6(a2 + l).
l-14asa: + 49a8a:2
= (l-7a*a>)2.
50. y2-4y-117
= (y-13)(y + 9).
51. x« + 6a;-135
= (a;+15)(a;-9).
52. 4a2-12a6 + 962-4c2
= (4a2-12a6 + 962)-4c2
= (2a-36)2-4c2
= (2a - 36 + 2c)(2a- 36 -2c).
53. (a + 3&)2-9(6-c)2
= {(a + 36) + 3(6-c)}{(a + 36)-3(6-c)}
= (a + 3& + 3&-3c)(a + 3&-36+3c)
= (a + 6&-3c)(a+3c).
54. 9x2-4y2 + 4yz-z2
= 9x2-(4y2-4yz + z2)
= 9x2-(2y-z)2
= (Sx + 2y-z)(Sx-2y + z).
55. 662x2-76a8-3x*
= x2(6&2-7&x-3x2)
= x2(3& + x)(2&-3x).
56. a*-&-Sab(a-b)
= a8-3a26 + 3a62-6»
= (a-6)8.
57. x* + y* + Zxy(x + y)
= x8 + 3x2y + 3xy2 + y2
= (x + y)8.
58. tf-6»-a(a» — F)+6(a — &)*
= a8-&8-a8 + a^ + a26-2a&2 + &8
= a26 — afc2
= a& (a — 6).
59. 9x*y2-Sxy* — 6y*
= 3y2(3x2 — xy — 2y2)
= 3y2(x-2/)(3x + 22/).
60. 6x2 + 13xy + 6y2
= (3x + 2y)(2x + Sy).
TEACHERS EDITION.
63
61. Ga262-a68-126*
= 62(6a2-a6-1262)
= 62(3a + 46)(2a-36).
62. a2 + 2ad + <22-462 + 126c-9c2
= (a2 + 2ad + d*) - (462- 126c T 9c*)
= a + d)2-(26-3c)2 V*.
- {(a + d) + (26 - 3c)}{(a + d) - (26 -
= (a + rf + 26-3c)(a+d-26 + 3c)
= (a + 26 - 3c + d) (a - 26 + 3c + d).
-k
3 c)}
63. x*-2xiy + 4xy2-$yi 64. 4a8*2 -Sabx + 362
= a? (a; - 2y) + 4y2 (a; - 2y) = (2ax - b)(2ax - 36).
= (a* + 4y2)(z-2y).
65. 18ar2-24zy + 8y2 + 9a;-6y
= (18s2 - 24 ay + 8^) + (9x - 6y)
= 2(9^ - 12xy + 4y2) + 3 (3a? - 2y)
= 2(3s-2y)24-3(3a;-2y)
= (6a;-4y + 3)(3a:-2y).
66. 2ar2 + 2sy-12y2 + 6sz + 18yz
= 2(3* + xy - 6y2 + 3a» + 9yz)
= 2(a?*+ xy - 6y2) + 3z (x + 3y)
= 2(x + 3y) (* - 2y) + 3z (x + 3y)
= 2(a;+3y)(a;-2yH-3z).
67. (x + y)»-l-ary(a: + y + l)
= (x + y + l)(x + y - 1)- xy (x + y + 1)
= (x + y + 1) (x + y - ary - 1).
= (x + y+l)(l-x)(y-l).
68. .r2-y2-z2 + 2yz + a? + y-z
= x2 — (y2 — 2yz + z2) + x +y — z
= x*-(y-z)2 + (x + y-z)
•■(x + y-z)(x-y + z) + (x + y-
z)(x-i
*(x + y-
-y + z + 1).
2x* + ±xy + 2y2 + 2aa; + 2ay 72. 12 az2 - 14 axy -6 ay:
70.
* 2(x2 + 2xy + y2) + 2a(a;+y)
= 2(x + y)2 + 2a(a; + y)
= 2(x + y + a)(x + y).
16a26+32a6c + 126c2
= 46(4a2+8ac + 3c2)
= 46(2a + 3c)(2a + c).
= 2a (6a* -7
= 2a(3a; + y)
xy — 3y2)
(2*-3y).
71. m2» — w?q — ri*p -\-n2q
= (m2 p — m2q) — (n2p — n2 q)
= f(p-q)-n*(p-q)
= (m2-n2)(p-q)
«= (m + w) (ra — n J ( p — q).
73. 2x* + 4x*-10x
= 2x(x2 + 2x-35)
-2*(* + 7)(*-5).
74. 16a3z-2aT*
= 2z(8aJ-ss)
= 2x(2a-x)(4a2 + 2az + arJ)
75. 326ar»-46y»
= 46(8ar8-y8)
= 46(2a,--y)(4s2 + 2*y+y2)
64 ALGEBRA.
76. a- 27**
-3aO(l + 3aj + 9s2).
-« (1-27 Xs)
-*(i-r * A
77. a^-y12
- C^+y*) (*4-aV+y*)(*+y) (^-sy +y2) (* -y) (*»+3y+y2).
78. 49m2- 121 na 81. a^-^ + oj-l
- (7m + lln) (7m - 11 n). - (a8 - a2) + (x - 1)
-rf(*-l) + (*-l)
79. 16-81 y* -(rf + l)(»-l).
= (4 + 9y2H4-9y2)
= (4 + 9y2)(2 + 3y)(2-3y). gg. ^ + 2* + 1 _y*
= (z + 2a: + l)-y2
80. 12«*-22-6 -(s + ip-y*
-(a? + l+y)(*+l-y).
= (3z2 + 2)(422-3)
83. 49(a -6)2- 64 (m-n)*
= {7(a-6) + 8(m-7i)}
- (7a - 76 + 8m - 8n) (W - 76 - 8m + 8n)V
84. 1
it* l/y Ut ^7/» ">f
7 (a - 6) + 8 (m - n)}{7(a - 6) - 8(m - n)}
w' " _8n)(7a-r " " " x
V 2a6 J
-fl I g' + ^-^Vi a' + y-^N
V 2a& A 2a& /
= /2a6 + a2 + 62 - <*\ (2ab - a* - 6' + c*\
\ 2ab )\ 2ab J
_ /(a2 + 2a6 + 62) - c*\ (<* - (a2 - 2a6 + 62)\
V 2a6 )\ 2ab )
_ ((a + 6)« - <A /V -(a - 6)*\
V 2a6 JV 2a6 )
- /(« + 6+c)(o + 6-c)\ / (c + a-6)(c-a + 6)\
V 2a6 JV 2a6 /
86. a^-53a; + 360 87. 2a6-2&c-ae + c6 + 2&2-6«
= (x - 8) (x - 45). - (2a6-26c+262)-(ac-c« i-fte)
= 26(a-c + 6)-e(a-c + &)
86. x*-2x*y + z2-4.z + 8y-4 = (26-e)(a + 6-c).
- (38-2z2y+z2)-(4z-8y+4)
- x^(*-2y+l>-4(*-2y+l) 88. 125a* + 350r»y2 + 245 w4
- (a*-4)(jp-2y+l) = 5x(25x* + 70a*y2 + 49y*)
- (aM-2)(a-2)(»-2y+l). = 5ar(5a^ + 7y2)2.
TEACHERS* EDITION. 65
89. a8 + a5 6 + a*62 + a863 + a264 + aP
= a(ab + a46 + a362 + a2 6s + a&* + 6s)
= a{a3(a2 + ab -k6*) + 6s (a2 + a6 + 6*)}
= a(as + 68)(a2 + a6 + 62)
- a (a + 6)(a2 - ab + 68)(a2 + a6 + 6»).
90. 2a*x-2a3cx + 2acix-2c4'x
= 2a8a?(a — c) + 2c8 a; (a — c)
= {2a*x + 2(?x){a- c)
= 2a;(as + cs)(a-c)
= 2x (a + c) (a2 — ac + <?)(a — c).
91. 6a?-5a#-6y2 + 3a:z+15yz-9z2.
6a2 - bxy - 6y2 - (3a? + 2y)(2a; - 3y),
6a? + 3a;z -9Z2 -(3o?-3*)(2* + 3«),
- 6y2 + 15yz - 9z2 = (2y - 3z)(- 3y + 3z).
3a; + 2y, 3x-3z, 2y-3z;
2x-3y, 2a; + 3z, _3y + 3z.
(3a: + 2y - 3z)(2s - 3y + 3z).
92. 4z2-9zy + 2y2-3a;z-yz - z2.
4ar* - 9a?y + 2y2 - (4a; - y) (a? - 2y),
4a? - 3a?z - z2 = (4a; + z) (a? - z),
2y2-ys-z2 = (-2y-z)(-y + z).
4a; — y, 4a; + z, — y + z; s
x — 2yy x — z, — 2*y — z.
(4a;-y + z)(x-2y-z).
93. 3a* -7a6 + 26s + 500-560 + 2^.
3a2-7a6 + 262 = (3a-6)(a-26),
3a2 + 5ac + 2c2 - (3a + 2c) (a + c),
2^2 _ 5bc + 2ct = (_ 26 + c) (- b + 2e).
3a-b. 3a + 2c, -6 + 2c;
a — 26, a+ c, -26 +c.
(3a-6+2c)(a-26 + c).
94. a^-2a? + a?-8a;+8 95. 5a?-8a^ + 3y2-5a; + 3y
= ** - 2a? + a? - (8a; - 8) = (pat-Sxy + 3y2)-(5a;-3y)
= x*(x2-2x + l)-S{x-l) = (5a;-3y)(a;-y)-(5a;-3y)
= a?(a;-l)2-8(a;-l) = (5a;-3y)(a;-y- 1).
= (ar»^a?-8)(a;-l).
96. a2-2a^H-rf2-462 + 126c-9c«
= (a1 - 2ad+ d*) - (462 - 126c + 9c2)
= (a-a7-(26-3c)2
- {(a - d) + (26 - 3c)}{(a - d) - (26 - 3c)}
= (a-d + 26-3c)(a-d-26 + 3c)
= (a + 26-3c-a*)(a-26 + 3c-d).
97. (a?-a;-6)(a?-a;-20)
-(a;-3)(a;+2)(a;-5)(a:+4).
66 ALGEBRA.
Exercise XLVII.
1.
18a62c2rf=32x2a62c*d,
5. a2-62 = (a + 6)(a-6),
36a26cd2 = 32x22a26cd2.
a8-68 = (a-6)(a2+a6 + 62).
.-. H.C.F. = 18 abed.
.: H.C.F. = a - 6.
2.
Ytvej* = 17 ©a2,
6. a2 — a? = (a + a?) (a — x),
(a — 3;)*= (a — art (a — x).
.*. H.C.F. = a — x.
7. c^ + x8 *=(a + x)(a2— ax + x2)
(a + x^^a + x)8.
/. H.C.F. = a + x.
Zlfq - 17 X 2/fy
51pV - 17 x 3oV.
/. H.C J1. - 17^.
3.
gxV^-^xxVz4,
20x*yV = 22x5xVz2.
8. 9X2 -1 = (3x + l)(3x-l)
(3x + l)2=(3x + l)2.
.-. H.C.F. = 4x2y2z2.
.-. H.C.F. = 3x + l.
4.
30a; V = 2 X 3 X 5xV,
90*V-2 x32x5xV,
9. 7x2-4x = x(7x-4),
120arly* = 2Jx3 x 5*V.
7a2x-4a2=a2(7x-4).
/. H.C.F. = 30arV.
.\H.C.F. = 7x-4.
10. 12a8x2y-4a8xy2 = 4a8xy(3x-y),
30a2ary - 10a?x*if = 10a2xy (3x - y).
.-.H.C.F. = 2
2a2xy(3x — y).
11. 8a362c-12a26c8 = 4a26c(2a6-3c2),
6ab*c + 4a68c2 = 2a63c (36 + 2c).
.-. H.C.F. = 2a6c.
-2z-3 = (x-3)(x + l), 13. 2a8-2a62 = 2a(a + 6)(a-6),
-x-12 = (x-3)(x + 4). 46(a + 6)2 = 46(a + 6)(a + 6).
.-. H.C.F. = x - 3. .'. H.C.F. = 2 (a + 6).
14. 12x8y(x-y)(x-3y) = 22x3x8y(x-y)(x-3y),
18x2(x-y)(3x-y) = 2x32x2(x-y)(3x--y).
/. H.C.F. -^(x-y).
15. 3x8 + 6x2-24x = 3x(a,-2 + 2x-8)
= 3x(x + 4)(x-2),
6a*-96x = 6x(x*-ie)
= 6x(x-4)(x + 4).
.-.H.C.F. = 3x(x + 4).
16. ac (a — 6) (a — c) = ac (a — b) (a — c),
6c (6 - a) (b - c) = 6c (a - 6} (c - 6).
.-. H.C.F. = c (a -6).
17.
10x3y-60x2y2-f5xy8 = 5xy(2x2-12xy + y2),
Sa^y^-Say-lOOy* = 5y*(x2-xy-20y2)
= 5y2(x-5)(x + 4).
.-. H.C.F. = 5y.
TEACHERS EDITION.
67
18. x(x + lf = x(x + l)\
a?(a?-l) = a?(a;-fl)(a;-l),
2a;(a?-a:-2)= 2a;(a;-2)(ar+l).
.-. H.C.F. = a;(a;+l).
19. 3a?-6a: + 3=3(a:-l)2,
6a? + 6a:-12 = 6(a;+2)(a;-l),
12a?-12 = 12(a:-l).
.-. H.C.F. = 3(a;-l).
20.
21. x*-y2 = (x + y)(x-y).
(x + yf = (a; + yft
x*+'dxy + 2y*=:(x+y){x+2y).
.-. H.C.F. = a; + y.
22. ^-j£-(* + y)(*-y).
x* - r = (x-y)(xi+xy+y2) ,
a?-7ay+6y2 = (a?-y)(ar-6y).
.-. H.C.F. = x-y.
6(a-6)* = 6(a-W
8(a8-&2)2 = 8(a + &)2(a-&)!
10(a*-&*)=10(a2+&2)(a+&)(a-&).
.-.H.C.F. = 2 (a -6).
23.
a?-l = (s-l)(a; + l),
a?-l = (a;-l)(a? + a: + l),
a? + a;-2Ji(a;-l)(a:+2).
.-.H.C.F. = a;-l.
Exercise XLVIII.
1.
5a? + 4a;— 1
5a?- x
5x-i
5a; -1
20 a? + 21 a; -5
20a? + 16a;-4
4
5x-l
x + 1
1
.-. H.C.
F. = 5a;-1.
2.
2a?-4a?-13a:-7
2a? + 4a? + 2a;
-8a? -15a; -7
-8a?-16a;-8
6a? -11a? -37a: -20
6a? -12a? -39a; -21
3
a? + 2ar+ 1
a? + x
2a;-8
a;+ 1
a; + l
a;+ 1
x + 1
.-.H.(
3.F.«a>+l.
3.
a)6a*+25a3-21a2+4a
6a3+25a2-21a+4
6 a8— 5a2+ a
2a)24a4+112as-94a2+18a
Reserve a.
12as+ 56a2-47a + 9
12as+ 50a2-42a + 8
2
30a2-22a +4
30a2-25a +5
6a2- 5a + 1
6a2- 2a
a + 5
2a-l
3a -1
- 3a+ 1
- 3a + 1
.-. H.C.F. .
«a(3a-l).
4.
9a? + 9a? -4a; -4
9a? -4a;
9a? -4
9a? -4
45 a? + 54 a? -20 a: -24
45 a? + 45 a? -20 a; -20
5
9a:2 - 4
x + 1
1
.-. H.C.]
F. = 9a?-4.
68
ALGEBRA.
5.
3a?)27a?-3a? + 6a?-3a?
9s4- x* + 2x -1
9s4 + 6s8 + 3s
_6a?- a?-a;-l
-6a?-4a? -2
3a?-a; + l
6.
10) 20 a? -60 a? + 50 a; -20
2a?- 6a? + 5a?- 2
2a?- 6a? + 4a;
a;- 2
*- 2
6a;)162a?+48a?-18a?+6g
27a? + 8a?-3a; +1
27a?-3a? + 6a?-3a;
3a? + 2a? +1
3a?- a? + a;
3a?-a; + l
3a?-a? + l
Reserve 3 x.
Sx
3a;-2
x + l
,H.C.F. = 3a;(3a?-a; + l).
4s)32a?-92a?+68a?-24a;
8a?-23a?+17a?- 6
8a?-24a?+20a;- 8
a?- 3a; + 2
a?- 2a;
- x + 2
.-. H.C.F.
Reserve 2.
4
2x
x-1
= 2(a;-2).
4a?- 8a; -5
4a? -10a?
2a; -5
2a;-5
12a?- 4a;- 65
12a? -24a; -15
10)20a?-50
2a;- 5
2a; + 1
H.C.F. = 2a;-5.
8.
q)3g8-5a*a;-2aa?
3a2-5oa; - 2a?
3a*-6aa;
ax — 2a?
ax - 2a?
g)9as - 8a» a; -20 oa?
9a*- 8ax-20a?
9a*-15aa;- 6a?
7a;)7aa?-14a?
a - 2x
Reserve a.
3
3a + a?
, H.C.F. = a(a-2a?>
10a? + a?-
10a? -5a;2
9a; + 24
+ 15
3)6a?-9a; + 9
2a?-3a; + 3
20a? - 17a? + 48a; -3
20 a? + 2a?- 18a? + 48 a;
-2a? +
-2a? +
a? -3
3a?-3a;
-2a? + 3a;-3
-2a? + 3a;-3
2x
-5
— x ■
H.C.F. -2a? -3a; + 3.
TEACHERS EDITION.
69
10.
2)8a?-4a? -32s -182
4a? -2a? -16a;- 91
4a? -2a? -42a;
13) 26 s- 91
2a;- 7
3)36a?-
-84a?-llla;-126
12a?-
12a?-
-28a?- 37a;- 42
- 6a?- 48a;-273
-11)-
-22a? + liar + 231
2a?- x- 21
2a?- 7a;
6a;- 21
6a;- 21
2x
x + 3
H.C.F.-2a;-7.
11.
5a?(12a? + 4a? + 17a;-3)
12a? + 4a? + 17a; -3
12a? + 4a?- x
3) 18a; -3
6a;-l
.H.C.F. = 5a;(6a;-l).
10a;(24a?-52a? + 14a;-l)
24a?-52a? + 14a;-l
24a? + 8a? + 34a;-6
-5)-60a?-20a; + 5
12a? + 4a;-l
12a;2- 2x
6a;-l
6a;-l
Reserve 5x.
2
x
2x + 1
12.
2y) 18a?y-18a;V-2ayy3-8,v*
9a?-9a?y-xy2~ 4^
9a? -xy2-20yi
y)-9a?y + 16y3
-9 a? +16 y2
-9a?+12a?y
-12ay+16y2
-12ajy+16y2
xy)9x*y — a?^8 — 20ayy*
9a?- sy2-20y8
9a?-16ary2
5y*)15xy*-20yi
3a; - 4y
Reserve y.
-3a;-4y
•.H.C.F. = y(3a;-4y)
13.
6a?_ a;-15
03?-10a;
9a; -15
9a; -15
9a?-
2
-3a;-
20
18a?-
18a;2-
-6a;-
-3a;-
40
45
"
-3a; +
5
■2a; -3
H.C.F = 3a;-5.
70
ALGEBRA.
14.
12a?-
2
- 9a? +
5a; + 2
24 a?- 18a? + 10a? + 4
24a? + 10a? + x
-28a? +
-6
9a; + 4
168 a?-
168 a? +
54 a: -24
70a? + 7
-31)-
124 a; -31
15.
3)6a?
+ 15a?-
4a; + 1
■6a? + 9
2a?
11
+ 5a?-
-2a? + 3
22a? + 55 a?
22 a? + 56 a?
- 22a? + 33
-30a;
-a?
-a?-
+ 8a? + 33
- 3a?
11a; + 33
11a; + 33
16.
4a?— a?y — xy1 — 5y*
4a? + 4a?y + 4a?y2
— 5a?y — 5a?y2 — 5y*
— 5a?y — 5a?y2 — 5y*
17.
2a8
2
-2a2-
3a-
2
4a3
4a3
-4a2-
+ 5a2-
6a-
26 a
-4
-9a2 +
4
20a-
-4 -
-36 a2 h
-36 a2-
-80a- 16
-45 a + 234
125)
125 a
-250
24a? + 10a; + 1
24a? + 6a;
4a? + 1
4a; + 1
a?+7
6a? + l
.-. H.C.F. = 4a? + l
3)9a? + 6a?-51a? + 36
3a? + 2a? -17a; + 12
2
6a? + 4a?- 34a; + 24
6a? + 15a?- 6a; + 9
- 11a?- 28a; + 15
- 11a? + 88 a; + 363
-116)-116a?-348
x+ 3
Reserve 3.
-2x
11
-a? + ll
•. H.C.F. = 3(x + 3).
7.t? + 4a?y +
4
4a?y2-
- 3y»
28a? + 16a?2y +
28a?- 7a?y-
16 xy2 -
7a?y2-
-35y«
23y)23a?y +
23a?y»-
f23y3
a? +
xy
+ f
3a3-
- a2
-2a-
■16
2
6a3-
-2a2
-4a-
-32
6a3-
-6a2
-9a-
- 6
4a2
+ 5a-
-26
4 a2
-8a
13a-
-26
13a-
-26
4a?~ 5y
.-. H. C.F. = a? + a?y + y*.
a
-9
4a + 13
.-. H.C.F. = a-2.
TEACHER8 EDITION.
71
18.
2)12ys + 2_y2-94y-60
2)48y3-
-24y2-
-348.y+ 30
Reserve 2.
6y3+ y2_47y_30
8
24y3-12y2-
24y3 + 4y2-
-174y+ 15
-188y-120
4
48y* + 8y2-376y-240
48y»-42y2-405y
-16y2 + 14y + 135
-16y2- 40y
-3y-25
50y2 + 29y-240
54y + 135
54y + 135
-8y + 27
400y2 + 232y-1920
* 400y*-350y-3375
291)582y + 1455
2y + 5
.-. H.C.F. = 2(2y + 5).
19.
9a;(2a?-6a?-a? + 15a;-10)
2a?-6a?-a? + 15a;-10
_9
18a?-54a?- 9a?+135a;-90
18a?- 2a?-45a?+ 5a;
2)-52a?+36a?+130a;-90
-26a?+18a?+ 65a;-45
9
-234a?+162 a?+585 a;-405
-234a?+ 26 a?+585 x- 65
68) 136 a;2- 340
2 a?-
20.
15a? + 2a?- 75a? + 5a; + 2
15a? - 75a? + 15a;
6 a? (4 a?+6 a?-4 a?-15 x-15)
4a? + 6a?-4a?-15a;-15
4a? - 12a?- 2a? + 30 a;- 20
18a?-
18a?
-2a?-45a; +
-45 a;
-2a;2
-2a?
Reserve 3a;.
2
a?-13
9aj-l
H.C.F. = 3a;(2a?-5)
2a?
2a?
-lOaj + 2
-10a; + 2
.-. H.C.F. = a?-5a; + l.
21.
21a?-32a?-54a;-7
5
105 a? -160 a? -270a; -35
105a? + 99a? + 12a;
35a?+ a?-175a?+30a;+ 1
3
105a?+ 3a?-525a?+90a; + 3
105a?+14a?-525a?+35a; +14
-ll)-lla?
+55a;-ll
- 5a; + 1
7
15a; + 2
-259 a?-
5
282a; -35
-1295 a? -1410a; -175
-1295 a? -1221a; -148
-27)- 189a;- 27
7a; +
H.C.F. = 7a;+l.
21a?-
21a?-
- 4a?-
-32a?-
15a? - 2x
54a? -7*
28 a? + 39a? + 5a;
3
84 a?
84 a?
+ 117a?
-128 a?
+ 15 x
-216a;-
28
7) 245 a;2
+ 231 x + 28
35 a?
35 a?
+ 33a; +
+ 5a;
4
28 a; +
28 a; +
4
4
3a;-37
5a: + 4
72
ALGEBRA.
y)9a?y-22a?y3-3a;y4+10y&
9a;4 -22a?y2-3xy3+10y4
J
18a?-44a?y2- 6ay»+2(y
18a?- 9a?y2+105a?y8-69a?y
y)69a?y-35a?y2-l lla^^Oy4
69a?
2
-35a?y -llla#2+20y3
138a?- 70a?y-222ay2+ 40^
138a?-529a?y- 693^+805^
153y) 459a?y-153a3/2-765y»
3a? - ay — 5y2
23.
4a? + 2a?-18a? + 3a;-5
4a?-8a? + 2a?-2a;
10a? - 20 a? + 5a; -5
10a? - 20 a? + 5a; -5
xy) 9xby-Qxiy2+xiya—25x^
9a?-0a?y+a?y2 -25y4
9j4 -22afy2-3jy3+10y4
-y)- ear'y +23a?y2+3sy3-35y4
6a? -23a?y -3ay»+35y"
63^ - 2a?y -10a?y*
-7y)-21a?y +7ay2+35y8
3a? — xy — by*
3a? — xy — 5y2
Res.y.
1
3x
23
2ar
24.
3a?-7oa? + 3a2a;-2as
2
6a?-
6a?-
14aa? +
9oa?-
6a2a;-
6a2a;
-4a3
-a)-
-5aa? +
12a2a;-
-4a3
5a?-
2
12az +4 a2
10a?-
10a; -
24 ax +
\bax~
8 a2
10 a2
-9^
-9aa; +
18 a2
x- 2a
H.C. F. = y (3x8 - xy - by*).
6a?-4a;4-lla?- 3a?-
2
3a;-l
12a?-8a?-22a?- 6a;2- 6a;-2
12 a?+6 a?-54a?+ 9a?-15a;
-14a?+32a?-15a?+ 9a;-2
2
-28a?+64a?- 30a?+18a;- 4
-28a?-14a?+126a?-21a;+35
39)78a?-156a?+39a;-39
2a?- 4a?+ x- 1
3a?
-7
2a; + 5
H.C.F. = 2a?-4a?+a;-l
a?— ax3— o¥- a8a?— 2 a4
3
3 a?-3 aa?-3 oV-3 a9x-6 a*
3a?-7aa?+3a2a?-2a8a:
4oa?-6a2a?- a8a?-6a4
3
12aa?-18a2a?- 3a8a;-18a4
12aa?-28a2a?+12a3a;- 8a4
5a2)10a2a?-15a3a;-10a4
2a?- 3aa;- 2a2
2a?- 4a.x
ax— 2a2
ax— 2a*
4a
3a;
5
2aj + a
, H.C.F. = a;-2a
TEACHERS EDITION.
73
Exercise XLIX.
1. 2x*+x-l = (x+l)(2x-l),
x*+5x+4 = (x+l)(x+4),
a^+l = (a;+l)(ir2-a:+l).
.% H.C.F.= x+1.
2. y8-y2-y+i = ;/i(y-i)-(y-l)
= (y2-i)(y~i),
3y2-2y-l=(y-l)(3y+l),
y'-y'+y-i « y*(y-i)+(y-i)
= (y2+i)(y-i).
.-. H.C.F.- y-1.
3. 3s -4a? + 9* -10
x*-2x* + 5x
-2x2 + 4x-10
-2x2 + 4x-10
x*-2x + 5
. H.C.F.= x*-2x + 5.
Xs + 2a;2- 3a; + 20
x*-4x2 + 9a; -10
6)6art-12x + 30
a?- 2x + 5
a^ + Sx2- 9x + 35
x*-2x* + 5x
7x2-14x + 35
7x2-14x + 35
x-2
x + 7
4. Xs- 7x*-f 16x-12
_7
7xs-49xa + 112x-84
7x8-32x2 + 36x
-17x»+ 76x-84
7
-119x2 + 532x-588
-119x* + 544x-612
-12)- 12x-f 24
x- 2
.%H.C.F.= x-2.
5. ys_5y* + lly-15
y3- y2+ 3y + 5
_4W4t/2 + "
-*)-4y2
+ 8y-20
2y+ 5
3x»-
3x»-
-14x« + 16x
~21x2 + 48x-36
5x»-
5x»-
7x2-32x + 36
7x2-14x
-18x + 36
-18x + 36
10x* + 7x-14
-lOx2
7x-14
7x-14
-2y + 5
, H.C.F.-y2-2y + 5.
y8- y* + 3y + 5
y»-2jy2-f 5y
y2 - 2y + 5
y'-2y + 5
2y»_7y2 + 16y
2y8-4y2 + 10y
-15
-3y2 +
- 3y* +
6y-15
6y-15
3
x — 17
7x-18
6x* + 7
1
y + l
2y-3
74
ALGEBRA.
6. 2x2+3_-5 = (2x + 5)(x-l).
3x2-x-2 = (3x + 2)(x-l),
2x2+x-3 = (2x + 3)(x-l).
.-. H.C.F. = x~l.
7. x»-l
X8 + X2 + X
-a^-x-l
-x*-x-l
7? + X + 1
x*-x*-x-2
x9 -1
_aJ._a._l
2x*- .ra- x-
2x*+2x2 + 2x
-3
-3^-32!-
-Sa^-Sx-
-3
-3
3 + 1
2x-3
H.C.F.-^ + x-fl
8. x»-3x -2
_2x2-4x-2
_2x2-4x-2
x2 + 2x + l
x2-t- x
x + 1
x + 1
2xs + 3x2
2x*
-1
6x-4
3)3x* + 6x + 3
x2 + 2x + l
x« + l
x5 + 2x2 +
-2a,-2- x + 1
-2a?-4a?-2
3)3x + 3
x+1
x-2
x-2
x + 1
H.C.F. = x + l
9. 12(*-y*)-12fa* + jfl(a*-y")
= 12(xa + ya)(x + y)(x-y);
10(a» - y«) _ 10(x» + y3) (x8 - y3)
: 10(x + y)(xa - xy + y2)(x - y)(x* + xy + y2)
8(x*y + xy4) =
SxyfaJ + y8)
(* + y)(a
\ H.C.F.
8 xy (x
xy + y2).
2(x + y).
10. a^ + xy^x^ + y3)
= x(x + y)(x* — xy + y2)
x»y + y* =
a,4 + x*y* + y* =
y(x + y)(x*-xy +
(x4 + 2x2y2 + y*)--x2y2
(aJ + yX^-xv + y2);
ar'+y^-xy
x2 + xy + y2) (x2 - xy + y2).
.-. H.C.F. - x*-xy+y\
teachers' edition. 75
11. 2(x2y-xy2) = 2xy(x-y)f
A)ty ~ *$ = ^i* + y )(* - y),
i ly - «yj) = 4zy (s - y)(x* + ^ + ^
5(afy ~ sy5) = 5xy (x + y)(x - y) (x2 + y«).
.-. H.C.F. = xy(x-y).
Exercise L.
i. 4asx=22xa8xa:, 10. x2- 1 = (x + l)(x- 1)
6a2x2 = 3X2xa2x*2, *-*-*(* + 1)
2ax2 = 2XaXs2. *■- l-(»-l)(a> + * + 1).
.-. L.C.M. = 12 a8 x2. .-. L.C.M. = a?(a? + l)(x»-l)
2. 18ax2 = 32x2xaxa:2, n 2a +1-2/14-1
W-8»X?X.xy, 4a.-l-(2a++i)(2a-l),
12^ = 3 x 9 X x X y. 8a» + j . (V2a+lX&'-2a+l).
••• LC.M. = 72<w»y«. ... L.C.M. = (8a»+l)(2a-l).
..L.C.M.- **(« + *). .-.L.C.M. = (« + &)" (a -6).
.•.L.aM.-.(,+i)(—i,. Sii-StSii+Sa-.).
5. a2-&' = (a + &)(a-&), .'. L.C.M. = 4(1 + x)(l -x>
a2 + a& = a(a + 6). ^ ^_1
.•.L.C.M.-a(a+6)(a^). x2 + x +*1 = a-2 + s + 1,
6. 2x-l = 2x-l, a3- 1 = (s- l^x2 + x + 1).
4s2 - 1 = (2s +'l)(2x - 1). .\ L. C. M. = x8 - 1.
.•.L.C.M. = (2x4-l)(2x-l). 15, *>_y. = (iC + y)(iC__y),
7. a-f-6 = a4-6, (* + y)' - (* + y)a.
a8+J3 = (a + 6)(a2- a& + 62). (* - tf* " (* ~ V) •
.-. L.C.M. = (a+6)(a2-a&+&»). •*• L- C- M. - (* + y)2(* - y)2.
8. x2 - 1 = (X + l)(x - 1), 16. * \- y* « (* + y)(« - y),
rf + i-Wi, * 3(*-y)8 = 3(x-v)2,
x* - 1 = (x2 + l)(x + 1) (* - 1). 12(x*4yj= ^(x+yX^-xy+y2).
.'. L.C.M. = (x2+l)(x4-l)(x-l). •'• L.C.M. = 12(x»+y')(x-y)2
9. x2-x = x(x-l), 17. er^ + xyJ^Sxfx + y),
^-l-(*-l)(a- + af + l)f 8(xy-y2) = 8v(a-y),
xs+l = (x4-l)(x2-x+l). 10(x2-y2)=l0(x + y)(o;-y).
.%L.aMf = x(x8 + l)(r,-^J). .-. L.C.M. = 120 xy(x + y)(x-y)
76 ALGEBRA.
18. x» + 5x + 6 = (* + 3)(x + 2), 20. x' + llx + 30 = (x + 6Kx+5),
x* + 6x + 8 = (x + 2)(x + 4). x»-f 12x+35 = (x+5)(x+ 7).
.-. L.C.M. = (x + 2)(x + 3)(x + 4). .-. L.C.M. = (x+5)(x+6)(x+7).
19,
>. a*-a-20=-(a-5)(a+4), 21. x»-9x-22 = (x+2)(x-ll),
a* + a - 12 = (a + 4)(a - 3). x* - 13* + 22 = (a- 2) (x-11).
.% L.C.M. = (a-3)(a+4)(a-5). .% L.C.M.-(x + 2)(x-2) (x-11).
22. 4a&(a2-3a&+2&*)=4a&(a-26)(a-6),
5a* (a2 + a& - 65*) - 5a* (a + 3i)(a - 25).
.-. L. C. M. - 20a*6 (a - b) (a - 2&) (a + 36).
23. 20(xa-l) = 20(x + lXx-l),
24x«-x-2) = 24(x-2)(x + l), 26 (a„&)(a_c)== (<*_&)(«- c),
(6-a)(M = -(a-5)(6-c),
.(c-a)(c-&) = (a-c)(6-c).
L.C.M. = (a-b)(a-c)(b-cy
XS~ X i rfS> i^X+ 1 ' 26- (a~5)(a-c)=(a-6)(a-c,
16(x*+x-2) - 16(x+2)fx-l). ?*-«) (i-c) - -\a-b)Q>-c),
.-. L.C.M. = 240 (x+1) (x-l)(s+2) (x-2). (c - a) (c - 6) = (a - c) (6 - c).
24. 12xy(xs-y*)=12xy(x+y)(x-y),
2x*(x+y)*=2x»(x+y)(x+y),
3y'(x-y)'=3y*(x-y)(x-y). ^ rB»_4^+3x = x(x»-4x+3)
.-. L. C. M. - 12xy(x-y)*(x+y)*. = x(x-3)(x-l),
x*+x»-12x*= x*(x*+x-12)
25. (a - b) (b - c) - (a - 6) (6 -c), - x*(x+4) (x-3),
(6_c)(c-a)=-(a-c)(5-c)f x5+3x*-4xs=x3(x«+3x-4)
v(c_a)(a_&) (a-i)(a-c). = x»(x-l)(x+4).
.-. L.C.M. = (a-&)(6-c)(c-a). .-. L.C.M. = xs(x-l) (x-3) (x+4).
xty-xy* = xy(x~y\
3x(x-y)* = 3x(x-y)*t
4y(x--y)s = 4y(x-y)8.
.-. L.C.M. = 12xy(x-y)8.
29.
,. (a + &)« _ (c + d)2 = (a + 6 + c + d)(a + b - c - a*),
(a + c)» _ (6 + df = (a + b + c + d)(a-b + c-d\
(a + df- (6 + c)* = (a + i + c + d ) (a - & - c + d).
.-. L.C.M. = (a+b+c+d)(a+b-c-d)(a-b+c-d)(a-b-c+d).
(2 x - 4) (3 x - 6) = 2 (x - 2) X 3 (x - 2),
(x-3)(4x-8) = (x-3)x4(x-2),
(2x- 6)(5x- 10) - 2(x- 3) X 5(x- 2).
.-. L. C. M. = 60(x - 2)* (x - 3).
TEACHERS EDITION.
77
Exercise LI.
1. 6x*-x-2 = (3x-2)(2x + l),
21 x* - 17* + 2 = (3* - 2) (7x - 1),
14x* + 5x - 1 = (2x + l)(7x - 1).
.-. L.C.M. = (3x-2)(2s + l)(7x-:i)
2. ^-l-(* + l)(s-l),
x2 + 2 x - 3 - (x + 3) (x - 1),
6x2-x-2 = (3x-2)(2x + l).
.-. L.C.M. - (2x + l)(3x -2)(x- l)(x + l)(x + 3).
3. xs-27 = (x-3)(x* + 3x + 9),
x* - 15x + 36 - (x - 3)(x - 12),
a? _ 3x2 _ 2x + 6 = x2 (x- 3)-2(x - 3)
= (x2-2)(*-3).
.-. L.C.M. = (x - 3) (x - 12) (x» - 2)(x* + 3x + 9).
4. 5x2 + 19x-4 = (5x-l)(x + 4),
lOx2 + 13x - 3 = (5x - l)(2x + 3).
.-. L.C.M. = (5x-i)(x + 4)(2x + 3).
5. 12x* + xy-6y2 = (4x + 3y)(3x-2y),
18x2+ 18xy-20y* = 2(3x-2y)(3x+5y).
.-. L. C. M. = 2 (3 x -7 2y) (3x + by) (4x + 3y).
6. x)x*-2x* + x
x3-2x2 + l
x8-2x -1
~2)-2x2-f2x +2
X2— X — 1
2)2x4-2x8-2x -2
X4— Xs— X — 1
x4-2x8+ x
x8-2x -1
Xs— X*— X
X2- X -1
X* — X — 1
1
x + 1
Hence,
and
x4 - 2x» + x = (xa — x - 1) X x(x — 1),
2x4-2x3-2x-2 = (x2-x-l)x2(xa+l).
.-. L.C.M. = 2x(x*-x-l)(xHl)(x-l).
7. 12x2 + 2x- 4 = (6x + 4)(2x-l) = 2(3x + 2)(2x-l),
12x2 - 42x - 24 = (6x + 3)(2x - 8) = 6(2x + 1) (x - 4),
12x* - 28x - 24 = (6x + 4)(2x - 6) = 4(3x + 2) (x - 3).
.-. L.C. M. - 12(3x + 2)(2x - l)(2x + l)(x - 4)(x - 3).
78
ALGEBRA.
8. a* - 6a;2 + 11a? -6
a? — 5a? + 6x
- a?+ 5a;-
— ar2 + 5a;-
-6
-6
X*
a?2
-5a? + 6
-3a;
-2a; + 6
-2a; + 6
x3
X3
-9a* + 26a?-
-6a? + lla?-
-24
- 6
-3)
a*-
X3-
-3a* -f 15a;-
-18
a*-
-8a* +
-5a? +
- 5a? + 6
19a; -12
6a;
3a* +
3a* +
13a?-
15a?-
12
18
-21-
2a? +
6
X —
3
a;-l
a?-3
a;-2
Hence, x*- 6a* + 11a;- 6 = (a;- 1) (x -2) (a? -3),
a? - 9a? -f 26a; - 24 = (a; - 2)(x - 3)(a? - 4),
a*-8a? + 19a?-12 = (a;-l)(x-3)(x-4).
.-. L.C.M.=(a;-l)(a;-2)(a;-3)(a;-4).
9. x*-4a*-(x + 2a)(x-2a\
x9 + 2ax* + 4a2 a; + 8 a8 = x*(x + 2a) + 4a* (a; + 2a)
= (a? + 4a2)(a; + 2a)l
a? - 2oa,J + 4a2 a; - 8a5 = x*(x - 2a) +4a2 (a; - 2a)
- (a? + 4a2) (a; -2a).
.-. L.C.M.«(a; + 2a)(a>-2a)(a* + 4aI).
10. ar, + 2a?y-a^2-2y8
x*-2x2y-xy* + 2y*
= a?(a; + 2y)-y2(a;-f 2y)
= (^-y2)(a; + 2y),
= a*(a;-2y)-y2(a?-2y)
-(*-t/)(*-2y).
.-. L.C.M. = (a*-y*)(a? + 2y)(*-2y).
11. l+p+p*=*l+p+p*, 12. 1-
1 -p + p2 = l-7> + 7>2,
l+^+p' = (1 +p+p>) ( l-p+p*).
.•.L.C.M. = l+jpI+/
(l-ar-tt-aVl-a),
(l-a)8=(l-a)(l-a)(l-a).
.-. L.C.M. =(1- a)8.
13. (a + c)J
(a + c)2 - b1 = (a 4- 6 + c)(a - b + c),
(a + i}2 - c2 = (a + 6 + c)(a + 6 - c),
(6 + cf - a2«= (a + b -f c)(-a + 6 + c).
.-. L.C.M. = (a + & + c)(a + &-c)(a-6 + c)(-a+6 + c)
TEACHERS EDITION.
79
14. cH<*-ch/-Sci/*
4c* -cy -3y*
3cs-3c2y + cy2-ys
9
36c2- 9ey-27y*
12c3-12c*y+ 4cy2-4y8
12c8- 3c2:y- 9cy2
3c
36c2-52cy + 16y*
-V)~ 9«2.V + 13cy2 - 4y*
4
43_y)43cy-43.y2
c — y
9c2 -13cy + 4y*
9c* - 9cy
9c-4y
- 4cy + 4y*
— 4cy + 4y*
Hence 4c8 — c*y— 3cy* = (c — y)(4c* + 3cy),
3 c8- 3c*y + cy*-tf = (c- y^c2 + y2).
.-. L.C.M. = c(c-y)(4c + 3y)(3c* + y2).
15. 7n8-8m + 3 = (m+3)(m2-3m+l),
m6 + 3m6 + m + 3 = m5 (m + 3) + (m + 3)
= (m* + l)(m + 3).
.-. L. C. M. - (m + 3)(ro* - 3m + l)(m* + 1).
16. 20n4 + n* - 1 = (5n2 - l)(4n* + 1),
25n* + 5n8 - n - 1 = (5n* - l)(5n2 + n + 1)
.-. L.C.M. = (5n2- l)(4n* + l)(5n* + n + 1).
17. 46s-
7
126* + 96-1
286s- 846*+ 636-7
286s- 1606* + 1326
766*-
7
696-7
6*-268 + 6*-86 + 8
4
4M_ 86a+ 45»_
46*-1263+ 96*-
326+32
6
468-,56*-316+32
468-1262+ 96- 1
532 62- 4836- 49
5326* -30406 + 2508,
76*-406+33
76*- 76
2557)25576-2557j
6- 1|
Hence, 46s -126* + 96
6*
-336+33
-336+33
6 + 1
46 + 76
76-33
l-(6-l)(46*-86 + l),
268 + 62 - 8 6 + 8 = (6 - l)(b* - 6* - 8).
18.
2r)2r5-8rA+12r8-8r2+2r
r*-47*+ 6r*-4r +1
L.C.M. = (6-l)(462-86 + l)(68-6*-8).
- 2r*
+1
-4r)-4r3+ 8r*-4r
r2- 2r +1
3
r)3r*-
-6r» +
3r
r4-
r4-
-27^ + 1
-27*+ ra
27*-
27*-
37^ + 1
4r* + 2r
r2-2r
r*-2r
+1
+1
Reserve r.
1
r* + 2r + l
Hence, 2r>-87** + 127* -8r* + 2r= 2r(r-l)*,
3r*-6r8 + 3r = 3r(r*-l)2.
.-. L.C.M.
= «r(r-l)*(r+l)2.
80
ALGEBRA.
Exercise LII.
1.
a?»-l
4a;(a? + 1)
4a;(a?+l)
_a?-l
4a?
a?2-9a? + 20
a?2-7a?+12
(a; -3) (a? -4)
%-3*
3.
a?-2x-3
x1- 10a- + 21
_(*-3)(* + l)
(*_7)(*-3)
= aM-J.
a; -7"
a?4 + a;2 + 1
x2 + .T +1
_(x2 + x + l)(x2-x + l)
a^ + x + l
-aj»-a: + l.
6.
a» + l
as + 2a2 + 2a + l
_(g + l)(aa-a + l)
(a + l)(a2 + a + l)
a* — a + 1
a* + a + 1
5 x* + 2x*y* + if
x*-tf
(x8 + ^)(x» + .v»)
(^ + ys)(^-y»)
^*» + .y»
x'-y*
a»-.a_20
a2 + a - 12
_(a-5)(a + 4)
(a_3)(a+4)
a — 5
~a-3*
8. x*-4x* + 9a;- 10
x* + 2x*- 3a? + 20
-6)-fijg + 12y-30
x2 - 2a? + 5
a?8
a?
+ 2x»-
-2a?1
-3* + 20
+ 5a?
4a?*-
4a?2-
-8a? + 20
-8a? + 20
H.C.F. = arl-2a; + 5.
y»-4a?»+9a;-10 _ (x2 - 2x + 5)(x~2) _ x- 2
xs + 2x2-3x + 20 (x2-2x + 5)(x + 4) x + 4*
TEACHERS EDITION.
81
9.
a*-5o? + lla;-15
x*-2x2 + 5x
-3a;2 + 6a; -15
-3a* + 6a; -15
x3- x*+ 3x+ 5
^-5a;2 + lla;-15
1
4)4 a1- 8* + 20
a;2- 2a; + 5
a;-3
.-. H.C.F.= x*-2x + 5.
z3_5a? + lla;-
a^ — ar, + 3o? + «
15_(a?-3)(xa-2a; +
3 (x + l)(x*-2x +
*) = •
5) .
c-3.
r + l'
10.
X4-
X*
-tfy-xyt-y*
+ afy*
- x*y — x2y2 — xy* — y4
-x*y -xy*
-*Y -y4
— x^y1 — y4
x* + x*y + xy* — y4
x* — x*y — a^y8 — y4
1
2xy)2x*y + 2xyi
X2
-r
.-. H.C.F.= a* + y2.
. a?4 + x*y + ary8 — y4 _
X* — afy _ ay8 — y4
(x2 + y2) (x* + a?y - y2) .
(«■ + y2) (*■ - *v - y%)
_x* + xy — y2
«* - ^y - y*
11.
o8-3a + 2
4
a* + 4a2— 5
a8- 3a + 2
4a2 + 3a - 7
-3
i
4a3 -12a + 8
4a8 + 3a2- 7a
-3a2- 5a + 8
-3a2 + 3a
- 8a + 8
- 8a + 8
a
-12a2- 9a +21
-12a2 -20a +32
4
ll)lla -11
a - 1
-3a
.\H.C.F. = a-l.
-8
. a3 + 4a2-5_(a
-l)(a2+5a + 5)__a2
+ 5a
+ 5
"a8 -3a + 2 (a
&-l)(a2 + a-2) a
■ + a
-2
12.
x* + x* — x — 1
3
3a;2 + 2a;-l
3o?-6a;-9
2C
3
3x* + 3x*-$x-'S
3a? + 2a;2- x
x*-2x-S
X2 + X
8)8a; + 8
a? + l
ar-3
-3a;-3
-3a?-3
.\ H.C.F.= o; +
1.
. 3a;2 + 2x-l
_(* + l)(3a>-l)_3
r-1
a^ +««-*_ 1 (i + l)(^_l) a^-1
82
ALGEBRA.
13. a8- a2 -2a + 2
a8-3a2 + 2a
2a*-4a + 2
2a?-6a + 4
2)2a-2
a-1
.-. H.C.F.= a-l.
'. a8-3a* + 4a-2
a8-a2-2a + 2 "
a3-3a2 + 4a-2
a3- a?-2a + 2
-2)-2a2 + 6a-4
a2-3a + 2
a2— a .
-2a? + 2
-2a + 2
x-2
(a - Vjjx2 - 2x + 2) _ a* - 2x + 2
(a _!)(**_ 2) a* -2
14.
4a*-12aa + 9a*
8a8-
(2 a
27 a8
3a)(2a?-3a)
16.
(2a-3a)(4a* + 6aa+.9a2)
, 2a-3a
"4a2 + 6aa + 9a2*
a*-bi-2bc-ci
aa + 2ai + &2-c*
_ q«-(6' + 25c+c*)
(a2 + 2aZ> + i2)-c2
_ (a + & + c)(a— 5 — c)
(a + i + c)(a + 6 -c)
a — ft — c
a + 6 — c
15.
15a2 + a&-2&»
9a* + 3a&-2i2
(5a + 26)(3a-^
(3a + 26)(3a-W
5a + 2&
3a + 26
17.
x*-x2-2x + 2
2x*-x-\
(g-l)(z8 + j8-2)
= (a-l)(2a8 + 2a + l)
x* + x*-2
= 2a*-f 2a + l
18.
ar"-2x2- a + 2
a2-3a* + 2a
-3a + 2
-3a + 2
ar,-6x* + lla;-6
a8 -2a*- x + 2
-4)-4a*-f 12a-8
x1- 3a + 2
H.C.F. = a*-3a + 2.
x9 - 6sa + 11a - 6 _ (a2 - 3s + 2)(x - 3) ,
a8 - 2** - x + 2 (a2 - 3* + 2){x + 1) "
x + 1
a-3
a + 1
19. 6x8-17a8 + lla;-2
6a?8— 5a2 + a;
-12a2* 10a
- 12a2 + 10a;
6x* - 23a;8 + 16a; -3
6x*-17x* + 11a; -2
-1)- 6x2+ 5a; -1
a-2
6a2- 5a + l
Qx3 - 23 a* + 16a - 3 _ (6a2 - 5a + l)(a - 3) a-3
' Sa8-17aJ2 + lla-2 (6a2- 5a+ l)(a-2)=Sa-2
teachers' edition.
83
20.
-x* — x + l
a*-2x*-x2-2x + l
s»(g-l)-(s-l)
W-D(x-l)
(^ + * + l)(as"-3a + l)
_(g-l)(a' + a? + l)(g-l)
=3(a>2-3a; + l)(:B2 + a; + l)
a*-3x + l
21. tt)q4-q8ft-a2ft2 + a&3
a*-a*b-ab* + ft3
a5-a4ft
a5-a4ft
- aft* + ft*
- a8ft2 + a26»
a* + 6*
a8ft2-a2ft8-aft* + ft*
a*b2-a2b*-ab* + &
. H.C.F. = a3-a2ft-aft2 + ft3.
a*ft __ a»* + fts (fli + y) (as __ a*h _ ah* + y ^ aa + 52
' a4_ aSJ _ a2&3 + aft3 a(aS _ a2b _ ab2 + JS)
22.
(a + &)2
24.
a2 -aft -2ft2
_ (a + b)(a + b)
(a-2ft)(a + ft)
= a + b
a -2b
3 aft (a2 -ft2)
4(a2ft-aft2)2
3aft(a4-ft)(a-&)
4a2ft2(a-ft)(a-ft)
3(a+ft)
= 4aft(a-ft)'
a2 + 2aft + ft2-c2
a2 + ab — ac
(a2 + 2aft + 62)-c2
a2 -f aft — ac
(a + b + c){a + b-c)
a(a + b — c)
^a + b + c
a
25.
26.
27.
<$x*-\\x2y + Zxy2
§x2y-bxy2-§tf
xjGx^-Uxy + Sy2)
yijM-bxy-Qy2)
= ar(2ar-3y)(3g-y)
y(2a-3y)(3a? + 2y)
a?(3a; — y)
y(3s+2y)'
o« - (ft + c + J)»
(a_ft)a_(c + d)2
(a + ft + c + a')(a — ft — <
^)
(a-5 + c + d)(a-
a — 6 -f c + d
6a?-5s-6
8z2-2a?-15
_(3a? + 2)(2g-3)
(4» + 6)(2s-3)
3s + 2
^40; + 5
-b-c-d)
84
ALGEBRA.
28. s* + x V + .V4
' (x-y){?*-y*)
_(a? + xy + y*)(x* - xy + i/*)
{x-y){x-y)(x* + xy +y*)
_ g* — xy + y*
(*-yy '
x6 + y*
x* — afy2 + y*
_(s2 + .yW-a?y2 + y*)
a?* — afy* + y4
= a*+y2.
' (a8 - #») (a2 - afc + &*)
_ (a+&)(q2-a&+&a)(q»+a&+&2)
q + ft
Exercise LIII.
, 3?-2a; + l m t
1. — =*a — l.
a — 1
2. 3a? + 2a?+ l|« + 4
3a2 + 12a;
3z-10
-\Ox+ 1
-10a; -40
,.^il=Lio+iL
a; + 4 <r+4
3. 3a? + 6a; + 5
3a^ + 12a;
a; + 4
3a;-6
-6a; + 5
-6a?-24
+ 29
3a;2 + 6a; + 5 ~ . 29
= da; — bH --
a; + 4 x+4
5. 2a? +5
2a?-6a;
5-3
2a; + 6
6x + 5
6a; -18
+ 23
2* + 5 >2a; + 6+-23
a;-3
a;-3
6. 10a2-17aa; + 10a?|5q-3;
10a2- 2ax |2a-3a;
-15aa; + 10a?
— 15oa; + 3a?
+ 7a?
lo^-nos+io*2 2a_Sx+JL*_.
5a — x 5a— x
4. a2
aa; + a?
a2 + ax
-2x
-2aa;+ a;2
-2oa;-2a?
+ 3a?
a2 — ax + x1 ~ 3t*
-- -a-2a;+-
a + x
a + a;
7. 48 a?
48 a?-
+ 16]4a;-l
12a;
112x4-3
12a; + 16
12a;- 3
+ 19
4a; — 1 4x— 1
8. 2s2 -5s- 2
2s2-8s
teachers' edition. 85
s -4 9. a2 + &2ja-5
2s + 3 a2 - ab \a + b
3s- 2 a& + 62
3s -12 a&-&2
+ 10 +26*
s — 4 s— 4 a — & a — 6
10. 5s8- s2 + 5|5s2 + 4s-l
5s8+4s2— s |ag — 1
— 5s2+ s + 5
— 5 s2— 4s + 1
+5s + 4
. 5s8-s2 + 5_/w _j 5s + 4
5sa + 4s-l 5s2+4s-l
Exercise LIV.
1. l-*^2 4. a-x +
a? + x*
x + y a — s
_ s + y - (s - y) _ a2 - 2ax + s2 + (a2 + Xs)
x + y a— s
= s+j/j- sj-^ _ 2(o2 — ax + x2)
x +y a— x
^y 5. 5a-26-3a'-4iy
5a — 6o
a._ _ 25a2-40a&+12&2-(3o2-4y)
2' 1+rTt 5a-66
_22a2-40a5 + 16&2
5a-66
a + y
= xjLy^+j[s-1y)
* + y
= s_+y_+s-^ fi a + ^al+62
a+y a+6
= 2x ■ _a2 + 2a5 + 52-(a2 + &2)
*+? a + 6
s
3s2-(l+2s2) 7 7g 2-3a + 4a2
s 5 — 6a
_3s2-l-2s2 ^ 35a -42a2 - (2 -3a + 4a2)
s 5 — 6a
_s*-l _ 38a -46a2 -2
s ' "" 5-6a
86
ALGEBRA.
2a 14. . + «-&=i?
_6az-(5as-3) s» + 2s-15- 2x + 15
2a ""
_gg + 3 jpj
2a
* + y
x-3
a?-3
a-±* + l 15. 2a-b-™L
a — b a + b
_a + b + (a-b) _2a* + ab-b*-2ab
a-b <* + &
__ 2a 2a»-a&-6«
a—b a+b
10. a-ft i le- 3a?-10 + 41
a + 6
_ a - & - (a + b)
a + 6
_-26
a + 6
s + 4
_3s* + 2x-40 + 41
17.
» + 4
_3ar* + 2a; + l
» + 4
x* + a. + i+_L^
a — i
_ar»-l + 2
_2x*~(x* + 2xy+f) x-1
*»+!
_ g* — 2sy -- y» « — 1
* + * 18. **-Zx-3x(*-*)
x-2
12. 5a~12? + 6a + 3s ^ s*-2s»-3s»+6s-9s+3s»
x — 2
_ 5a- 12a? + 24a + 12s x*-2x*-3x
4 x-2
,29a
4
_g(s»-2ar»-3)
s-2
13. a-l+_L- 1Q ^.o^j.a-j 6a?»
"^+\ 19- a2-2aa;+4a?»-
, , , . a + 2a?
- <*' - 1 + 1 _as + 8ar»-6ar»
« + l a + 2»
_ <*' _a' + 2a»
a + 1 " a + 2*'
TEACHERS EDITION.
87
20. x-g + y + o'-W+y'
x + a
_x2 — a2 + xy + ay + a* — ay + y*
x + a
_ x9 4- xy + y2
a;4- a
3a;-7 4s-9
Exercise LV.
6 18
L.C.D. = 18.
The multipliers are 3 and 1
respectively.
3s-7_9s-21.
6 18 *
4x-9 Ax-9
18
18
4a — 5c 3a — 2c
5ac ' 12a2c
L.C.D. = 60a2c.
The multipliers are 12 a and 5
respectively.
4a- 5c _ 48a2- 60 oc,
5ac
3a-2c
12 a2 c
60a2c
15a -10c
60a2c
2 2g~4y Sx-Sy
5a? ' 10a;
L.C.D. = 10a*
The multipliers are 2 and x
respectively.
2x — Ay = 4a? — 8y .
5s* 10z*
3s-8y _3g»-8apy
10a; 10a;2
6
1-a; 1-a*
L.C.D. = l-a^.
The multipliers are l+a?and 1
5 _5+5a;.
l_a. I-**1
6 6_
l_ar» i-aJ*
5.
1
1
(a-b){b-c) (a-b)(a-c)
L.C.D. = (a-6)(a-c)(6-c).
The multipliers are a — c and b — c.
1 a — c #
(a_6)(6_c) (a-6)(a-c)(6-c);
1 __ b-c
(a-b)(a-c) (a-b)(a-c)(b-c)
88 ALGEBRA.
4a? xy 7 8 x 4-2 2a?-l 3a; 4-2
3{a + b) 6(a'-&2)' ' x-2* 3x-6 5a;- 10'
L.C.D. - 6(a2-&»). L.C. D. - 15(a; - 2).
The multipliers are 2(a—b) The multipliers are 15, 5
and 1. and 3.
4a* 8a?(a-fc).
3(a + 6) 6(a*-&2)'
ay ay
8a; + 2_ 30(4 x + 1)
a; -2 15(a?-2)
2x-l 5(2a;-l).
6(a»-62) 6(a*-&»)
3a;-6 15(a;-2)'
3a; + 2 _3(3a; + 2)
5a; -10 15(a?-2)
8 a~6m 1
c — bn
mx
nx
L.C. D. «=mnaj.
The multipliers are n, mruc, and m.
a — bm an
— bmn.
Tax
mnx
- mnx m
mnx
c — bn cm
— bmn
Exercise LVI.
1 Sx-2y 5a? -7y 8a? + 2y
5a; 10* 25
L.C.D. = 50 a;.
The multipliers are 10, 5, and 2 a?.
30a; — 20 y = first numerator,
25 a; — 35 y = second numerator,
16a? +4ay = third numerator.
16a?2 + 55a; + 4a?y — 55 y = sum of numerators.
.-. Sum of fractions = 16** + 55s + 4sy-55y,
50 a;
teachers' edition. 89
4a? -7y2 3x-8y 5 - 2y
3a? 6x 12
L.C.D. = 12x2.
The multipliers are 4, 2 a;, and a?.
16 x2 — 28 y2 = first numerator,
6x* — 16 xy = second numerator,
5a? — 2a?y = third numerator.
27a? — 2x2y — 16xy — 28y2 = sum of numerators.
. Sum of fractions = 27,^-2^-16^-28^
12a?
4a* + 562 ( 3a + 26 > 7-2o
26* 56 9
L.C.D. = 9062.
The multipliers are 45, 186, and 106*.
180a2 + 225 62 = first numerator,
366s + 54 a6 = second numerator,
706^ -20o62 = third numerator.
180a2 + 331 6* + 54a6 - 20a62 = sum of numerators.
,. Sum offractions^180^ + 54^-20^+331^
90 62
^ 4x + 5 3x-7 9
3 5x 12a;2
L.C.D. = 60a?.
The multipliers are 20 a?, 12 x, and 5.
80 a? + 100 a? = first numerator,
— 36 a? + 84 x = second numerator,
45 = third numerator.
80a? + 64a? + 84 x + 45 = sum of numerators.
... Sum of fractions = 80x» + 64a? + 84x + 45,
60 a?
5 4x — 3y 3x+7.y 5a; - 2y 9x + 2y
7 14 21 42
L.C.D. = 42.
The multipliers are 6, 3, 2, and 1.
24 x — 18 y = first numerator,
9x -f 21 y = second numerator,
— 10 x -f 4y = third numerator,
9x + 2y = fourth numerator.
32 x + 9y = sum of numerators.
.-. Sum of fractions - 32x + 9V.
42
90 ALGEBRA.
6. 3*y-4 5y» + 7 6s* -11
*V ay8 ar*y
3 a? y2 — 4 xy = first numerator,
— 5 a? y2 — 7a^ = second numerator,
— Qx2?/3 -flly2 = third numerator.
— Sa^y2 — 4ary + lly2 — 7a2 = sum of numerators.
. .. Sum of fractions « 11^-4^-8^-7*.
a*y>
7 g2-2gc + c2 y-2ftc + c?
gV frV
L.C.D.-a»ftV.
a262 — 2ab2c + 62 c2 = first numerator,
— of6f -f2a2frc — g2^ = second numerator,
— 2 g£>2 c+2 a2 6c+62 c2— g2 c2 = sum of numerators.
... Sum of fractions = ** -2aVc + 2a*bc -a>*
a2b2<*
8 5g8-2 3a2-a
8a2 8
L.C.D. = 8a2.
5 a* — 2 = first numerator,
g8 — 3 g4 = second numerator.
6gs — 3 g4 — 2 = sum of numerators.
6a3-3g4-2
Sum of fractions =
8g2
a — b b — c c — a ab2 + 6c2 + ca2
cab abc
L.C.D. = g6c.
g2 b — ab2 = first numerator,
b2 c —be2 = second numerator,
gc2 —a2c= third numerator,
ab2 + bci + a2c= fourth numerator.
g2 b + b2 c + gc2 = sum of numerators.
.-. Sum of fractions = a' 'b + * 'e + "*
abc
teachers' edition. 91
10.
I 1 1 2x — zy — 2z
2x*y 6y2z 2xz* 4a?2z2 taPyz
L.C.D. = 12sVz*.
6yz* = first numerator,
— 2x2z = second numerator,
— 6xy2 = third numarator,
6xy2 — 3 y2z = fourth numerator,
— 6 yz2 + 3 y2z = fifth numerator.
— 2 x2 z = sum of numerators.
.
0 2- "I
Sum ot fractions = -— - = — — — •
12zV2a 6y2«
Exercise LVII.
1.
j_+^_.
a?-6 z + 5
L.C.D.=(a;-6)(a; + 5).
The multipliers are jc + 5 and a?— 6 respectively.
x + 5 = first numerator,
a? — 6 = second numerator.
2x — 1 = sum of numerators.
Sum of fractions = - x~
x*-x-20
_1 1__.
z-7 z-3*
L.C.D. = (s-7)(3-3).
The multipliers are * — 3 and x — 7 respectively.
a; — 3 = first numerator,
— x + 7 = second numerator.
4 = sum of numerators.
4
, Sum of fractions =
s2-10a: + 21
-x
L.C.D ^1-s2.
The multipliers are 1— x and 1 + x respectively.
1 — x = first numerator,
1 + x = second numerator.
2 ^= sum of numerators.
o
, Sum of fractions — - — — •
1 — ar
92 ALGEBRA.
6.
I
l-x l-x*
L.C.D.= l-x*.
The multipliers are \+x and 1.
1 + x — first numerator,
—2 >= second numerator.
— 1 + x = sum of numerators.
— (1-*).
.\ Sum of fractions = ~* "*"*' = — •
l-x" 1+a?
c 1 x
6. +
*-y ' (*-y)"
L.C.D. = (a-y)2.
The multipliers are a? — y and 1.
x — y = first numerator, I
x- = second numerator.
2 x — y = sum of numerators. \
Sum of fractions = x~*.
(*-y)*
1 1
2a(a-f:e) 2a(a — a)
L.C.D. = 2a(a + aO(a-x).
The multipliers are a— a; and a+c.
a — x = first numerator,
o + a? = second numerator.
2 a = sum of numerators.
, Sum of fractions = -— ^- x ........
2a(a-Kr)(a— aw a2 — or
(a + &)6 (a — 6)a
L.C.D. = a6(a2-62).
The multipliers are a(a — b) and J (a + b)
a8 —a2 6 = first numerator,
— a62 — 6s = second numerator.
az — a%b — db% — 6s = sum of numerators.
. Sum of fractions - *-*h-&-».
teachers' edition. 93
8.
10.
2x{x-\) 4x{x-2)
L.C.D. = 4a;(a?-l)(a:-2).
The multipliers are 2 (a?— 2) and (a — 1).
10 a: — 20 = first numerator,
—3s + 3 = second numerator.
7x — 17 = sum of numerators.
tj I It
. Sum of fractions = - .
r 4s(xa-3a; + 2)
1 +x 1 — a;
1 + x + x* 1 — a? + x*
L.C.D. = l+x* + a;4.
The multipliers are 1— x + x* and 1 + x + xa.
1 + x8 — first numerator,
— 1 +s* = second numerator.
2 3* = sum of numerators.
. Sum of fractions — •
l + a* + x*
2ax — Zby 2ox + 35y
2xy{x-y) 2xy(x + y)
L.C.D. = 2xy(x*-y*).
The multipliers are x+y and x—y.
2 ax* + 2axy — 3 bxy — 3 by9 = first numerator,
—2ax* + 2 gay — 3 bxy + 3 &y* *= second numerator.
4 oxy — 6 fay = sum of numerators,
or 2 ay (2a — 3 6) = sum of numerators.
•\ Sum of fractions — -^^
x*-y2
Exercise LVIII.
1.-L.+ x • 2a
1 + a 1 — a 1 — a*
L.aD.-1-a".'
The multipliers are 1 — a, 1+a, and 1.
1 — a «= first numerator,
1 + a — second numerator,
2a =» third numerator.
2 + 2a=*2(l + a) = sum of numerators.
, Sum of fractions - 2^ + a) ?—
(l + a)(l-o) 1-a
94 ALGEBRA.
__1 1 , 2 s
1-x 1 + x 14-a*
L.C.D.*(l-x)(l + x)(l + x»).
1 +x + x* + Xs = first numerator,
— 1 + a? — x* + x8 = second numerator,
2x —2a8 = third numerator.
4x = sum of numerators.
4*
\ Sum of fractions^
1-x*
x x* _ X
1-x 1-x 1+x*
L.C.D.= (l-x)(l + x*).
x + X8 = first numerator,
— x* — x* = second numerator,
g —x1 = third numerator.
2x — 2x*+x8 — x* = sum of numerators.
= 2x(l-x)+x»(l-x).
, Sum of fra*tions = (2* +J<l-«), *g±g.
(1 + 3*)(1 -x) l+s»
?+_y_ +
y x+y X* + xy
L.C.D. = xy(x + y).
x» + x*y = first numerator,
+ ajy* = second numerator,
•f x*y = third numerator.
x* + 2x*y + xy* =■ sum of numerators.
=«x(x + y)2.
... Sum of fractions = x-^±^ = ^±£
ajy(a?+y) y
x — 2 x — 3 x — 4
L.C.D. = (x-2)(x-3)(x-4).
X8 — 8x2 + 19x-12 = first numerator,
aJ— 8x*+20x — 16 = second numerator,
a*_ 8xa + 21x — 18 = third numerator.
3 a? — 24 x* + 60 a; — 46 = sum of numerators.
3x8-24x» + 60x-46
, Sum of fractions =
x.s_ 9x* + 26x- 24
teachers' edition. 95
3 4a 5a2
x — a (a—a)2 (x— a)s
L.C.D.^x-a)8.
3X2 — 6ax + 3a2 = first numerator,
4 ax — 4 a2 = second numerator,
—5a2 — third numerator.
3s2 — 2ox — 6 a2 = sum of numerators.
, Sum of fractions =33x2-2ox-6a2
(x-af
1 1
x-1 x+2 (x + l)(x + 2)
L.C.D. = (x- l)(x + l)(x + 2).
x2 + 3 x + 2 = first numerator,
— x2 + 1 = second numerator,
— 3x + 3 — third numerator.
6 = sum of numerators.
o
, Sum of fractions »
8.
(x2-l)(x + 2)
a — b , b — c
(b + c)(c+a) (c + a)(a + b) (a + &)(6 + c)
L.C.D. = (6 + c)(a+6)(c + a).
a2 — J2 — first numerator,
+ 6s — c2 = second numerator,
— q2 4- c2 = third numerator.
0 = sum of numerators.
.*. Sum of fractions — 0.
9 x — a x— b (a — b)1
x— b x — d (x — a)(x — b)
L.C.D. = (x-a)(x-6).
x2 — 2 ax + a2 «= first numerator,
x2 — 2 6x + J2 — second numerator,
-a2 + 2o6 — 62 — third numerator.
2X2 — 2bx + 2ab — 2ax = sum of numerators.
= 2(x-a)(x-6).
.-. Sum of fractions - 2(s-a)(x-6) _ 2.
(x-a)(x-6)
96 ALGEBRA.
11.
12.
13.
10. s + y 2j afy-a8
y * + y yO^-y*)'
a* — sy* + ofy — y8 = first numerator,
2ajy* — 23*y = second numerator,
— a* + xhj = third numerator.
ay* — y* = sum of numerators.
.-. Sum of fractions = y'(*~# = -y—
y(jc*-y2) jr + y
(6_c)(c_a) (c-a)(a-b) (a-b)(b-e)
L.C.D. = (&-e)(c-a)(a-&).
a* — 6* = first numerator,
+ 6* — c* = second numerator,
— o* + c1 = third numerator.
0 = sum of numerators.
• \ Sum of fractions = 0.
q* — be , 62 — oc . c* + qft
(q + 6)(q + c) (& + a)(6 + c) (c+b)(c + a)
L.C.D. = (a + 6)(6 + c)(a + c).
q26 — 6'c -f a2c — 6c* = first numerator,
aft* + ft*c — a*c—ae* = second numerator,
a*b + qc1 -f 6c* 4- ob* — third numerator.
2q*o + 2 aft* — sum of numerators.
= 2aJ(a + 6).
Sum of fractions ~ 2«K«+*) ^
(a + b)(b + c)(a + c) (b + c)(a + c)
a x a*+ x* *
a — x a + 2x (a — x)(a + 2x) l
L.C.T). = (a-x)(a + 2x).
a2 + 2 ax — first numerator,
— ax + x* = second numerator, ,
— a* — a2 = third numerator. I
ax = sum of numerators. |
.•. Sum of fractions = —
(q — s)(q + 2a:)
14.
16.
teachers' edition. 97
6
(a-6)(6-c) (g-6)(g-c) (g_c)(6-c)
L.C.D. = (g-6)(g-c)(6-c).
3 a — 3 c = first numerator,
— 4 6 + 4 c = second numerator,
6o —66 = third numerator.
9 g — 106 + c == sum of numerators.
.-. Sum of fractions 9a-10b+e >t
(g-6)(g-c)(6-c)
15 x — 2y 2x+y 2x
' x {x - y ) ~~ y (x + y) ~" x* - y2'
L.C.D^a^-y2).
x*y — xy2 — 2yi = first numerator,
— 2 a? -f afy + &y2 = second numerator,
— 2x*y = third numerator.
— 2a? — 2y* = sum of numerators.
= -2(x + y)(x*-xy+y*).
... Sum of fractions = "2(g +y)(a'"ay + y*).
oy(a;+y)(aj-y)
_ 2(a?-sy + y2)
ay(o;-y)
a — 6 (g — 6)(a; + y)
a; (g + 6 j y (g + 6) xy (a + 6)
L.C.D. = ajy(g + 6).
ay — ty = first numerator,
— gar + 6a; = second numerator,
— gy + 6y — ga; + 6a; = third numerator.
— 2 ga; + 2 6a; = sum of numerators.
= 2a;(6-g).
... Sum of fractions - 2x(b~«) = *!k=J$.
xy(a + b) y(g + 6)
17 3a; x + 2y 3y
' {x+yf x*-y^{x-y?
L.C.D. = (a; + y)2(a;-y)2.
3a£ — 6a?y + 3 ay2 = first numerator,
—Xs — 2x*y + xy% + 2y* = second numerator,
+ 3x2y + 6xy2 + Zy* = third numerator.
2a£ — 5x*y + 10 xy3 + 5y* = sum of numerators.
.-. S«m of fractions = **-***+}<>*? + *?.
98 ALGEBRA.
18 a—c a—b
• (a + bY-<* (a + cY-b*
L. C. D. - (a + b + c)(a + b - c) (a - b + c).
a* — c* «= first numerator,
— a* -f oc -f 5* — ab = second numerator,
ac + 6* — o6 — c* ■» sum of numerators.
.-. Sum of fractions = ac-ab + b*-<*
(a + b + c)(a + b — c)(a + c—b)
19.
a + b _ a — b ab(x — y)
ax + by ax — by a V — &y
L.C.D. = (ax + 6y)(oa; - ty).
a*x + ate — aby — b*y = first numerator,
— a'a; + o&c — aby + 6*y — second numerator,
+ qfcg — o6y = third numerator.
+ Sabx—Saby — sum of numerators ;
or, 3 ab (a? — y) = sum of numerators.
•. Sum of fractions = 3,°^*~y)-
a*ar — by2
Exercise LIX.
i. x | a?~~y_ g g-y
x-y y-x x-y x-y
L.C.D. = x-y.
a? »= first numerator,
y — a; = second numerator,
y = sum of numerators.
. \ Sum of fractions = — 2L. .
x-y
2 3 + 2a; Sx-2 16x-x*
' 2-s 2 + s s*-4
^3 + 23? 3a:-2 16a?-ar«
" 2-aj 2 + x 4-s2 '
L.C.D. =4 -a*
6+ 7aj + 2xI = first numerator,
— 4+ 8 a? — 3 a? = second numerator,
— 16a? + x* = third numerator.
2 — x = sum of numerators.
n -I
••. Sum of fractions =- —
4-a1 2 + a;
teachers' edition. 99
3* _£ X ^ X* X X
a?-l x+1 1-x a?-l x + 1 a? — 1
L.C.D. = a?-l.
a? = first numerator,
a? — x = second numerator,
g* rha; = third numerator.
3 a? »= sum of numerators.
3a?
, Sum of fractions = - — - •
ar — 1
4.
1
3_3y* 2-2y 6y + 6
4 ,1,1
3(l+y)(l-y) 2(1 -y) 6(1 + y)
L.C.D. = 6(l+y)(l-y).
8 => first numerator,
3y H- 3 = second numerator,
— y + 1 = third numerator.
2y + 12 = sum of numerators ;
or, 2(6 + y) = sum of numerators.
.*. Sum of fractions a •—- — 2l_-
3(1 -y*)
(2-ro)(3-m) (m-l)(m-3) (m-l)(m-2)
1 2,1
(2-m)(3-m) (l_w)(3-m) (l-m)(2-ro)
L.C.D. = (l-m)(2-m)(3-m).
1 — m = first numerator,
— 4 + 2 m = second numerator,
3— m = third numerator.
0 == sum of numerators.
.•. Sum of fractions = 0.
6.
(b — a)(x + a) (a-b)(x + b)
1 1
(b-a)(x + a) (b-a)(x + b)
L.C. D. = (b - a)(x + a)(x + 6).
x + b — first numerator,
— a? —a = second numerator.
6 — o = sum of numerators.
, Sum of fractions * ~~a —
(b-a)(x + a)(x + b) (ar + a)(x + b)
100 ALGEBRA.
a* + 6* 2a6* ^ 2a'& a* + h1 2a&' , 2a'6
L.C.I). = (a8- &»)(<*» + 6s).
a* +2a4&*+2a'&* +6* = first numerator,
— 2a*6* — 2aJ* = second numerator,
+2a?b -2a*b* = third numerator.
8.
a«+2a*&
—2abP+& = sum of numerators.
Sum of fractions =
(a»-&»)(a8 + &»)
b — a a— 2b
x — \ b + x
3x(a-b)_ b-a a-26
&* — z2 a? — b x + b
3x(a-
Xs-
-b)
b*
L. CD. = **-&».
— ab — ax + bx ■
ab — ax + 2bx ■
Sax-Zbx
+• 6s = first numerator,
-26* = second numerator,
= third numerator.
■ 6* = sum of numerators.
Sum of fractions = -aaj
x* -b*
S + 2x 2-3a? 16g-ar»^3 + 2s 2-3.r 16S-S1
2-a? 2 + x s*-4 = 2-a? 2 + x 4 — ar*
L.C.D.-4-x".
6+ 7 a? -f 2 s* = first numerator,
— 4+ 8 a? — 3 e* = second numerator,
— 16 s + s*«° third numerator.
2 —x = sum of numerators.
2-x 1
. Sum of fractions =
4-ar* 2+a;
10 3 7 4 -20s 3 7 4- 20a?
' 1-2* 1 + 2* 4z*-l,Bal-2s 1+2* l-4a*'
L.C.D. = l-4ar*.
3 + 6 a? = first numerator,
— 7 + 14 a? = second numerator,
4 — 20a; =» third numerator.
0 = sum of numerators.
•\ Sum of fractions = 0.
teachers' edition. 101
11. fl + fr , 6 + c , c + a
(J_c)(c_a) (6-a)(a-c) (a-6)(6-c)
_ a + b b + c a + c
(b-c){a~c) (a-6)(a-c) (a-6)(6-c)
L. C. D. = (a - 6) (a - c)(b - c).
— a2 + 62 = first numerator,
_ b2 + c2 = second numerator,
qa — c* = third numerator.
0 = sum of numerators.
.•. Sum of fractions = 0.
12 <*-bc b2 + ac <? + ab
' (a-6)(a-c) (6 + c)(6-a) (c-a)(c + 6)
_ a2 — be ac + b2 ab + c*
(a-6)(a-c) (6 + c)(a-6) (a-c)(6 + c)
L.C.D.= (a-6)(a-c)(6+c).
a*6— 62c+a2c— 6c* =■ first numerator,
l?c—a2c 4-ac2— a6a = second numerator,
— a*b +bck—a&+alP=z third numerator.
0 = sum of numerators.
. \ Sum of fractions = 0.
i3. y+2 i z+x , g+y
(x-y)(x-z) iy-x){y-z) (z-x){z-y)
^ y + z g + z x + y
(x-y)(x-z) (x-y)(y-z) (x-z){y-zj
L.C.T>. = (x-y)(y-z){x-z).
y2 — z2=* first numerator,
— x2 -f z2 =» second numerator,
Xs — y2 = third numerator.
0 = sum of numerators.
.\ Sum of fractions = 0.
14. 3 4 6
(a_6)(6_c) (6_a)(c-a) (a-c)(c-6)
3 4,6
(a_6)(&_c) (a-6)(a-c) (a-c)(6-c)
L.C.D. = (a-6)(a-c)(6-c).
3 a — 3 c = first numerator,
— 46 + 4c = second numerator,
6a— 66 ' = third numerator.
9a — 106 + c = sum of numerators.
,\ Sum of fractions =- ~
(a-6)(a-c)(6-c)
102
ALGEBRA.
15.
x(x-y)(x-z) y(y-x){y-z) xyz
1 1 1_
x{x-y){x-z) y{x-y){y-z) xyz
L. C. D. ■» xyz(x — y)(x — z) (y — z).
y*z —yz* = first numerator,
— a^z+as* = second numerator
—xht+xyt—yh+sPz—xzi+yz* = third numerator.
—x*y+xy* = sum of numerators ;
or, — xy {x — y) = sum of numerators.
.•. Sum of fractions = ■
-qy(*-y)
xyz(x-y)(x-z)(y-z)
1
2(*-«)(y-2)
Exercise LX.
1.
a C*
5.
8oW 15sy*
45**y 24a86*
Cancelling common
factor x,
Cancelling 8, 15, a\ &», x,
ac
andy,
bd
= -&.
9 ox
2.
2*v3o5v3oc
a c 26
A.
9xY« 20aWc
Cancelling 2a6c,
10 aWc 18 sy'z
= 9aa;.
Cancelling 9xy*z, 10 cWc,
and 2,
3.
3j> 2j>
21>-2'<>-l
=» — ax.
« 3j> »-l
2(j>-1) 2^
7.
3^yx5^x_12£
4xz» 6rry 2sy*
Cancelling p and j) -
-1,
Cancelling 2, 6, a?, y8, and z,
"i
15 a?
4z'
4
Sx*y 2x*
15 aP Sab*
g
9mVx5pVx24xV
8**y 3a&*
&jp<f 2xy 90 mn
15a&» 2a*
Cancelling 9, 5, 8, mn, j?1^,
Cancelling 2»" and 3 ab7,
and xy,
4xy
3mnxy
5b
W
TEACHERS EDITION.
103
25Pm2
70nV 3pm
14 nV 75p*m 4#n
Cancelling 25 £2ra, 14n8g', and
5*ro2
11.
41>2
10.
'-62
a-b
a* + ab a*—ab
= a-b x(a + b)(a-b)
a(a + b) a(a — b)
Cancelling a — b and a + 6,
a-6
a2 + 6* . a — b
a2-b2^ a + b
^ a2 + 6* a 4- 6
(a—b)(a + b) a — b
Cancelling a + b,
a2 + 62
12.
(a -ft)2
3* + X - 2 xf_
-13a: + 42
x2 — 7x x2 + 2x
(s+2)(x-l)x(x-6)(x-7)
x(x-7) x(x+2)
Cancelling a? — 7 and x + 2
,(x-l)(—fl).
13 g'-Hg-f 30 a»-3a?
a^-Bx + S a^-Sx
(x-5)(x-6) *(g-3)
(x-3)(x-3)- x(x-5)
"** — 3*
14. <*-*x(a + xf
a3 + x* (a — a;)2
_ (a - x)(a2 + ax + x2) (a + a;)2
(a + x)(a2 — ax + x2) (a — x)2
(a + x) (a2 + ax + x2)
(a — x)(a2 — ax + X2)
16 2a(x2-y2)2x x»
ex (x-y)(x + y)2
2a(x + y)2(x-y)\
*(.T-y)(x + y)2
2ax* (x — y)
104 ALGEBRA.
16 aa+2q&xa6-26a 17 s»-4 xs»-25
a3+461 a1-46» ' x* + bx 3*+2x
_a(a + 26) 6(a-26) _ (x+2)(s-2) (g+5Xs-5)
a* +46* (a-26)(a+26) s(ay+5) x(x+2)
ab (x-2)(x-5)
a*+461" " x*
is. a^ + ay>/(*-y)1
a;-y a?*-y*
_x(* + y)y (s-y)Qg-y)
*-y («, + y,)(* + y)(a?-y)
a;
V + y*
19.
c8 + d8 c + d
_ (ro + n) (ro — n) c + d
(c + rf)(c* — cd+ d*) m — n
m + n
<?-cd + d*
-4a + 3 g»-9g + 20 a* - 7a
a1 -5a + 4 a* -10a + 21 a* -5a
(a-3)(a-l) (a-5)(a-4) a(a-7)
(a-4)(a-l) (a-7)(a-3) a(a-5)
= 1.
6»-76 + 6 6» + 106 + 24 . 6* + 66
" 6* + 36-4 6*-146+48T68-868
(6-6)(6-l) (6 + 6)(6 + 4) P(6-8)
(6 + 4) (6 - 1) (6 - 8)(6 - 6) 6(6 + 6)
= 6.
22 s*-y» x.a?y - 2y8 x x*-xy
x2 — 3xy + 2yi x2 + xy {x—yf
_ (s+y)(s-y) x y(g-2y) x a;(a?-y)
(x-2y)(x-y) x(x + y) (*-y)(*-y)
y
as-y
teachers' edition. 105
03 as-3q»& + 3ay-6s 2ab- 2b* „ a* + ab
a*-6» + 3 X a-b
_a3-3a86 + 3a6'-y 3 a* + ab
a8-6* 2a&-26» a-b
_(a-b)(a-b)(a-b) 3 „a(a + b)
(a + b)(a-b) 2b(a-b) a-b
^Sa
2 b
2± (a+ftP-c* . <*-(a + bY
a*-(b-c)*^c>-{a-bY
_^ (a + b + c)(o + 6 - c)(c -0 + 6)(c + a-b)
(a — 6 + c)(o + 6 - c)(c - a — 6)(c + a + 6)
_ c — a-f 6
(x-&)»_a* ^-(a-ft)*
_ (a; - a -f 6) (a; - a - 6) (a + 6 - a) (x - b + a)
(3 — b + a)(» — b — a) (x + a — &)(s — a + 6)
a? + a — 6
(a + 6)»-(c + aT (a-c)»-(<*-6)»
* (a + «)■-(& + <*)»- (a -ft)1 -(<*-«)■
c (0 + b + c + <?)(a + ft — c - a*)(a - 6 + d~ c)(a -b-d + 6)
{a + c + b + d)(a + c — b — d)(a — c + d — ft) (a — c-d + 6)
= 1.
27 a? — 2ary + y* — z*x + y — z
x* + 2xy+y* — z* x — y + z
_(x-y)i-z*yx + y-z
(x + yf-z* x-y + z
^ (a - V + z)(x - y - z)(x + y - z)
*° (x + y + z) (x + y - z) (x - y + z j
a; + y + 2
106 ALGEBRA.
Exercise LXI.
3s x-l
2 3
Multiply both terms by 6,
9a; + 2a; -2
13a? + 13 -2a -15
__ 11 x — 2 _-. ax+bx + cx— be
~lla;-2~
X
— a
x
(*-
-»)(•-«>
a? + a
a? — a
x2 + ax— x* + bx + cs-
-6c
(«■
ar + a
-a)(s + a)
-1+A
\x a/\x a J
x-2 +
x-6
—' l-x~a
x + a
Multiply both terms by * - 6, ( a'-^Ug+^N
2a
a + a?
_g»-7a? + 12
"V-Ss + lS
(a.-5)(a-3) ^-j
a>-4
3.
3
21 + 1 *'
2a;-l
+ 5-1
2 2
Multiply both terms of sec-
ond fraction by 2,
3
x + 1
4x-2
2x* + x-l
3
B + l
2(2*- 1)
(2*-l)(*+l)
3
2 1
a?-y a?-y»
a; y
xy+y* x* + xy
x + y — x
s*-ya
y *y(x + y)
"a*-tfX **-y»
jc + 1 x + l a:+l (a;+y)(x-y)»
teachers' edition. 107
x+l+x— 1
~ x — \ x + 1
x + 1 a;— 1
ar — 1 x + 1
10.
1
x_
(a + 1)2 (x-Xf
1
a^-l x*-\
1 £_
(x + 1? (x-llP
x + 1
x + 1
(ar + 2x + l) + (x»-
(x* + 2x + l)-(x*-
-2x + l)
-2x + l)
X + 1 —X
= x+l
2x* + 2 x* + l
4x 2x
1
-x + 1.
11.
1
1 1 *
X
1 , 2ar
l+x + -^-
1— X
1
x + 1
1+l+%
X + 1 —X
1-x
x + 1
1
1
x + 1
i X — X*
1 + x2
^l+ar|
"l + x*
9. 1 + .
l+x + n
1 — x
e .„ e+He+H
= 1 +
x(l-x)
(l+x)(l-x) + 2ar»
(fir
(l+x)(l-x) + 2x3 /g»-2ax + x»\ /raa + 2ox + xa\
1 + g(l ~ g) V ax jy ax /
" " * /a8-xa\/a2-ar\
\ ax J \ ax J
l + x»
1 1 + X* + X — X*
1 +x» (q — x)(a — x)(a + x)(a + x)
1 + a? (a + x)(a - x)(a + x)(a - x)
'l+x*' «1.
108 ALGEBRA.
g' + y1 2a? (xy-& x + y)
13# g'-y* »+yt(^-y)> g-y)
»-y
g»+y» 2a? r aj_ + g_+y J
g'-y* g + y( g~y x-y\
= g-y
a?-y* g+yjg-yj
*-y
g* + y» 2ay
g-y
g» + 2gy + y»
g'-y*
g-y
_(g+y)(*+y)x i ^ * + y
(g-y)(g + y) g-y x*-2xy + y*
(g»-y»)(2g«-2a!y)
14. 4<*-y>a
_gy_
g + y
_ (g + y)(g-y)2a?(a?-y)(a? + y)
4(a?-y)(g-y)gy
2y
aft oc
jg s* +(a + ft)a? + aft a? + (a + c)x + ac
b — c
g*+(ft + c)a? + ftc
(qfta; + aftc) — (aca; -f qftc)
(a? + q)(g + ft)(s+c)
(a? + ft)(a; + c)
ax(b — c) (x + ft) (x + c) qg
(a? + a)(g + ft)(g + c)(6 — c) g + a
teachers' edition. 109
16. -
1
x 1
— Y + 1--
X
1
' + 1
x+l
1_
x+l
x* + x
x + l
'X.
a + b b
j~ b a + b
19.
i+i+I
ab ac be
a b
a2-(6 + c)J
ab
(a + b)* + b*
b(a + b)
a + b
Multiply the terms of the nu-
merator by abc, and factor
the denominator,
~aT
c + b + a
_(a + b? + b*x ab
b(a + b) a + b
abc
(a + b + c)(a-b-c)
ab
{(a + b)* + b*\a
{a + b?
_(a* + 2ab + 2b*)a
_ c+b+a ab
abc " (a+b+c)(a-b-c)
1
(a + &)*
c(a—b — c)
1
m
2m-l
20.
2m-3 + — i . o
18. 2 i , 3
2m-i 1+rr;
2m'-3m + l 3(1 — ag)
™ l-s + 3
3
"l-±x
_(2m-l)(ro-l) 4-x
2™-1 __3(4-aQ
-»-l. "" 7- 4a'
110 ALGEBRA.
Exercise LXIL
x*-9x8 + 7x* + 9x-8 0 ot + 6a-cs + 2o5
1.
(x-8)(g*-x*-x + l) ^16 + }-l+4
=a(x + 8)(x*-xa-x + l) ~16-J-1 + 1
s-8 ^19}
~* + 8* =15}
Multiply both terms by 4,
3. 3«. + ^-^
c b1
■tt-l|-
= 48 + 2-4 = 46.
2 1 1
(3*-l)* 2s»-4x + 2 1-s*
(x»-l)» 2(x-l)a ar»-l
L.C.D.=.2(x»-l)2.
4 = first numerator,
—Xs — 2a: — 1= second numerator,
2s* —2 = third numerator.
x1 — 2s + 1 =» sum of numerators.
.-.Sum of fractions- **-2* + l _ 1
2(x*-2x + l)(x + l)» 2(x + l)2
Vb + 1 x + 1/ \x—l x-lj
/ x* {x + l l\/a* x»-x 1_\
\x + l x + 1 x + l) \x-l x-1 s-lj
_ x(x + 1) X - 1
X + l X — 1
teachers' edition. Ill
6. (x-<*\* (x-2a + b\
\x-b) \x + a-2b)
(a + b-2a\* fa±b-^a±2b\
BS\a + b-2b) \a + b + 2a-4b)
\a-b) \Za-n)
= (-!)»-(- 1) = 0.
7 f a + b a-b 2b* \a-b
' \2{a-b) 2(a + b) a»-W 26
L.C.D. of fractions in brackets — 2(o*— 6*).
a* + 2 aft + 6* *= first numerator,
— a2 + 2 oft — 6* = second numerator,
46* = third numerator.
4 ab + 46* = sum of numerators -,
or, 4 6 (a + b) = sum of numerators.
.\ Sum of fractions in brackets = — *a +, '■ — -^—, -•
2(a,-6») a- 6
— ,x^ = i.
a-6 26
Va^-y1 x*+y*)^\x-y s + yy
_ / s*+2 x*y*+y* x*-2 afy* +y*\ # (x*+2xy+y* s*-2sy+y2\
\ a^-y4 a^-y4 y \ a?-y2 ""iT^y2-" /
_ 4aj*y* 4:cy
x* — y* x*—y*
_4xyx*»-y'
o^— y* 4ary
„ *y
a^+y8
112 ALGEBRA.
V ya J\*-y) \ y8 A* + *y + W
y2 s-y y8 o^ + ajy+y*
y y
= 2.
U8-W\ <* + & y Us + ^ A a*+ab + b*J
(a(a-b)\ (a2 + ab + b*\ /a8-&8\ /q'-gft + yx
= U8-^A <* + & J U8 + &vU, + <** + &V
= a a — 6
a + 6 a + 6
= 2a-_6
a + 6
.... a + jc a3 + x2
- a — <c - a'
a + x a2 + x2
Multiply both terms of first fraction by a+x, and both terms of
the second by a2 + x2,
a + x + a — x a2 + X2 + a2 — x2
a + x — a + x a2 + a? — a2 +x*
^2a 2X2 x
2x 2a2 a
=(-4)+3K)+4KVH)
-(^-l+3l) + 8(-l) + 4
X X2
teachers' edition.
113
1-
13.
2xy
1-'-
1 +
2ay
1 +
(»-yJ*
g' + y1 ' /» + y'
(^J
= 1.
14.
15.
x + 2a x — 2a
2b-x 2b + x'
ab
tab
U*-x*
a + b
+ 2a
ab
a + b
2a
~2b--?L
a + b
Sab + 2a*
~2b + ^-
a + b
2a* + ab
tab
tab (a+b)*
ab + 2b* Sab + 2b* b*(Sa* + Sab + U*)
_a(3b + 2a) a(2a + b) 4ab(a + b)*
b(a + 2b) b(Sa + 2b) b*(Sa + 2b)(a + 2b)
L.C.D. = b*(3a + 2b)(a + 2b).
Ga'b + 13a*b* + 6aP = first numerator,
— 2 a*b — 5 a*b* — 2 a&* -» second numerator,
— 4 a*b — 8 a262 — 4ag » third numerator.
0 = sum of numerators.
0
Sum of fractions = 0.
x + y—1
x-y + 1
a + 1 ab + a «
ab + 1 ab + 1
a + 1
ab+1
ab + a *
ab + 1 +
2a
ab + 1
2
ab + 1
— a.
114 ALGEBRA.
16.
1.1,1
a(a-b)(a-c) 6(6 - c) (6 - a) c(c -a)(c- 6)
1 1,1
a{a - b)(a - c) 6(6 - c) (a - 6) c(a - c)(6 - c)
L.C.D. = a6c(a-6)(a-c)(6-c).
6'c — 6c* — first numerator,
— a^c + ac3 = second numerator,
+ ga6 — a6* = third numerator.
6*c — 6c* — a*c + ac2 + a2b — ab* = sum of numerators.
a -. .. 6*c - 6c* - a*c 4- ac2 + a26 - a6*
.•. Sum of fractions = — — ~
a6c(6*c — 6c2 — a*c + ac* + a*b — a6s)
= J-.
abc
17.
a_l [ 5_i | c-1
3a6c a 6 c
6c + ca — ab 1,1. 1
a 6 c
Multiply both terms of the second fraction by o6c,
3o6c o6c — 6c + o6c — ac + o6c — 06
6c + ca — ab bc + ca — ab
3 abc 3 a6c — 6c — ac — ab
be + ca — ab be + ca — ab
be + ac + ab
bc + ac — ab
m2 + n*
18.
n
-xro!
-n»
1
w
m2-
1
m
-mn
+ n8
mn
y (m + n)(m-n)
" m — n (m+ n) (m* — mn + n*)
^teachers' edition. 115
U-L-
1Q a b + ef-, , b2 + <?-a'\
ml ~^")
a b + c
_(b + c + a)(2bc + b* +<*- a?)
(b+e — a)2bc
_(b + c + a){(b + cy-a*}
(b + c—a)2bc
__ (b + c + a)(b + c + a)(b + c — a)
{b + c — a)2bc
^(5-fc + a)2
2bc
20. 3a-[&+{2a-(&-c)}] + * + ^^l
2c-f 1
= 3a-[& + 2a-& + c] + J+|^=i
= 3 a - & - 2 a + 6 - c + } + ?£=i
2c+l
1
= a-
-•♦S3
= a-
C + i + 2(2c + 1)
— a~
-c + J + c-J
= a.
21.
1
a— a?
1 X
a — y (a— a;)2 (a
y
-y?
1 1
(a-y){a-x)* (a-x)(a-yf
(a-x){a-yy-(a-y){a-xy+x(a-yf-y{a-xf
= (a-a;)2(a--y)>
s-y
(a-x?{a-y)*
«a(2a — a? — y).
o(2o - x - y)(x -y)^(g- a;)* (a - y)2
{a-x)t{a-yf x-y
116 ALGEBRA.
1
xl 1
X I ■ -
1+£±1
3-x
1
* + 3-^
3 + 1
4
4x + 3 — x
4
3(x + l)
/c_5 <J_&r
23 (x»-y»)(2^-2gy)
v y/ *+y
(g + y)(s-y)(g-y)2s
" *(*-y)(*-y)(«+y)
xy
2
L. C. D. 1st expression — c8+6*. L. C. D. 2d expression =■ <?— ft*.
c»-2c26+2cft*-6, = lstnnm. c*+2c6+ 6s -1st num.
— c8 +6f = 2d num. c* + 6* = 2d num.
— 2c*6+2c6* = sum of nums. 2c*+2c6+2o* = sum of nums.
or, — 2 c6 (c — 6) *» sum of nums. or, 2 (c*+co+62) — sum of nums.
-2c6(c-6) (c + 6)(c-6)
" (c + 6) (c* - cb + b*) X 2 (<* + c6 + &*)
" <<* + <jp + b*
-bc(b-c)*
"M + W + /
25. y + * + *+y
(x-y){x-z) (y-s)(y-«) (*-*)(*-y)
= y x x+y
{x-y){x-z) (x-y)(y-z) (*-f)(y~s)'
L.C.D.-(»-y)(*-*)(y-«).
^ — y« = first numerator,
— jc* + xz = second numerator,
a* — y* » third numerator.
rrz — yz «- sum of numerators ;
or, z(x—y)=> sum of numerators.
.•. Sum of fractions — -. v ~~ V)
(x -y)(y - *) (* - *) (* - *)(y - *)
teachers' edition. 117
26. I + I J-
a{a — b)(a — c) b(b — a) (b — c) abc
i i l.
a(a — b)(a — c) b(a — b) (b — c) abc
L.C.D. r abc(a - b)(b - c)(a - c).
— be2 + b*c = first numerator,
— a?c + ac* = second numerator,
— be9 — a*b + ab2 — b*c + a*c — ac* = third numerator.
— a2b + ab* = sum of numerators.
.-. Sum of fractions ab(a- b)
abc(a — b)(b — c) (a — c)
1
c(b — c){a — c)
,-4 + JL i-!L+!
27 1+1 x ^
, 6_ (*-l)(*-2)
* + l «■-!
s'-s-e (x-1) (as -2)
x~l
(s-l)(s-2)(s-l) (g-3)(g + a
> + l)(*-3)(* + 2) A (x + 1)(»- 1)(«- 1)(»- 2)
1
" (* + !)»'
Exercise LXIII.
1.5«-*±*-71. 2. «-£n»-H
3 3
Multiply by 3 ; then
3a;- 3 + x = 17,
4* =20,
x = 5.
2
Multiply by :
2; then
10x-x-2 =
-142,
9a? =
= 144,
x =
= 16.
118
ALGEBRA.
■ + Z = x —
4 " 2
Multiply by 4 ; then
5 - 2a; + 8 = 4ar - 12a? + 16,
6a?=3,
5. 2s-5g-4-7-J-^
6 5
Multiply by 30 ; then
60 a; -25 a; + 20
= 210-6 + 12a?,
23 x =184,
a; = 8.
4.
5a; «r>*_9 3-g
2 4 4 2
Multiply by 4 ; then
10a?-5a? = 9-6 + 2a?,
3a; = 3,
x + 2_14 3 + 5a;
2 9 4
Multiply by 36 ; then
18a; + 36 = 56 -27- 45 a?,
63a? = -7,
7 5a? +3 3-4a? a;^31 9-5a?
8 3 2*2 6
Multiply by 24 ; then
15a? + 9 - 24 + 32a; + 12a? = 372 - 36 + 20a?,
39a? = 351,
a?«9.
Q 10a? + 3 6a?-7 in, 1X ,A 7a; + 5 5a;-6 8-5a;
8. — __10(*-1). 10. — — .
3 2
Multiply by 6 ; then
20 a; + 6- 18a; + 21
= 60 a; -60,
58 a; = 87,
a;=l*.
5a?-7 2a; + 7
= 3ar-14.
Multiply by 12 ; then
14a?+10-15a; + 18 = 8-5a;
4* =-20,
a; = ~5.
11. s + 4 x~* = 2 | 3s-l
2 3
Multiply by 6 ; then
15a;-21-4a;-14 = 18a;-84,
7a; = 49,
a? -7.
3 5 " ' 15
Multiply by 15 ; then
5a; + 20- 3a; +12
= 30+3a;-l,
- x - - 3,
a?-3.
12. 3a; + 5 2a; + 7 iq_3jc^0
7 3 5
Multiply by 105 ; then
45a- + 75 _ 70a;- 245 + 1050 - 63a; - 0,
-88 a; = -880,
* = 10.
teachers' edition. 119
13. }(3s-4)+£(5aH-3) = 43-5a;. 14. }(27-2a?)-f— ^(7*-54).
Multiply by 21 ; then Multiply by 10 ; then
Pa; - 12 + 35s + 21 135 - 10a? = 45 - 7x + 54,
= 903 -105s, -3* = -36,
149* = 894, x = 12.
s = 6.
15. 5a;-{8aj-3[16-6a;-(4-5a;)]} = 6,
5a;-{8a;- 3 [16 -6a;- 4 + 5a;]} = 6,
5 x - {8 x - 48 + 18 x + 12 - 15 a;} - 6,
5a? - 8a; + 48 - 18a; - 12 + 15a; - 6,
~6a; = -30,
a; = 5.
16. 5a;-3_9zi = 5j 19( 4)
7 3 2 6 v '
Multiply by 42 ; then
30a> - 18 - 126 + 14a; = 105 a; + 133 a; - 532,
-194 a? = -388,
a? = 2.
l7 2a? + 7 9a?-8_a?-ll lg 8a?-15 lla?-1^7a? + 2
' 7 11 2 '3 7 " 13
Multiply by 154 ; then Multiply by 273 ; then
44a. + 154 _ 126a; + 112 728a; - 1365 - 429a; + 39
= 77a; -847, = 147 a; + 42,
- 159a; = - 1113, 152a; - 1368,
*-7. a = 9.
1Q 7a;+9 3a; + 1 __ 9a;- 13 249 -9a;
™' —* 7 4 14~
Multiply by 56 ; then
49a; + 63-24a;-8 = 126a;-182-996 + 36a;,
-137 a; = -1233,
a? = 9.
120
ALGEBRA.
Exercise LXIV.
x 9* + 20 4(*-3) *
' 36 " 5*-4 4
Multiply by 36 ; then
ox— 4
144(x-3) .gp
5a; — 4
144* -432 = 100* -80,
44 a; = 352,
* = 8.
« 9(2*-3) 11a? -1 9 a; +11
14 3* + l = 7
Multiply by 14 ; then
154 x -14
3* + l
49.
18*-27 + -
- 18a: + 22,
154* -14^
3x + l
Divide by 7,
22a?-2_H
3* + l
22*-2 = 21x + 7,
x = 9.
10a; + 17 12* + 2_5*-4
18 13a;- 16 9
Multiply by 18 ; then
216 a; + 36
10a; + 17-
= 10* -8,
216 a; + 36
13a;- 16
13a; -16
= 25,
325 a; -400 = 216* +36,
109* = 436,
* = 4.
6*+ 13 3a? + 5 ^2*
15 5a; -25 5
Multiply by 15 ; then
8« + 13-£*±??-8*
5*-25
13,
45* + 75_
5*-25 "
45 a; + 75 = 65* -325,
-20* = -400,
* = 20.
5.
18* -22 ^0^1 + 16*
+ 4S +
■*A-
101-64*
39-6* 24 l* 24
Reduce the mixed number to an improper fraction,
18*-22 ,o-.l + 16a? 53 101-64*
12 24
3(13-2*) ' 24
Multiply by 24 ; then
8(18* -22) 4ga, 1 16a; = 106_ 101 + 64a.
13-2*
144* - 176 d
13-2*
144* -176 = 52 -8*.
152* = 228,
* = 1J.
teachers' edition.
121
6_5x 7-23*
l + 3s 10x-ll 1
21 30 105*
15 14(x-l)
Multiply by 210 ; then
84-70x- 105 ~ *0x* = 10 + 30x- 70a; + 77+2,
x-1
105-30x»
. 30a; + 5,
x-1
- 105 + 30 a? = 30 x1 - 25 a; - 5,
25 a;- 100,
x = 4.
7 9x+5 8a;-7^36a;+15 41,
14 6X+2" 56 56*
Multiply by 56 ; then
36x + 20+224*-196
3a; + 1
= 36 a; + 15 + 41,
224 a; -196 ™
3x + l =36'
224 a; -196 = 108 a? +36,
116 a; = 232,
a; = 2.
9 6a? + l 2x-4
15 7x-16 =
2x-l
Multiply by 15 ; then
ft^u.1 30 a; -60 ~ «
6aJ + 1-7^i6- = 6aJ-3'
30 a; -60
7a; -16
- 30x + 60 = -
-2x = 4,
x = -2,
28 a; + 64,
6x+7 2x-2^_2x + l
15 7x-6 5
Multiply by 15 ; then
6, + 7-^^=6, + 3,
7a; — 6
30a; -30^ ^
7x-6
-30x + 30 = -28x + 24,
-2x = -6,
x = 3.
10.
7x-6 x-5
35 6a; -101 5
Multiply by 35 ; then
7x-6-35*-175 = 7x.
6x-101
Transpose, and clear of frac-
tions,
-35 a; + 175 = 36a; -606,
-71 x = -781,
x =11.
Exercise LXV.
1. ax + bc = bx + ae%
ax—bx = ac — bc1
x(a — b)^c(a~ b),
x-c.
2a — ex = 3 c— 56x,
5bx — cx = 3 c — 2a,
x(56-c)=3c-2a,
3c-2a
x = •
56-c
122 ALGEBRA.
3. a*x + bx — c = b*x + cx— dt
a*x — b2x + bx — cx = c — dt
x(a2 — b* + b — c) = c — d,
— c~d
a* _ &» + b - c
4. — aci + b*c + obex = abc -f cmx — ac*x + b*c — mc,
obex — cmx + ac*x = abc + b*c — mc+ac* — 6*c,
x(abc — cm + ac*) = abc — mc + ac2,
x = l.
5. (a + a; + 6)(a + b - a) « (a + x) (b - x) - abt
— x2 + a2 + 2a& + 62 = ab -f 6x — ax — x2 — ab,
ax — 6x = - a2 — 2ab — 6*.
ax - bx = -(a2 + 2ab + 62),
a-6
6. (a2 + x)2 = x2 + 4a2+a4, q affls+s8) , ax2
a* + 2a2x + x2 = x2 + 4a2 + a*l * te -«*+y
2a2x = 4a2, ... -
x = 2 Divide by a ; then
62s+x» ^ , i»
— frx"^'
7. (^-.rXa2+x) = (a*+2ax-x2)f Uxxltl^ ** **•
a* -x2 - a* + 2ax~ x2, 62x + x8 = frcx2 + x8,
- 2 ax = 0, 62x = fccx2,
*-0. x = 0or_.
ax — b x-\- ac
■ + a =
c c
Multiply by c ; then
ax — 6 -f ac = x 4- ac.
ax — x = 6,
•(a-l)-J,
a — 1 2a + 6
3a -bx 1
aa- _ — =_.
2 2
Multiply by 2,
2ax — 3 a + 6x =
2ax + 6x =
x(2a + 6) =
1,
3a + 1
3a + 1
x =
3a+l
teachers' edition. 123
11 a„ 4as — 26__ ab+x V*—x x—b ab—x
18a-4ax + 26 = 3aj, a86 -f a2x — b* + bx
-3s- 4ax = -18a~26, - b2x - 68 - a*b + a*x,
*(3 + 4a) = 2(9a + b), -b*x + bx = - a*b -a86,
_2(9a + 6) 6x(6-l) = 2a86,
3 + 4a ' Xr_ 2a8
6-1
12.
a—x 2x a
/-a-ax + * = 2*-db, 17. ax - ^±2 _ ^^H
— aa> = — a6 + a,
x x
jP-aft — 1. arc2 — 6a; — l = aa^ — a,
_ 6x = — a -f 1,
1Q 3 ab—x2 ±x — ac « = —-—.
lo. = &
c bx ex
3 bx — abc + ex2 = 4 6a? — a6c,
ex2 = 6x, -^ __
■ b 18. -^L+a + ^-O.
a? = Oor-. b-cx c
acaP + abc—aePx + abx—acx2
= 0.
14. am- b - y + - = 0. Divide by a,
a6m'-6'm-amlC + 6a; = 0> ^ + 6c-A + te-rf-0,
6x — a7nar = b2m — a6m*
6*771 — a6m*
a? = —
6 — am
— bm.
be
c*-6
15.
19. ^-fc + rf + L
3as-26 oa;-a_aa; 2 * *
36 26 6 3' . o6-to + <fo + l,
— 6ca? — ax = — a6 + 1,
6 ax— 4 6—3 aa;-f 3 a = 6 oa;— 4 6,
— 3aa:=— 3a,
a6-l
x=l. be + d
20. «Ml±^) = ac + ^.
ax a
ad2 + ax* = acda? + ox*,
aedx = ad2,
d
c
124
ALGEBRA.
x-5 1
s-3
4(s-l) 6(*-l) " 9
Clear of fractions,
9x- 27 = 6a; -30+ 4s -4,
_« — 7,
*=7.
Exercise LXVI.
A 1
1
x-1
2(x-3) 3(x-2) (*-2X*-3)
Clear of fractions,
3*-6-2x + 6 = 6x-6,
-5x = -6,
x=l*.
x + _*_ _ («-2)(g + 4)
x — 1 x+1
Clear of fractions,
x*-x+x*+x = xN-a^-lOx+S,
10a; = 8,
g , 2(2g+3) J> 5s+l
9(7-*) = 7-x 4(7-*)'
Clear of fractions,
252-36x-16x-24
= 216 -45 a; -9,
-7x=-21,
*=3.
7 _^6s+l 3(l+2x»)
a; — 1 x + 1 x* — l
Clear of fractions,
7x+7 = 6 **+*-6 *-l-3-6 a*,
12*= -11,
6. -iL-.4 = 5(21 + 2*)-10.
x + Z 3x + 9
Clear of fractions,
51-12x-36
= 105+ 10a; -30*- 90,
8a; = 0,
* = 0.
7.
*-7_^2x-15 1
a; + 7=* 2x-6 2(x + 7)
Clear of fractions,
2x*-20x + 42 = 2a? -x- 105-s + 3,
-18x = -144,
x-8.
8.
x + 4 ,, 3x + 8
3x + 5 * 2x + 3
Clear of fractions,
12*« + 66* + 72 + 36 *, + 114*+90 + 6x, + 19s + 15
= 54x* + 134x + 240,
-35* -63, ^
*»-lf \
teachers' edition. 125
9 132£+1 8aH-5 = 52 n 3x-l 4x-2_l
3x+l x-l ' 2x-l 3x-2 6*
Clear of fractions, Clear of fractions,
132x*-131x-l+24x*+23x+5 54 x*-54x+ 12-48 x»+48x-12
= 156x*-104x-52, = 6x*-7x+2,
-4x = -56, x = 2.
x = 14.
1 6 12.
3 x + l x*
10* o 5 + o = 5— To ' x-l X-l 1-3*'
2x — 3 x— 2 3x + 2
.„ 3 x + l _ — x2
Clear of fractions, or> x_\ "~ x_i "" a?— l'
6x*-8a;-8 + 6aj2-5a;-6 * ni ,r ..
= 12a? - 42* + 36, Clear of f™tlons.
29x = 50, 3aj + 3-aa-2a?-l = -a!1,
s=ljf " x = -2.
13 g~4 _ ft — 5 __ x — 7 _ x — 8
x — 5 x— 6 x — 8 x — 9
Then
(x-4)(x-6) (x-5)(x-5)_(x-7)(x-9) (x-8)(x-8)
(x-5)(x-6) (x-5)(x-6) (x-8)(x-9) (x-8)(x-9)'
- 1 _ - 1
(x-5)(x-6) (x-8)(x-9)'
Clear of fractions,
-x2 + 17x-72 = -x2 + llx-30,
6x = 42,
«-7.
14. (x-a)(x-b) = (x-a-b)2,
tf-ax — bx + ab^x2- 2ax — 2bx + a2 + 2ab + 6\
ax + bx = a2 + ab + b2,
m a2+ab + b2
x = _
a+b
15. (a- b){x- c)-(b - c)(x-a)-(c - a)(x - b) = 0,
ax — bx — ac + 6c — bx + ex 4- ab — ac — ex + ax + be — ab = 0,
ax— bx — bx + ex — ex + ax = ac — be — ab + ac — be + ab,
2ax — 2bx = 2ac — 26c,
2x(a-6) = 2c(a-6),
x = c.
126 ALGEBRA.
16. ^-2+1 , ** + x + l-2x 17. 4 | 7 - 37
a;-l a; + l " ' " s + 2 a; + 3 aj* + 5a; + 6
Clear of fractions, Clear of fractions,
a* + 1 + a* - 1 - 2ar*- 2x, 4a; + 12 + lx + 14 = 37,
2a; = 0, 11* = 11,
a; = 0. » * 1.
18. (*+!)». *[6-(l-a0]-2,
(a;+l)2 = a;(6-l + a;)-2,
a? + 2a;+l = 6a;-a; + a?-2,
-3a; = -3,
*-l.
19. ^-i^ + ^ + H— JL + 5.
a; + 1 3a;+ 2 x + 1
Reduce the complex to simple fractions,
75 - x , 80a; 4- 21 23 . K
— ■ + 5.
20.
3(a; + l) 5(3a; + 2) x + l
Clear of fractions,
1115a; - 15s2 + 750 + 240a? + 303a; + 63
= 1035 a; + 690 + 225 x2 + 375 a; + 150,
8a; = 27,
* = 3}.
Sabc a262 {2a-\-b)bix = 3ca, , &£
a + 6 (a + 6)8 a(a+6)2 a
Clear of fractions,
3 a46c+6 a862c+3 a268c+a8&2+2 a262a;+3 a&8a;+&4a;
= 3 a4ca;+9 as6ez+9 a262ca;+3 aP'caj+a^aj+S cWg+SoPoH-ftc,
3 a4ca; + 9 a3te + 9 a262ca; + 3 ab*cx + a36a; + a2b2x
= 3 a46c + 6 a*b2c + 3 cWc + a862,
aa; (3 a8c + 9 a2bc + 9 ab2c + 3 63c + a2b + ab2)
= a26 (3 a2c + 6 a&c + 3 b2c + aft),
a;{3c(a + 6)3 + a&(a + &)}=a&{3c(a+&)2 + a&}.
a; = -•
a+b
21 -±-+_? 29 = ^— 22.' 5-J7-gV*-3aH4-5'>
a>-8 2x-16 24 3x-24 V2 */ 2 4
Clear of fractions, 5_??_i_2 = - — 3a?~ 4 +5 a;
96 + 36 -29a; + 232 = 16, 2 2 4
- 29 a; = - 348 , clear of fractions,
x =12.
20-14a;+8 = 2a?-3a+4-5a?J
-8 a; = -24,
x = 3.
teachers' edition. 127
1 3 1 - ar,
5 x-1 3
Multiply both terms of right member by l—x\ then
1 3 6-s
5 x-1 3(1- a)'
1 3 _ s-6
5 a?-l 3(a;-l)'
Clear of fractions,
3aj-3-45 = 5s-30,
-2a; = 18,
x = - 9.
24. x~i + «-f ,1 + I -
|(*-1) f(x + l) 15(l-I^
Reduce the complex to simple fractions,
2aj-3 , 2o-5 = 1 a?
3a;-3 dx + 5 15a?-15
Clear of fractions,
103s -5s -15 + 6a* -21 a; + 15 = 15a?-15 + a*
-26 a; = -15,
Exercise LXVII.
1. Find the number whose third and fourth parts added to-
gether make 14.
Let
x = the number.
Then
- = one-third of the number,
3
and
- = one-fourth of the number,
4
and
- + - = sum of the two parts.
3 4 r
But
14 = sum of the two parts.
•"
-.5 + 5-14. Whence, x = 24.
3 4
128 ALGEBRA.
2. Find the number whose third part exceeds its fourth part
by 14.
Let
x = the number.
Then
- = one- third of the number,
3
and
- = one-fourth of the numbe
4
and
X
3
- 7 = the excess.
4
But
14 = the excess.
.^-^H. Whence, x =168.
3 4
3. The half, fourth, and fifth of a certain number are together
equal to 76 ; find the number.
Let x — the number.
Then - = one-half of the number,
2
and - = one-fourth of the number,
4
- = one-fifth of the number,
5
- + - + -*= sum of the parts.
2 4 5
But 76 — sum of the parte.
...* + !+ 5 - 7& Whence, x - 80.
4. Find the number whose double exceeds its half by 12.
Let ' x — the number.
Then - = one-half the number,
2
and 2x = double the number,
2a; — - = the excess.
2
But 12 = the excess.
.-. 2x - 1 = 12. Whence, x - 8,
2
teachers' edition. 129
5. Divide 60 into two such parts that a seventh of one part
may be equal to an eighth of the other.
Let x = one part,
and 60 — x = the other part.
Then - = one-seventh of one part,
7
and — ^-^ = one-eighth of the other part,
o
60 - x _ x
"'* 8 7
Whence, re =28,
and 60-x=32.
6. Divide 50 into two such parts that a fourth of one part
increased by flve-sixths of the other part may be equal to 40.
Let x = the smaller part.
Then 50 — x — the larger part,
7 + J(50—») — J of one part increased by 4 of the other.
4
But 40 = J of one part increased by J of the other.
■'.f + 1(50-*) -40.
4
Whence, x — 2$,
and 50-a; = 47f
7. Divide 100 into two such parts that a fourth of one part
diminished by a third of the other part may be equal to 11.
Let x = one part.
Then 100 - x « the other.
jP^ = J of one part diminished by J of the other.
But 11 = J of one part diminished by J of the other.
• X 1Q0 ~ X r- 11
' "4 3
Whence, x =76,
and 100-a; = 24.
130 ALGEBRA.
8. The sum of the fourth, fifth, and sixth parts of a certain
number exceeds the half of the number by 112. What is the
number ?
Let x = the number.
Then - = one-half of the number,
2
- — one-fourth of the number,
4
\ — one-fifth of the number,
5
- = one-sixth of the number.
6
...? + fE + ? = ii2 + ?.
4 5 6 2
Whence, a? = 960.
9. The sum of two numbers is 5760, and their difference is
equal to one-third of the greater. What are the numbers ?
Let x =» the greater number.
Then 5760 — x — the smaller number,
a? -(5760- a;) = |-
.\ 3a; - 17,280 + 3x = x.
Whence, x = 3456,
and 5760- x =2304.
10. Divide 45 into two such parts that the first part divided
by 2 shall be equal to the second part multiplied by 2.
Let x = first number.
Then 45 — a: = second number,
- = first divided by 2,
2 J
90 — 2 a; = second multiplied by 2.
Then - = 90-2ar.
2
.-. x = 180- 4a;.
Whence, x = 36,
and 45 — x = 9.
TEACHERS* EDITION. 131
11. Find a number such that the sum of its fifth and its
seventh parts shall exceed the difference of its fourth and its
seventh parts by 99.
Let x = the number.
Then - = one- fifth of the number,
5
- = one-fourth of the number,
4
^ = one-seventh of the number,
? + ^ =» sum of i and \ of the number,
5 7
? _ ? = difference between J and \ of the number.
(-+-]—(-— -J138 the excess of the sum of its fourth and
* '/ \ '/ seventh parts over the difference of its
fourth and seventh parts.
But 99 = this excess.
-H)-(h)-w-
Whence, x «= 420.
12. In a mixture of wine and water, the wine was 25 gallons
more than half of the mixture, and the water 5 gallons less than
one-third of the mixture. How many gallons were there of
each?
Let x = number of gallons in mixture.
. Then - + 25 = number of gallons of wine,
- — 5 = number of gallons of water,
- + 25 + - — 5 = number of gallons in mixture.
It O
.• * + 25 + £-
2 3
Whence, x = 120,
and ^ + 25 = 85, £-5 = 35.
2 3
132 ALGEBRA.
13. In a certain weight of gunpowder the saltpetre was 6
pounds more than half of the weight, the sulphur 5 pounds less
than the third, and the charcoal 3 pounds less than the fourth of
the weight. How many pounds were there of each?
Let x = number of pounds in mixture.
Then - + 6 = number of pounds of saltpetre,
- — 5 = number of pounds of sulphur,
and --3 = number of pounds of charcoal.
4
2 3
Whence, z = 24, |+6 = 18, |-5 = 3, | — 3 = 3.
14. Divide 46 into two parts such that if one part be divided
by 7, and the other by 3, the sum of the quotients shall be 10.
Let x = first part.
Then 46 — x = second part,
and^46-*--
3 7
Whence, x = 18, and 46 - x « 28.
15. A house and garden cost $ 850, and five times the price of
the house was equal to twelve times the price of the garden.
What is the price of each?
Let x = number of dollars the house cost,
and 850 — x = number of dollars the garden cost.
Then 5& = five times cost of house,
10,200— 12 & = twelve times cost of garden.
.-.5 x =10,200 -12 x.
Whence, x = 600, and 850 - x = 250.
16. A man leaves the half of his property to his wife, a sixth
to each of his two children, a twelfth to his brother, and the
remainder, amounting to #600, to his sister. What was the
amount of his property?
teachers' edition. 13S
Let x — number of dollars the property amounted to
Then - = number of dollars left to wife,
2
- = number of dollars left to each child,
6
^- = number of dollars left to brother.
12
* + * + * + JL+600 = number of dollars in all.
2 6 6 12
But x = number of dollars in all.
...! + § + § + JL + 600 = *.
2 6 6 12
Whence, x = 7200.
17. The sum of two numbers is a and their difference is b ;
find the numbers.
Let x = the smaller number.
Then x + b = the larger number,
2 x + b = the sum of the numbers.
But a = the sum of the numbers.
.•. 2x + b = a.
Whence, rr = ^, andz + & = ^±i.
18. Find two numbers of which the sum is 70, such that the
first divided by the second gives 2 as a quotient and 1 as a
remainder.
Let x = first number,
and 70 — x = second number.
Then -^1 = 2.
70 -a?
Whence, x = 47, and 70 - x = 23.
19. Find two numbers of which the difference is 25, such that
the second divided by the first gives 4 as a quotient and 4 as a
remainder.
Let x = smaller number.
Then x + 25 = larger number,
x + 25 4
= 4 + -
x x
Whence, x =7, and x + 25 = 32.
134 ALGEBRA.
20. Divide the number 208 into two parts such that the sum
of the fourth of the greater and the third of the smaller is less
by 4 than four times the difference of the two parts.
Let x = the greater part.
Then 208 — x = the smaller part,
- + — ^ = sum of \ the greater and £ the small""".
4 3
x — (208 — x) =- difference of the parts.
...£ + 20^-? + 4 = 4(iE_208+a!).
4 o
Whence, x = 112, and 208 - x = 96.
21. Find four consecutive numbers whose sum is 82.
Let x = first number.
Then x -f 1 = second number,
x+ 2 = third number,
x + 3 = fourth number.
Then x + x + l + x + 2 + x+3*= sum of the numbers.
But 82 = sum of the numbers.
.-. x + x + 1 + x + 2 + x + 3 = 82.
Whence, s=19, s + l = 20, s + 2 = 21, s + 3 = 22.
22. A is 72 years old, and B's age is two-thirds of A's. How
long is it since A was five times as old as B?
Let " x = number of years since A's age was five times
that of B.
J of 72 = 48 = B's age at present,
72 — x = A's age x years since,
48 — x = B's age x years since.
Then 72 - x = 5 (48 - a).
Whence, a; = 42.
23. A mother is 70 years old, her daughter is half that age.
How long is it since the mother was three and one-third times as
old as the daughter?
Let x = number of years since.
Then 70 — x = mother's age x years since,
35 — x = daughter's age x years since.
.-. 70-s = 3J(35-3).
Whence, x = 20.
24. A father is three times as old as the son ; four years ago
the father was four times as old as the son then was. What is
the age of each?
teachers' edition. 135
Let x — number of years in son's age.
Then 3x = number of years in father's age,
x — 4 = number of years in son's age 4 years since,
3 a; — 4 = number of years in father's age 4 years since.
.-. 3s- 4 = 4a; -16. Whence, x = 12, and 3a; = 36.
25. A is twice as old as B, and seven years ago their united
ages amounted to as many years as now represent the age of A.
Find the ages of A and B.
Let x = number of years in B's age.
Then 2x = number of years in A's age,
x — 7 = number of years in B's age 7 years since,
2x — 7 = number of years in A's age 7 years since.
.\ a;- 7+ 2a;- 7 = 2a?. Whence, a; =14, and 2a; = 28.
26. The sum of the ages of a father and son is half what it
will be In 25 years ; the difference is one-third what the sum will
be in 20 years. What is the age of each?
Let x — number of years in father's age.
Then 50 — x = number of years in son's age,
x — (50 — x) — difference of their ages.
But (x + 20)+(50-*)+20 _ difference of their ageg
. „ CA , _ x + 20 + 60- x + 20
a . X ~~ tJ\J "T X — '
Whence, x = 40, and 50 - x = 10.
27. A can do a piece of work In 5 days, B in 6 days, and C in
7\ days ; in what time will they do it, all working together?
Let x — number of days required for A, B, and C,
together.
Then - = part all can do in one day.
But - = part A can do in one day,
5
- = part B can do in one day,
2
— = part C can do in one day.
15
112
Then- + - + — = what all can do in one day.
But - = what all can do in one day.
a?
.•.i + I + A = l. Whence, x = 2.
5 6 15 x
136 ALGEBRA.
28. A can do a piece of work in 2\ days, B in 3} days, and C
in 3 J days ; in what time will they do it, all working together?
Lot x ■--- number of days required for A, B, and C,
together.
Then - = part they can do in one day.
Now — = part A can do in one day,
—- = part B can do in one day,
-— = part C can do in one day.
Then ~- + — + -- = part all can do in one day.
2y 3£ 3J
But - = part all can do in one day.
x 2^3J^3|
Whence, x = l^y.
29. Two men who can separately do a piece of work in 15
days and 16 days, can, with the help of another, do it in 6 days.
How long would it take the third man to do it alone?
Let x = number of days required for third man.
- + — 4- — - = part all can do in one day.
x 15 16 r J
But - = part all can do in one day.
.-. i + — + -i = i- Whence, x = 26J.
x 15 16 6 ' *
30. A can do half as much work as B, B can do half as much
as C, and together they can complete a piece of work in 24 days.
In what time can each alone complete the work?
Let x •-= number of days C works.
Then 2x = number of days B works,
4 x = number of days A works.
Then - + — -\ = part all can do in one day.
x 2x 4a:
But — = part all can do in one day.
.-. - + — + i = — . Whence, s= 42, 2x = 84, and 4x - 168.
x 2x 4tx 21 '
teachers' edition. 137
31. A does $ of a piece of work in 10 days, when B comes to
help him, and they finish the work in 3 days more. How long
would it have taken B alone to do the whole work?
Let x = number of days required for B.
Then - = part B can do in one day,
— - = part A can do in one day,
- — part left to be finished,
9 r
14 4
- of - or — =» part both can do in one day.
3 9 27 r J
But -r + - = part both can do in one day.
18 x
...±+I = ±
18 a 27 #
Whence, x = 10}.
32. A and B together con reap a field in 12 hours, A and C in
16 hours, and A by himself in 20 hours. In what time can B and C
together reap it? In what time can A, B, and C together reap it?
— = part A and B can do together in one hour,
and — = part A can do in one hour;
20 r
.: or — = part B can do in one hour,
12 20 30 r
— = part A and C can do together in one hour.
16
.-. or — = part C can do in one hour.
16 20 80 v
Let - = part A, B, and G can do together in one hour.
Then ' I-JL + 2. + JL
x 20 30 80
Whence, x =* 10$}. B and C together in 21T\ hours.
138 ALGEBRA.
33. A and B together can do a piece of work in 12 days, A and
C in 15 days, B and C in 20 days. In what time can they do it,
all working together?
Let x — number of days required working together.
— = part A and B do in one day,
— « part A and C do in one day,
— = part B and C do in one day.
Then — + — + — = part-all do in two days.
2
But - = part all do in two days.
...Ll + i + JL
x 12 15 20
Whence, x = 10.
34. A tank can be filled by two pipes in 24 minutes and 30
minutes respectively, and emptied by a third in 20 minutes. In
what time will it be filled if all three are running together?
Let x = number of minutes required for all running
together,
- =» part filled by all in one minute,
— = part filled by first in one minute,
24
— = part filled by second in one minute,
30 l J
— = part emptied by third in one minute,
— + = part filled by all in one minute.
24 30 20 * J
But _ = part filled by all in one minute.
. i = j, J J_
" x 24 30 20*
Whence, x = 40.
teachers' edition. 139
35. A tank can be filled in 15 minutes by two pipes, A and B,
running together. After A has been running by itself for 5 min-
utes, B is also turned on, and the tank is filled in 13 minutes
more. In what time may it be filled by each pipe separately?
Let x = number of minutes required for A.
Then - = part filled by A in one minute,
18
and — = part filled by A in eighteen minutes,
x
= part filled by B in one minute,
15 x
13 13
=> part filled by B in thirteen minutes.
15 x
.^ 18 + 13_13_1
x 15 x
Whence, x = 37J.
Therefore, it can be filled by A in 37£ minutes, and by B in
25 minutes.
36. A cistern could be filled by two pipes in 6 hours and 8
hours respectively, and could be emptied by a third in 12 hours.
In what time would the cistern be filled if the pipes were all run-
ning together?
Let x = number of hours required for all running
together,
- = part all can fill in one hour,
x
- = part filled by first pipe in one hour,
6
% - = part filled by second pipe in one hour,
8
— - = part emptied by third pipe in one hour.
La
Then-+ — — - =part filled by all in one hour.
6 8 12 r J
But - = part filled by all in one hour.
. M 1__J_
" x 6 8 12
Whence, x = 4$ .
140 ALGEBRA.
37. A tank can be filled by three pipes in 1 hour and 20 min-
utes, 3 hours and 20 minutes, and 5 hours, respectively. In
what time will the tank be filled when all three pipes are running
together?
Let x = number of minutes required for all to nil it,
— = part first will fill in one minute,
80 r
= part second will fill in one minute,
200 r
= part third will fill in one minute,
300 r
- = part all will fill in one minute.
x
x 80 200 300
Whence, x = 48.
38. If three pipes can fill a cistern in a, b, and c minutes,
respectively, in what time will it be filled by all three running
together?
Let x = number of minutes required for all.
Then - — part first fills in one minute,
I •
- = part second fills in one minute,
o
- = part third fills in one minute,
- + r + - = part all fill in one minute.
a b c
But - = part all fill in one minute.
x a b c
abc
x = -
Whence,
ab + ac + be
39. The capacity of a cistern is 755 \ gallons. The cistern
has three pipes, of which the first lets in 12 gallons in 3 J min-
utes, the second 15 } gallons in 2 \ minutes, the third 17 gallons
in 3 minutes. In what time will the cistern be filled by the three
pipes running together?
teachers' edition. 141
Let x = number of minutes required for all.
7551
Then — I = number of gallons let in per minute by all,
12
— = number of gallons let in per minute by first,
-£& = number of gallons let in per minute by second,
17
— - = number of gallons let in per minute by third,
12 15i 77
— - + -— j* 4- -— = number of gallons let in per minute by all.
7551
But — i == number of gallons let in per minute by all.
. 755} = 12 15j 17
" x 3J 2J 3 '
Whence, x=48}.
40. A sets out and travels at the rate of 7 miles in 5 hours.
Eight hours afterwards, B sets out from the same place, and
travels in the same direction at the rate of 5 miles in 3 hours.
In how many hours will B overtake A?
Let x = number of hours A is travelling.
Then x — 8 = number of hours B is travelling,
1| = number of miles per hour A is travelling,
if = number of miles per hour B is travelling,
If x = number of miles A travels,
1§ (x — 8) = number of miles B travels.
.•.lf*-lf(*-8).
ice, a: =50, x- 8 = 42.
Whence,
41. A person walks to the top of a mountain at the rate of 2 J
miles an hour, and down the same way at the rate of 3J miles an
hoar, and is out 5 hours. How far is it to the top of the moun-
tain?
Let x = number of hours required to go up,
and 5 — x = number of hours required to go down.
Then 2£x = distance up the mountain,
and 3J(5 — x) = distance down the mountain.
.-.21* -31(5-*).
Whence, x = 3, and 2Jjc = 7.
142 ALGEBRA.
42. A person has a hours at his disposal. How far may he
ride in a coach which travels b miles an hour, so as to return
home in time, walking back at the rate of c miles an hour?
Let x = number of miles he may go.
Then ^ = number of hours he is riding,
b
and - = number of hours he is walking.
• • T + - = a.
b c
Whence, x =
43. The distance between London and Edinburgh is 360 miles.
One traveller starts from Edinburgh and travels at the rate of 10
miles an hour ; another starts at the same time from London,
and travels at the rate of 8 miles an hour. How far from Lon-
don will they meet?
Let x = number of hours both travel.
Then 10 a; = number of miles first travels,
and 8x = number of miles second travels.
10 a; + Sx = number of miles both travel.
.-. 18x = 360.
Whence, x = 20, and 8 x = 160.
44. Two persons set out from the same place in opposite
directions. The rate of one of them per hour is a mile less than
double that of the other, and in 4 hours they are 32 miles apart
Determine their rates.
Let x = rate of second in miles.
Then 2x — 1 = rate of first in miles,
and 3 x — 1 = number of miles apart in one hour.
12a? — 4 = number of miles apart in four hours.
.-. 12rr-4 = 32.
Whence, a; = 3, and 2a;— 1 = 5.
45. In going a certain distance, a train travelling 35 miles an
hour takes 2 hours less than one travelling 25 miles an hour.
Determine the distance.
Let x = number of miles.
Then — = number of hours first was travelling,
and — = number of hours second was travelling.
.% £ + 2 = £. Whence, x = 175.
35 2o
TEACHERS EDITION.
143
46. At what time are the hands of a watch together :
I. Between 3 and 4?
II. Between 6 and 7?
III. Between 9 and 10?
I. Let CH and CM denote the positions of the hour and minute
hands at 3 o'clock, and CB the position of both hands when together.
Then arc HB = * of arc MHB.
Then x = number of minute-spaces in
arc MB,
Then — = number of minute-spaces in
12 arc HB,
and 15 = number of minute-spaces in
arc MH.
Now arc MB = arc MH + arc HB.
That is, x = 15 + ~
Whence,
'16*.
II. Let CM and CH denote the positions of hour and minute
hands at 6 o'clock, CB the position of both when together.
Then arc HB = * of arc MHB.
Let a; — number of minute-spaces in
arc MHB.
Then — = number of minute-spaces in
12 arc#£,
and 30 = number of minute-spaces in
arc MH
Now arc MHB = arc MH+ arc HB.
That is,
Whence,
3 = 30 +
x = 32*.
12
III. Let BC and BA denote the positions of the hour and minute
hands at 9 o'clock, and BD the position of both hands when together.
Then CD = * of arc AECD.
Let x = number of minute-spaces in
arc AECD.
Then — = number of minute-spaces in
12 arcCA
and 45 = number of minute-spaces in
arc AEC
Now arc AECD = arc AEC + arc CD.
That is,
Whence,
x = 45 + -
3 = 49*
12
144
ALGEBRA.
47. At what time are the hands of a watch at right angles:
I. Between 3 and 4?
II. Between 4 and 5?
III. Between 7 and 8?
I. Let CB and CD denote the positions of the hour and minute
hands when at right angles.
Let x = number of minute-spaces in
arc MHBD.
Then — = number of minute-spaces in
12 arcJra,
15 = number of minute-spaces in
arc IflT,
15 = number of minute-spaces in
arc BD.
Now arc MHBD = arcs MH+HB+BD.
That is,
Whence,
x = 15+^- + 15.
x = S2^.
II. Let CE and DE denote the positions of the hour and minute
hands when at right angles.
Let x = number of minute-spaces in
arc ABCD.
Then 4:= number of minute-spaces in
12 arc BQ
and 20 = number of minute-spaces in
arc AB,
also 15 = number of minute-spaces in
arc CD.
Now arc ABCD = arcs BC+AB+CD.
That is,
Whence,
3= 20 + -£-+15.
x = 38^.
12
Let
Then
and
x *= number of minute-spaces in
arc AB.
— = number of minute-spaces in
12 arcCA
20 = number of minute-spaces in
arc ABC,
also 15 = number of minute-spaces in
arc BCD.
Now arc AB = arcs CD + AC- BD.
That is,
Whence,
x = ^-+20-15.
12
teachers' edition.
145
III. Let 2?Cand DC denote the positions of the hour and
hands when at right angles.
Let x = number of minute-spaces in
arc MHBD.
Then — = number of minute-spaces in
12 arc HB,
and 35 = number of minute-spaces in
arc MAH,
also 15 = number of minute-spaces in
arc BD.
Now arc MHBD = arcs MAH+HB+BD.
That is, x - 35 + — + 15.
12
Whence, x = 54^.
Let x = number of minute-spaces in
arc MB.
Then — = number of minute-spaces in
12 arcffl),
and 35 = number of minute-spaces in
arc MBH%
also 15 = number of minute-spaces in
arc BHD.
Now arc MB
= arcs HBM+ HD - BHD.
That is, x = 35 + -£ - 15.
minute
Whence, x = 21^,
12
48. At what time are the hands of a watch opposite to each
other:
I. Between 1 and 2?
II. Between 4 and 5?
III. Between 8 and 9?
I. Let CB and CD denote the positions of the hour and minute
hands when opposite.
Let x = number of minute-spaces in
arc MHBD.
Then -- = number of minute-spaces in
12 arc HB,
and 5 = number of minute-spaces in
arc MS,
also 30 = number of minute-spaces in
arc BAD.
Now arc MHBD
- arcs MH+ HB + BAD.
That is,
Whence,
.5 + ^ + 30.
■38*
146
ALGEBRA.
II. Let CB and CD denote the positions of the hour and minute
hands when opposite.
Let x = number of minute-spaces in
arc MHBD.
Then — = number of minute-spaces in
12 arc HB,
and 20 = number of minute spaces in
arc MH,
also 30 = number of minute-spaces in
arc BAD.
Now arc MHBD
= arcs MH+ HB + BAD.
That is, x = 20 + -| + 30.
Whence, x = 54^.
III. Let CB and CD denote the positions of the hour and minute
hands when opposite.
Let x = number of minute-spaces in
arc MD.
Then — = number of minute-spaces in
12 arc HB,
and 40 «= number of minute-spaces in
arc MDH%
also 30 = number of minute-spaces in
arc DHB.
Now arc MD - arcs MDH+HB-DHB.
That is,
Whence,
x = 40+^--30.
12
and I
49. It is between 2 and 3 o'clock ; but a person looking at his
watch, and mistaking the hour-hand for the minute-hand, fancies
that the time of day is 55 minutes earlier than it really is. What
is the true time?
Let CB and CD denote the positions of the hour and minute hands
id CE the 1 o'clock point.
Let a; »number of minute-spaces in
arc MED.
Then — = number of minute-spaces in
12 arc HB,
and 10 = number of minute-spaces in
arc MEDH, -
also 5 = number of minute-spaces in
arc DHB.
Now arc MED
= arcs MEDH+ HB - DHB.
Thatis,x=10 + ^--5.
teachers' edition. 147
50. A hare takes 6 leaps to a dog's 5, and 7 of the dog's leaps
are equivalent to 9 of the hare's. The hare has a start of 50 of
her own leaps. How many leaps will the hare take before she is
caught?
Let 6 a; = number of leaps taken by the hare.
Then 5 a; = number of leaps taken by the dog.
Also let a = number of feet in one leap of the hare.
Then -~ = number of feet in one leap of the dog.
... f—\5x = (50 + 6a?) a,
MM = 50a+6aa;,
7
45aa; = 350a + 42oaj.
Divide by a, 3 a; =350.
Whence, x = 116$,
6s = 700.
51. A greyhound makes 3 leaps while a hare makes 4 ; but 2
of the greyhound's leaps are equivalent to 3 of the hare's. The
hare has a start of 50 of the greyhound's leaps. How many leaps
does each take before the hare is caught?
Let 3a; = number of leaps taken by the greyhound.
Then 4 a; — number of leaps taken by the hare.
Also let a = number of feet in one leap of the hare.
o -
Then — = number of feet in one leap of the greyhound.
A
That is, 3 a; X -£■ = the whole distance.
150a
But v 4 ax = the whole distance.
2
9 ax 150a . A„„
2 2
Divide by a, i* = 1^ + 43;,
9a; = 150 + 8a;.
Whence, x = 150,
3a; = 450,
4 a; = 600.
148 ALGEBRA.
52. A greyhound makes two leaps while a hare makes 3 ; but
1 leap of the greyhound is equivalent to 2 of the hare's. The
hare has a start of 80 of her own leaps. How many leaps will
the hare take before she is caught?
Let 2x = number of leaps taken by the greyhound.
Then « 3x = number of leaps taken by the Hare.
Also let a = number of feet in one leap of the hare.
Then 2 a = number of feet in one leap of the greyhound.
That is, 2x X 2 a = whole distance.
But (80 + 'Sx) a = whole distance.
.-. (80 + 3a:)a = 4aa:.
Divide by a, 80 + Sx = 4x. Whence, x = 80, and 3 a; = 240.
53. A rectangle whose length is 5 feet more than its breadth
would have its area increased by 22 feet if its length and breadth
were each made a foot more. Find its dimensions.
Let x — number of feet in breadth.
Then x + 5 = number of feet in length.
x(x + 5) — number of square feet in area,
x + 1 = number of feet in breadth + 1,
x + 6 = number of feet in length + 1.
.-. {x + l)(a: + 6) - x(x + 5) = 22.
Whence, x = 8, and x + 5 = 13.
54. A rectangle has its length and breadth respectively 5 feet
longer and 3 feet shorter than the side of the equivalent square.
Find its area.
Le^ x — 3 = number of feet in breadth,
and x + 5 = number of feet in length.
Then (x— 3) (a; + 5} = number of feet in area.
But ar = number of feet in area.
.-. a* = x* + 2x - 15. Whence, x - 7£, and x* = 56J.
55. The length of a rectangle is an inch less than double its
breadth ; and when a strip 3 inches wide is cut off all round,
the area is diminished by 210 inches. Find the size of the rect-
angle at first.
Let x = number of inches in breadth.
Then 2x — 1 = number of inches in length,
and 6a;+12a: — 6— 36 = number of inches in area cut off.
But 210 = number of inches in area cut off.
.-. 6<c+12<c-6-36 = 210. Whence, x - 14, and 2«-l = 27.
teachers' edition. 149
56. The length of a floor exceeds the breadth by 4 feet ; if
each dimension were increased by 1 foot, the area of the room
would be increased by 27 square feet. Find its dimensions.
Let x = number of feet in breadth.
Then x + 4 = number of feet in length,
and a? + 4 x = number of feet in area,
x + 1 = number of feet in breadth + 1 foot,
x + 5 = number of feet in length + 1 foot,
x* + 6 a? + 5 = number of feet in area after addition.
But x2 + 4a; H- 27 = number of feet in area after addition.
.-. s2 + 6x + 5 = z2 + 4a; + 27. Whence, a; = 11, and a; + 4 = 15.
57. A mass of tin and lead weighing 180 pounds loses 21
pounds when weighed in water ; aud it is known that 87 pounds
of tin lose 5 pounds, and 23 pounds of lead lose 2 pounds, when
weighed in water. How many pounds of tin and of lead in the
mass?
Let x = number of pounds of tin.
Then 180 — x = number of pounds of lead,
— «= number of pounds x pounds of tin lose in
'37 water,
— (180 — a;) = number of pounds 180 — a; pounds of lead
23 lose in water.
But 21 = number of pounds tin and lead lose in water.
...*L? + JL(i80-aO = 21.
37 23 v '
Whence, x = 111, and 180 - x = 69.
58. If 19 pounds of gold lose 1 pound, and 10 pounds of silver
lose 1 pouud, when weighed in water, find the amount of each
in a mass of gold and silver weighing 106 pounds in air and 99
pounds in water.
Let x = number of pounds of gold.
Then 106 — x = number of pounds of silver,
-^- = number of pounds the gold loses in water,
= number of pounds the silver loses in water.
10 l
1 ^~ = number of pounds both lose in water.
But 7 = number of pounds both lose in water.
. x 106-a;_7
"19 10
Whence, x = 76, and 106 - x = 30.
150 ALGEBRA.
59. Fifteen sovereigns should weigh 77 pennyweights ; but a
parcel of light sovereigns, having been weighed and counted,
was found to contain 9 more than was supposed from the weight;
and it appeared that 21 of these coins weighed the same as 20
true sovereigns. How many were there all together ?
Let x = number in parcel,
77
— *= number pennyweights a good sovereign weighs,
15
x — 9 = number good sovereigns that weigh same as bad,
\~ ' = number pennyweights the good coins weigh,
15
on 77 44
— X — or —- =* number pennyweights a bad coin weighs.
2t\ 15 9
. 44s_77(s-9)
"9 15
Whence, x = 189.
60. There are two silver cups, and one cover for both. The
first weighs 12 ounces, and with the cover weighs twice as much
as the other without it ; but the second with the cover weighs
one-third more than the first without it. Find the weight of the
cover.
Let x = weight of cover in ounces,
12 + x = weight of first cover and cup in ounces,
2(16 — x) = double the weight of the second cup in ounces.
But 12 + x = double the weight of the second cup in ounces.
.-. 12 + x=2(16-x).
Whence, x — 6 j.
61. A man wishes to enclose a circular piece of ground with
palisades, and finds that if he sets them a foot apart he will have
too few by 150 ; but if he sets them a yard apart he will have too
many by 70. What is the circuit of the piece of ground?
Let x = number of feet in circuit of field.
Then a? —150 = number of palisades he had.
But - + 70 = number of palisades he had.
.-. <c-150 = | + 70.
Whence, x = 330.
teachers' edition. 151
62. A horse was sold at a loss for $ 200 ; but if it had been
sold for $250, the gain would have been three-fourths of the loss
when sold for $ 200. Find the value of the horse.
Let x = number of dollars the horse is worth.
Then 250 — x = number of dollars made if sold for $ 250,
re -200 = number of dollars lost if sold for $200.
.-. 250-a; = t(x-200).
Whence, x = 228f
63. A and B shoot by turns at a target. A puts 7 bullets out
of 12, and B 9 out of 12, into the centre. Between them they
put in 32 bullets. How many shots did each fire?
Let x = number of shots each fired,
-~ = number of centres made by A,
— = number of centres made by B.
12 J
But 32 = number of centres made by both.
... ^ + £? = 32.
12 12
Whence, x = 24.
64. A boy buys a number of apples at the rate of 5 for 2 pence.
He sells half of them at 2 a penny and the rest at 3 a penny, and
clears a penny by the transaction. How many does he buy?
Let x = number bought.
Then — - = number of pence paid,
5
_ -I x
and - X - or - = selling price of one-half.
£t & 4
But - X - or - = selling price of the other half.
2 3 6
■G+5)-
2x *
. 1.
5
Whence, x = 60.
152 ALGEBRA.
66. A person bought a piece of land for $6750, of which he
kept $ for himself. At the cost of $250 he made a road which
took tV °f tne remainder, and then sold the rest at 12} cents a
square yard more than double the price it cost him, thus clearing
his outlay and $500 besides. How much land did he buy, and
what was the cost-price per yard?
Let x = number of yards.
Then — - =- number of yards kept,
y
— = number of yards used for road,
- = number of yards sold.
.-. 6750 + -x-= 7500.
8 2
Whence, x = 12,000,
and $6750.00 -s- x = $0.56J.
66. A boy who runs at the rate of 12 yards per second starts
20 yards behind another whose rate is 10} yards per second.
How soon will the first boy be 10 yards ahead of the second?
Let x = number of seconds they are running.
Then 12 a; = number of yards first boy runs,
01 r
and — - = number of yards second boy runs.
12s-(l0 + ^\ = 20,
12jc_20 + 21*
2
24x-20-21z = 40,
3a: = 60,
z = 20.
67. A merchant adds yearly to his capital one-third of it, but
takes from it, at the end of each year, $5000 for expenses. At
the end of the third year, after deducting the last $5000, he has
twice his original capital. How much had he at first?
teachers' edition. 153
Let x = number of dollars he had at first.
4-rr
Then — — 5000 = number of dollars he had at the end of
3 the first year,
or g~~ ' = number of dollars he had at the end of
3 the first year,
4/4s-15,000\ 5(XX) = number of dollar8 he had at the end of
H * J foe second year,
or — ^~~Z — ! = numDer of dollars he had at the end of
9 the second year,
4/16s-105,000\ _5000== number of dollarB he had at the end of
3\ 9 / the third year,
or — x~ ' = number of dollars he had at the end of
27 the third year.
But 2x = number of dollars he had at the end of
the third year.
64 s -555,000 „
.*. : = Ax.
27
Whence, x = 55,500.
68. A shepherd -lost a number of sheep equal to one-fourth of
his flock and one-fourth of a sheep ; then, he lost a number equal
to one-third of what he had left and one-third of a sheep ; Anally,
he lost a number equal to one-half of what now remained and
one-half a sheep, after which he had but 25 sheep left. How
many had he at first?
Let x = number of sheep he had at first.
q r 1
Then — - — = number of sheep he had left after first loss,
'^/T ' = number of sheep he lost the second time,
~ = number of sheep he had left after second loss,
r 4- 1
— - — = number of sheep he lost the third time,
4
x~~ = number of sheep he had left after third loss.
But 25 = number of sheep he had left after third loss.
4
Whence, x = 103.
154 ALGEBRA.
69. A trader maintained himself for three years at an expense
of $ 250 a year ; and each year increased that part of his stock
which was not so expended by one-third of it. At the end of
the third year his original stock was doubled. What was his
original stock?
Let x = number of dollars in stock at first.
Then ^(s-250)
or a"~ = number of dollars in stock at the end of
3 first year,
|/4«-1000_2BO\
or — 2Lz = number of dollars in stock at the end of
second year,
-("
3\ 9
4 /16 a: - 7000 _ ^X = number of dollar8 in gtock at the end of
third year.
But 2x — number of dollars in stock at the end of
third year.
■ •■f(16*-7000-250)-2s.
Whence, x = 3700.
70. A cask contains 12 gallons of wine and 18 gallons of
water ; another cask contains 9 gallons of wine and 3 gallons
of water. How many gallons must be drawn from each cask to
produce a mixture containing 7 gallons of wine and 7 gallons of
water?
Let x = number of gallons drawn from 1st cask,
14 — x = number of gallons drawn from 2d cask,
2
- = proportion of wine to water in 1st cask,
o
- = proportion of wine to water in 2d cask.
.-. ^ + !(14-*) = 7.
Whence, x = 10,
and 14 — x = 4.
teachers' edition. 155
71. The members of a club subscribe each as many dollars as
there are members. If there had been 12 more members, the
subscription from each would have been $ 10 less, to amount to
the same sum. How many members were there?
Let x = number of members of the club.
Then x = number of dollars each subscribed,
x + 12 = number of members + 12,
and x — 10 = number of dollars each would have subscribed
in second case.
But x2 = number of dollars all subscribed.
.-. (x + 12)(x-10) = x*.
Whence, x = 60.
72. A number of troops being formed into a solid square, it
was found there were 60 men over ; but when formed in a col-
umn with 5 men more in front than before and three men less in
depth, there was lacking one man to complete it. Find the
number of troops.
Let x = number of men on one side.
Then x2 + 60 = number of men in the square,
x + 5 = number of men on a side + 5,
x — 3 = number of men on a side — 3,
and (x + 5)(x — 3) — 1 = number of men in the square.
.-. (x + 5)(x-S)-l = x* + 60.
Whence, x = 38,
and x2 + 60 = 1504.
73. An officer can form the men of his regiment into a hollow
square 12 deep. The number of men in the regiment is 1296.
Find the number of men in the front of the hollow square.
Let x = number of men in front.
Then 12 x = number of men in twelve lines,
and 24 x = number of men in twelve lines front and rear.
12 (x — 24) = number of men on a side,
12 (x — 24) x 2 = number of men on both sides.
Then 24 x + 12 (x - 24) X 2 = whole number of men.
But 1296 = whole number of men.
.\ 24x + 12(x-24)x2 = 1296.
Whence, x = 39.
156 ALGEBRA.
74. A person starts from P and walks towards Q at the rate
of 8 miles an hour ; 20 minutes later another person starts from
Q and walks towards F at the rate of four miles an hour. The
distance from P to Q is 20 miles. How far from P will they
meet?
Let x — number of miles first travels.
Then 20 — x = number of miles second travels,
- =s number of hours first travels.
3
— ^— =■■ number of hours second travels.
4
^. x^M-x 1 whence, x = 9f .
3 4 3 »7
75. A person engaged to work a days on these conditions :
for each day he worked he was to receive b cents, and for each
day he was idle he was to forfeit c cents. At the end of a days
he received d cents. How many days was he idle?
Let x = number of days he was idle.
Then a — x = number of days he worked,
and ex = number of cents he forfeited,
b (a — x) = number of cents he received,
(rib — bx) — cx = whole amount.
But d = whole amount.
. \ (rib — bx) — cx = d.
Whence, x = ab~d
6 + c
76. A banker has two kinds of coins : it takes a pieces of the
first to make a dollar, and b pieces of the second to make a dol-
lar. A person wishes to obtain c pieces for a dollar. How many
pieces of each kind must the banker give him?
Let x = number of pieces of first kind.
Then c — x = number of pieces of second kind,
- = the part of a dollar in one piece of first,
a
- = the part of a dollar in one piece of second.
b
. x c — x = -.
a b
Whence, i^4 and c-x = S(e ~a).
b —a b — a
TEACHERS EDITION.
157
Exercise LXVIII.
3
(1)
(2)
2a;+3y = 7
4a;-5y = 3
Multiply (1) by 2,
4z + 6y = 14
(2) is 4a;-5y = 3
Subtract, lly = ll
.-. y = l
Substitute value of y in (2),
4a;-5 = 3.
.\x=*V.
a;-2y = 4
2a;- y = 5
Multiply (1) by 2,
2a;-4y = 8
(2) is 2s- y = 5
Subtract, - 3y = 3
.-.y = -l.
Substitute value of y in (2),
x + 2 = 4.
.-. a; = 2.
7a; + 2y = 30
-3a; + y= 2
(l)is 7a; + 2y = 30
(2) by 2, -6a; + 2y = 4
Subtract, 13 a; = 26
.-. x = 2.
Substitute value of a; in (1),
14 + 2y - 30,
2y = 16.
.vy = 8.
3a;-5y = 51 (1)
2a; + 7y = 3 (2)
Multiply (1) by 2, and (2) by 3,
6a;-10y = 102
6a; + 21y« 9
Subtract, -31y= 93
.-.y 3.
Substitute value of y in (1),
Sx + 15 = 51,
3a; = 30.
.*•£*» 12.
8
5. 5a;+4y = 5S (1)
3a; + 7y = 67 (2)
Multiply (1) by 3, and (2) by 5,
15a;+12y = 174
15a? + 35y = 335
Subtract, -23y=-U>l
.-. y = 7.
Substitute value of y in (1),
5a; + 28 = 58.
.-. a; = 6.
6. 3a; + 2y = 39 (1)
3y-2a;=13 (2)
Multiply (1) by 3, and (2) by 2,
9a; + 6y = 117
-4a; + 6y= 26
Subtract, 13 a; =91
.\*-7.
Substitute value of x in (1),
21 + 2y = 39, 2y = 18.
.-.y = 9.
7. 3a;-4y = -5 (1)
4x-5y = l (2)
Multiply (1) by 4 and (2) by 3,
N12a;-16y = -20
12s -15y- 3
Subtract, - y = - 23
.-.y = 23.
Substitute value of y in (1),
3a; -92 = -5, 3a; = 87.
.-. a; = 29.
8. lla; + 3y = 100 (1)
4a;-7y = 4 (2)
Multiply (1) by 4 and (2) by 11,
44a; + 12y = 400
44a;-77y = 44
Subtract, 89y = 356
.-. y = 4.
Substitute value of y in (1),
11a; + 12 =100, 11 x = 88.
.-. x = 8.
158
ALGEBRA.
* + 49y =
49*+ y =
093(1)
357(2)
Add, 50a; + 50y = 1050(3)
Divide by 50, * + y = 21 (4)
Subtract (4) from (1),
48y = 672.
.-. y = 14.
Subtract (4) from (2),
48* = 336.
.-. * = 7.
12. 2x-7y= 8 (1)
-9* + 4y=19 (2)
Multiply (1) by 4 and (2) by 7,
8x-28y = 32 (3)
-63* + 28y = 133 (4)
Add,- 55 x =165
.\ x = -3.
Substitute value of x in (2),
27 + 4y=19,
4y = -8.
10. 17* + 3y= 57(1)
-3* + 16y = 23(2)
Multiply(l) by 3 and (2) by 1 7,
51*+ 9y = 171
-51* + 272y = 391
Add, 281y = 562
.-.y = 2.
Substitute value of y in (2),
3a; + 32 = 23,
-3s = -9.
.-. * = 3.
13. 69y-17* = 103 (1)
14*-13y = -41 (2)
Multiply (1) by 14, and (2)
by 17,
-238 a; + 966 y = 1442(3)
238*-221y = -697(4)
Add 745y = 745
.-.y = i.
Substitute value of y in (2),
14s -13 = -41,
14s = -28.
.-. * = -2.
11. 12* + 7y = 176 (1)
3y-19* = 3 (2)
Multiply (1) by 3 and (2) by 7,
36* + 21y = 528 (3)
-133* + 21y= 21
Subt., 169* =507
.\ * = 3.
Substitute value of * in (2),
3y-57 = 3,
3y = 60.
.-.y = 20.
14. 17* + 30y = 59 (1)
19*+28y = 77 (2)
Multiply (1) by 14, and (2)
by 15,
238* + 420y= 826(3)
285* + 420y= 1155(4)
-47* =-329
.-. *=7.
Substitute value of * in (2),
133 + 28y = 77,
28y = -56.
.-.y = -2.
TEACHERS EDITION.
159
Exercise LXIX.
3a?-4y = 2
7a?-9y = 7
Transpose — 4y in (1),
3x=2 + 4y.
Divide by coefficient of a?,
3
Substitute value of a? in (2),.
i^y°>-
■ 7.
Simplify,
14+28y-27y = 21.
.•■y-7.
Substitute value of y in (1),
3a;- 28 = 2,
3* = 30.
.-.a; = 10.
(1) 3. 3 a; + 2y = 32 (1)
(2) 20a;-3y = l (2)
Transpose 2y in (1),
3x = 32-3y.
Divide by coefficient of a?,
x=™-2y.
3
Substitute value of x in (2),
20(M^_3y = l,
3 *
Simplify,
640-40y-9y = 3,
-49y = -637.
.-.y = 13.
Substitute value of y in (1),
3a? +26 = 32.
.:x = 2.
7a;-5y = 24 (1)
4a;-3y = ll (2)
Transpose by in (1),
7a; = 24 + 5y.
Divide by coefficient of x,
Substitute value of x in (2),
4(M±S«)-s,-ii.
Simplify,
96 + 20y-21y = 77,
-y = _19.
.-.y = 19.
Substitute value of y in (1),
7* -95 = 24, '
7a? = 119.
.-.a? =17.
4. lla?-7y = 37 (1]
8a?+9y = 41 (2;
Transpose 7y in (1),
llx = 37+7y.
Divide by coefficient of x,
x-37 + 7y
11
Substitute value of x in (2),
8(^i^)4 9y = 41.
Simplify,
296 + 56y + 99y = 451,
155y = 155.
.-.J-l.
Substitute value of y in (2),
83 + 9 = 41.
.-. a? =-4.
160
ALGEBRA.
5. 7*+ 5y = 60 (1)
13a:-lly = 10 (2)
Transpose 5y in (1),
7& = 60-5y.
Divide by coefficient of x,
60 -5y
Substitute value of x in (2),
IS^-^-lly-lQ,
Simplify,
780-65y-77y = 70,
780-142y = 70,
142y=710.
...y = 5.
Substitute value of y in (1),
7x + 25 = 60,
7a; =-36.
.-.*=» 5.
7. 10*+ 9y = 290 (1)
12z-lly=130 (2)
Transpose 9y in (1),
10s = 290-9y.
Divide by coefficient of ar,
j,_290-9.y
10
Substitute value of x in (2),
12/290-9y\ lly = 130
Simplify,
3480 -108y-110y = 1300,
218y = 2180.
.-.y = 10.
Substitute value of y in (1),
10a? + 90 = 290,
10* = 200.
.-. x = 20.
6. 6s-7y = 42 (1)
7z-6y = 75 (2)
Transpose 7y in (1),
6x = 42 + 7y.
Divide by coefficient of x,
^12 + 7^.
6
Substitute value of x in (2),
Simplify,
294 + 49y-36y = 450,
13y = 156.
.-.y=12.
Substitute value of y in (1),
6* -84 = 42.
.\*-21.
8. 3s-4y = 18 (1)
3s + 2y = 0 (2)
Transpose 4y in (1),
3* = 18+ 4y.
Divide by coefficient of xt
o
Substitute value of x in (2),
Simplify,
54 + 12y + 6y = 0,
18y = -54.
.-.y = -3.
Substitute value of y in (2),
3s-6 = 0.
.\x-2.
TEACHERS EDITION.
161
9. 9a;-5y = 52
(1)
11. 9y-7z = 13 (1)
8y-3z = 8
(2)
15a?-7y = 9 (2)
Transpose 5y in (1),
Transpose — 7 a; in (1),
9a: = 52 + 5y.
9y=13 + 7a;.
Divide by coefficient of
*.
Divide by coefficient of y,
• 52 4
x = — -
S
,.. 13+ 7a:
y~ 9 *
Substitute value of x in
(2).
Substitute value of y in (2),
8y 3(5-H^) = 8-
15* 7(13 + 7*W
Simplify,
Simplify,
72y-156-15y = 72,
135 a; -91 -49a; = 81,
57y - 228.
86 a; = 172.
.-.y-4.
.-.a; = 2.
Substitute value of y in
(1).
Substitute value of x in (1),
9a? -20 =52,
9y-14 = 13.
9* =72.
.-.y-3.
.-.a; = 8.
10. 5a?-3y = 4
(1)
12. 5a;-2y = 51 (1)
12y-7a; = 10
(2)
19a;-3y = 180 (2)
Transpose — 3y in (1),
Transpose 2y in (1),
5a; = 4 + 3y.
5a; = 51 + 2y.
Divide by coefficient of
X,
Divide by coefficient of x,
* 5
*y
- 51 + 2y
—6
Substitute value of x in
(2),
Substitute value of x in (2),
12y 7(*+»jrt_10.
19pLt2jrt_8y-l80.
Simplify,
Simplify,
60y-28-21y = 50,
969 + 38y-15y = 900,
39y = 78.
23y = -69.
.-.y = 2.
.-.y = -3.
Substitute value of y in
(1),
Substitute value of y in (1),
5s- 6 = 4,
5* + 6-51,
5a? = 10.
5a; = 45.
.-.a: -2.
.-. x = 9.
162
ALGEBRA.
13. 4s + 9y=106 (1)
8x + 17y=198 (2)
Transpose 9y in (1),
4s=106-9y.
Divide by coefficient of x,
«_"»-»y.
Substitute value of x in (2),
8/106-9y\ + 17y=198
Simplify,
212 -18y + 17y = 198.
.-.y = 14.
Substitute value of y in (1),
43 + 126=106,
4s = -20.
.•. x = — 5.
14. 8x + 3y=3 (1)
12s + 9y = 3 (2)
Transpose 3y in (1),
8x = 3-3y.
Divide by coefficient of x,
8
Substitute value of x in (2),
12(^)+9y=3-
Simplify,
9-9y + 18y = 6,
9y = -3.
Substitute value of y in (1),
8s-l = 3,
8x = 4.
Exercise LXX.
1. a; + 15y - 53 (1)
3*+ y=-27 (2)
Transpose 15 y in (1) and y
in (5),
x = 53-15y(3)
3x = 27-y (4)
Divide (4) by 3,
x =
27 -y
Equate values of a,
27-y
3
(5)
(6)
2. 4*+ 9y = 51 (1)
8x-I3y = 9 (2)
Transpose 9y in (1), and— 13j/
in (fe),
4x = 51-9y (3)
8x = 9 + 13y (4)
Divide (3) by 4 and (4) by 8,
,-61-9y(5)
53 - 15y =
Reduce,
159 -45y = 27-y,
44y = 132.
.-.y-3.
Substitute value of y in (2),
3* + 3 = 27,
3x=-24.
.\ x = 8.
._9+13y
(6)
Equate values of x,
51-9y_9 + 13y m
4 8 W
Reduce,
102-18y = 9 + 13y.
.-.y = 3.
Substitute value of y in (1),
4x + 27-51.
.-. x-6.
TEACHERS' EDITION.
163
3. 4a; + 3y = 48 (1)
5. 5.*- 7y«33 (1)
5y-3a: = 22 (2)
lla;+12y=100 (2)
Transpose 3y in (1) and 5y
Transpose —7y in (1) and 12y
ln(2)' 4a; = 48-3y (3)
3z = 5y-22 (4)
in (2)» 5x = 33 + 7y (3)
llx = 100-12y(4)
Divide (3) by 4 and (4) by 3,
Divide (3) by 5 and (4) by 1 1,
, = 4^,
g-33 + *y,
a^-y-22.
,-100-i2y..
3
11
Equate values of x,
Equate values of x,
48-3y_5y-22
33 + 7y 100 -12y
4 3
5 11
Reduce,
Reduce,
144-9y = 20y-88,
- 29y = - 232.
.-. y = 8.
363 + 77y = 500-60y,
147y = 147.
.-.y = i.V. -'''
Substitute value of y in (1),
Substitute value of y in (1),
4* + 24 - 48,
5a;- 7 = 33,
4a? = 24.
5a; = 40.
.-.a; = 6.
.-. *-8.
L 2a: + 3y = 43 (1)
6. 5a; + 7y = 43 (1)
10a:- y = 7 (2)
lla: + 9y = G9 (2)
Transpose 3y in (1) and y in
Transpose 9y in (2) and 7y
(2)' 2a; = 43-3y (3)
lnW. 5x = 43-7y (3)
lla; = 69-9y (4)
10a? = 7 + y (4)
Divide (3) by 2 and (4) by 10,
Divide (3) by 5 and (4) by 11,
- 43 -3y
..H=U
«_*±*.
*-69-9.y.
10
n
Equate values of a;,
Equate values of x,
43 -3y 7+v
43-7.y G9-9y
2 10
5 11
Reduce,
Reduce,
215-15y = 7+y,
- 16y = - 208.
473-77y = 345-45y,
- 32y = - 128.
.-.y^l3.
.-. y = 4.
Substitute value of y in (1),
Substitute value of y in (1),
2x + 39 - 43.
5x + 28 = 43.
.-. x -2.
.\* = 3.
164
ALGEBRA.
7. 8x-21y = 33 (1)
6x + 35y=177 (2)
Transpose 21 y in (1) and 35y
in (2), 8x = 33 + 21y (3)
6*=177-35y (4)
Divide (3) by 8, and (4) by 6,
33 + 21y
X~ 8 '
x m -35y
6
Equate values of x,
33 + 21y_177-35y
8 6
Reduce,
99 + 63y=708-140y,
203y = 609.
-■-J -3.
Substitute value of y in (1),
801-63 = 33,
8a? = 96.
.\x = 12.
2ly + 20s=165 (1)
77y-30a; = 295 (2)
Transpose 20 s in (1) and 30*
in(2)' 21 y = 165 -20a? (3)
77y = 295 + 30a; (4)
Divide(3)by 21 and(4)by77,
= 165 - 20a:
Vs
21
295 + 30a;
77
Equate values of x,
165 -20s^ 295 + 30a;
21 77
Reduce,
1815- 220 a; = 885 + 90x,
-310a; = -930.
.-.a; = 3.
Substitute value of ar in (1),
21y + 60 = 165,
21y = 105.
8. 3y-7a; = 4
2y + 5a;=22
• Transpose 7 a; in (1) and
in<2)> 3y = 4 + 7s
2y = 22-5a;
Divide (3) by 3 and (4) by
y 3
22 -5a;
5a;
(3)
(4)
0
8
Equate values of y,
4+ 7a; ^22 -5a
3 2
Reduce,
8 + 14a; = 66 -15a;,
29x = 58.
.\ a; = 2.
Substitute value of x in (1),
3y-14 = 4.
y,.y = 6.
10. llx-10y = 14
5a; + 7y = 41
Transpose — 10 y in (1) and 7y
in(2)' lla; = 10y + 14 (3)
5a; = 41-7y (4)
Divide (3) by 11 and (4) by 5,
,-10y + 14,
5
Equate values of a;,
10y + 14_^41-7y
11 5
Reduce,
50y+70 = 451-77y,
127y = 381.
...y = 3.
Substitute value of y in (1),
lla;-30=14.
.-.a; = 4.
TEACHERS EDITION.
165
11. 7y-3a;=139 (1)
2a; + 5y = 91 (2)
Transpose 7y in (1) and 5y
™ (2), 3x = >7y-i$9 (3)
2a; = 91-5y (4)
Divide (3) by 3 and (4) by 2,
„ 7.y - 139
x = — * >
3
„. 91 -5y
Equate values of a;,
ly - 139 _ 91 - by
3 2
Reduce,
14y-278 = 273-15y,
29y - 551.
.-.y = 19.
Substitute value of y in (4),
2s = 91 -95,
2s = -4.
.\ x = -2.
13.
24* + 7y = 27
8a;-33y = 115
(1)
(2)
12. 17a; + 12y = 59
19a: -4y =153
s
Transpose 7y in (1) and 33 y
m W 24 a; = 27 -7y (3)
Sx = 115 + 33y (4)
Divide (3) by 24 and (4) by 8,
24
115 + 33v
8
Equate values of x,
27-7y_115 + 33y
24 8
Reduce,
27 - 7y - 345 + 99y,
- 106y = 318.
Substitute value of y in (3),
24a: = 27 + 21,
24 a; = 48.
.-. x = 2.
Transpose 12y in (1) and 4y +*
in(2), I7a; = 59_i2y (3)
19ar=153 + 4y (4)
Divide (3) bv 17 and (4) by 19,
", = 59-12y|
17
x_163+4£
19
Equate values of x,
153 + 4y ^ 59 - 12y
19 17
Reduce,
2«01+68y=1121-228y,
296y = - 1480.
.-. y = -5.
Substitute value of y in (1),
17a:-60 = 59.
.\»-7.
a; = 3y-19 (1)
y = 3a?-23 (2)
Transpose 3 x and y in (2),
3a;=23+y (3)
Divide (3) by 3,
*=23±2.
3
Equate values of x,
3y-19 = 23±i/.
y 3
Reduce,
9y-57 = 23 + y,
8y = 80.
.-.y = 10.
Substitute value of y in (1),
a; = 30-19.
.VJC-11.
166 ALGEBRA.
Exercise
LXXI.
1. z(y + 7) = v(*+l)
2x + 20 = 3y + l
$
3 2 _ 3
a: + 3 y-2
(1)
Simplify (1),
5(* + 3) = 3(y-2)+2(2)
xy + 7x = sy + y.
Simplify (1),
Transpose and combine,
2y-4 = 3a: + 9.
7x-y = 0
(3)
Transpose and combine,
Transpose and combine (2),
2y-3x = 13
(3)
2a:-3y = -19
(4)
Simplify (2),
Multiply (3) by 3,
5a+15 = 3y-6 + 2.
21x-3y = 0
Transpose and combine,
(4) is 2s-3y = -19
5x-3y = -19
(4)
Subt., 19 x =19
.-. x = l.
Multiply (3) by 3 and (4) by 2,
6y- 9a: = 39
Substitute value of x in (3),
-6y + 10x 38
7-y = 0.
Add, x=l.
Substitute value of x in
(3),
2y - 3 = 13.
.-.y-8.
2. 2a;-2^l3_4 = 0
5
(1)
3y + ^-9 = 0
(2)
4. ^r:i_2Mj = o
5 10
(1)
Simplify (1),
£ + 3tJ-3
6 4
(2)
10x-y + 3-20 = 0.
Transpose and combine,
Simplify (1),
10x-y = 17
(3)
2x-8-y-2 = 0,
Simplify (2),
2x-y = 10
(3)
9y + x-2-27 = 0.
Simplify (2),
Transpose and combine,
2x + 3y-6 = 36,
x + 9y = 29
(4)
2x + 3y = 42
«
Multiply (3) by 9,
Subtract (4) from (3),
90a: - 9y = 153
2s- y= -10
(4) is ar + 9y= 29
2x + 3y= 42
Add, 91a: =182
-4y 32
.-. x = 2.
.-.y-8.
Substitute value of x in (4),
Substitute value of y in
(3),
2 + 9y = 29.
2a -8 -10.
.-.y-3.
.-. z-9.
teachers' edition. 167
(* + l)(y + 2)-(* + 2)(y + l)--l (1)
3(a;+3)-4(y + 4) = -8 (2)
Simplify, (1), xy +y + 2x + 2 — xy — 2y — x — 2 = — 1.
Combine, x — y = — 1 (3)
Simplify (2), 3x + 9 - 4y - 16 = - 8.
Transpose and unite, 3 a — 4y = — 1 (4)
Multiply (3) by 3, 3s-3y = -3
Subtract,
Substitute value of y in (3),
Simplify (1),
Transpose and combine,
Simplify (2),
Transpose and combine,
Multiply (3) by 3,
Multiply (4) by 8,
Add,
Substitute value of y in (3),
-y- 2
.-.y — 2.
X+2 = -\.
.-. s = -3.
e x — 2 10-a; y-10
' 5 3 4
(1)
2y + 4 2x + y x + 13
3 8 4
(2)
12* - 24 - 200 + 20x - 15y - 150.
32s - 15y = 74
(3)
16y + 32 - 6a? - 3y = 6x + 78.
-12a> + 13y = 46
(4)
96a;- 45y=>222
-96a + 104y = 368
59y = 590
.-.y-10.
32a: -150 =74,
32a; =224.
.\a>-7.
7 x + 1 y + 2 2(x-y)
' 3 4 5
(1)
4 3 *
(2)
20a; + 20 - 15y - 30 = 24a; - 24y.
-4a; + 9y=10
(3)
3a; - 9 - 4y + 12 = 24y - 12x.
15a;-28y = -3
W
60a;-135y = -150
60a;-112y = - 12
Simplify (1),
Transpose and combine,
Simplify (2).
Transpose and combine,
Multiply (3) by 15,
Multiply (4) by 4,
Subtract, -23y = -138
.-. y = 6.
Substitute value of y in (3), — 4 a; + 54 = 10,
* -4a; = -44.
.\*-ll.
168 ALGEBRA.
8.
3x-
2^ + 5x-3^ = x + 1 (1)
5 3
2s-3y+4s-3y==y + 1 (2)
o A
Simplify (1), 9x-6y + 25x-15y = 15x+ 15.
Transpose and combine, 19 x — 21 y = 15 (3)
Simplify (2), 4x-6y + 12x-9y = 6y 4- 6.
Transpose and combine, 16 a: — 21 y = 6 (4)
Subtract (3) from (4), 19x-21y=15
-3x =-9
.\ x = 3.
Substitute value of x in (4), 48 — 21 y = 6,
-21y = -42.
.-.y = 2.
9.
2x-y + 3 x-2y+3_d ,.*
3 4 W
3x-4y + 3 { 4x-2y-9_1 ^
Simplify (1), 8 x - 4y + 12 - 3 * + 6y -- 9 = 48.
Transpose and combine, 5x + 2y = 45 (3)
Simplify (2), 9x - 12y + 9 + 16x - 8y- 36 = 48.
Transpose and combine, 25 x — 20 y = 75 (4)
Divide (4) by 5, 5x - 4y = 15
(3) is 5x + 2y - 45
Subtract, -6y = -30
.-. y = 5.
Substitute value of y in (3), 5x + 10 = 45,
5x = 35.
.-. x = 7.
10. li^ = ily+4A (l)
4Jx = Jy-21A (2)
Simplify (1), 18 x - 16y = 53 (3)
Simplify (2), 54 x- 4y = -259 (4)
Multiply (3) by 3, 54x-48y- 159
(4) is 54x- 4y = -259
Subtract, - 44y - 418
.-.y = -9J.
Substitute value of y in (3), 18 x + 152 = 53,
18x--99.
.-.x = -5J.
teachers' edition. 169
11. — — = * (1)
a; + 2y + 3 4a;-5y+6 '
3 19 (2)
6a;-5y + 4 3a; + 2y+l
Simplify (1), 55a;-59y = -87 (3)
Simplify (2), - 105 x + 101 y - 73 (4)
Transpose 59 y in (3) and 101 y in (4), and divide by 55 and 105
respectively,
x =
59y - 87
55 '
x =
^101y-73
105
59y-
55
87
_101y-73
105
Equate values of x,
Simplify, 1239y - 1827 - 11 1 1 y - 803,
128y = 1024.
.-.y-8.
Substitute value of y in (3), 55 a? — 472 = — 87,
55 a; = 385.
.*. x=* 7.
12. *±2«1§
y-x 8
(1)
9* 2*±ii-ioo •
7
(2)
Simplify (1),
Transpose and combine,
Simplify (2),
Transpose and combine,
8x + 8y = 15y-15a;.
23a;-7y = 0
63 a; -3y- 44 = 700.
63a;-3y = 741
(3)
(4)
Multiply (3) by 3,
Multiply (4) by 7,
Subtract,
69a>-21y=. 0
441x-21y = 5208
-372 a; =-5208
.-.a; =14.
Substitute value of x in (3),
322-7y = 0,
-7y--322.
.-. y=46.
170 ALGEBRA.
13.
3a?~5y + 3 = 2x+y (i)
Z o
Z__x-2y=ix y (2)
4 2 3
Simplify (1), 15a;- 25 y + 30 = 4a; + 2y.
Transpose and combine, 1 1 x — 27 y = — 30 (3)
Simplify (2), 96 - 3a: + 6y = 6a; + 4y.
Transpose and combine, — 9a; + 2y — — 96 (4)
Multiply (3) by 9, 99x - 243y - - 270
Multiply (4) by - 11, 99a;- 22y= 1056
Subtract, - 221 y = - 1326
.-. y = 6.
Substitute value of y in(4), - 9x + 12 = - 96.
.\x=12.
14 4a?-3y-7^3s 2y 5 m
5 10 15 6 W
3 2 20 15 6 10 w
Simplify (1), 24a; - 18y - 42 = 9a; - 4y - 25.
Transpose and combine, 15a;— 14y = 17 (3)
Simplify (2), 20y- 20 + 30a;-9y-60=:4y-4a; + 10a;+6.
Transpose and combine, 24a; + 7y = 86 (4)
Multiply (4) by 2, 48 x + 14y = 172 (5)
(3) is 15a;-14,y= 17
Add (3) and (5), 63 a; =189
.-. x = 3.
Substitute value of a; in (3), 45 — 14 y = 17.
.-.y = 2.
16. •ri-lgi (!)
I+V2=3 (2) !
Simplify (1), 2ar-8 = y + 2. I
Transpose and combine, 2x — y = 10 (3)
Simplify (2), 2x + 3y - '6 = 36.
Transpose and combine, 2a; + 3y = 42 (4)
2x- y = 10 (3)
Subtract, 4y = 32
.-.y = 8.
Substitute value of y in (4), 2a; + 24 = 42.
.-. s«9.
I
TEACHERS EDITION.
171
16. 3* + 12.v-9
(i)
1& 3x-0.25y = 28 (1)
11
0.12<r+0.7y = 2.54 (2)
l-3x_ll-3j
'(2)
Multiply (1) by 0 04,
7 5
0.12a; -0.01y= 1.12 (3)
Simplify (1) and (2),
0.12x+0.7 v = 2.54' (2)
3a; + 12y = 99
(3)
Subtract, — 0.71y = — 1.42
15a;-21y = -72
(4)
.-.y = 2.
Divide (3) by 3 and (4) by
15,
Substitute value of y in (1),
x-"-12
■V
3a: -0.5 =28.
3
.-. x = 9.5.
*--72 + !
21 y
15
Equate values of x,
99-12.7^-72+1
21 y.
3 15
Simplify,
495-60y = -72 + J
21 y.
-81y = -567.
.-.y-7.
Substitute value of y in (3),
3s + 84 = 99.
19. 7(*-l)-3(y+8)(l)
.-. x = 5.
4a; + 2 5y + 9 ((>)
9 " 2 W
Simplify (1) and (2),
17. 5a-i(5y + 2) = 32
(1)
7a;-7 = 3y + 24,
3y + J(* + 2) = 9
(2)
7a:-3y = 31 (3)
Simplify (1) and (2),
8* + 4-45y + 81,
20aj-5y = 130
(3)
8a:-45y = 77 (4)
a? + 9y = 25
(4)
Multiply (3) by 8 and (4) by 7,
Multiply (4) by 20,
56a- 24y = 248
20a: + 180y = 500
(5)
56a;-315y = 539
20a; - 5y = 130
(3)
Subtract, 291y = -291
.-.y — 1.
Subtract, 185y = 370
.:y = 2.
Substitute value of y in (3),
Substitute value of y in (3),
20 a?- 10 =* 130.
7* + 3-31.
t ** •? — I •
.\a?-4.
172
ALGEBRA.
20. 7s + J(2y + 4)=16
3y-J(a? + 2) = 8
Simplify (1),
35 s + 2y + 4 = 80.
Transpose and combine,
35a? + 2y = 76
Simplify (2),
12y-a;-2 = 32.
Transpose and combine,
12y-z=»34
. Multiply (4) by 35,
- 35 s + 420y = 1190
35a? + 2y= 76
422y = 1266
.-.y-3.
Substitute value of y in (3),
35a? + 6 = 76,
35 a; =70.
• \ a?= 2.
(1)
(2)
(3)
(4)
(3)
21. ^L=_^+3s = 4y-2 (1)
5s + 6y_3a;-2y ,
6 4 * W
Simplify (1),
5a? - 6y + 39a; = 52y-26.
Transpose and combine,
44a?-58y = -26 (3)
Simplify (2),
10a?+12y-9a;+6y = 24y-24.
Transpose and combine,
a;_6y = -24 (4)
Multiply (4) by 44,
44a; - 264y = - 1056
44a?- 58y = - 26 (3)
206y = 1030
.-.y = 5.
Substitute value of y in (4),
x _ 30 - - 24.
.-.a? =-6.
22.
5s-3 3a
2
2x+y
-19
2
9a;-
«4
_3jL
(1)
3(y + 3) 4a; + 5y ,9)
4 16 K"'
2 8
Simplify (1), 15 x - 9 - 9x + 57 - 24 - 6y + 2ar.
Simplify (2), 16a; + 8y-18a? + 14 = 12y + 36-4a?-5y.
Transpose and combine (1),
Transpose and combine (2),
Divide (1) by 2,
Substitute value of y in (4),
4x+6y = -24
(3)
2a; + y= 22
(4)
2a; + 3y = -12
2y = -34
.-.y = -17.
2a; -17 -22,
2a? = 39.
.-. a;=19J.
TEACHERS* EDITION. 173
23.
3y + ll = i*l=^L±^ + 3l-4a (1)
(a; + 7)(y-2) + 3 = 2z£-(y-l)(x + 1) (2)
Simplify (1), Zxy - 3 y2 + 12y + 11a— lly + 44
= 4a;2-zy-3ya + 31a;-31y + 124-4a? + 4.Ty-lG.T
Transpose and combine, 32y — 4a; = 80
Divide by 4, 8y-a; = 20
Simplify (2), xy + 7y — 2x — 14 + 3 = 2xy-xy— y + x + 1 (6)
Transpose and combine, 8y — 3a;=12
Subtract (5), 8y- a; = 20
-2a; = -8
.\a> = 4.
Substitute value of a; in (5), 8y — 4 = 20,
8y = 24.
.-.y-s.
24.
6a; + 9 3a? + 5y = «, 3a? + 4 ,,v
4 4a; — 6 2
8y + 7 6a;-3y , 4y -9 (2)
10 2y - 8 5 w
Multiply (l)by4, 6a? + 9 + 6* + *0V =13 + 6a; + 8.
Transpose and combine. — -* = 12.
^ 2a;-3
Divide both sides by (2), 3* + 5 / = 6.
2a; — 3
Multiply by 2a; - 3, 3a; + 5y = 12a;- 18.
Transpose and combine, — 9 a; + 5y = — 18 (3)
Multiply (2) by 10 8 y + 7 + S0x-]5y = 40 + 8y - 18.
y-4
Transpose and combine, — — * = 15.
y-4
Divide both sides by 15, ^~a ~ l'
Multiply by y — 4, z. x — y = y — 4.
Transpose and combine, 2ar— 2y= — 4.
Divide by 2, x -y = - 2 (4)
Multiply (3) by 1 and (4) by 9, - 9 x + 5y = - 18
9a;-9y = -18
Add, _4y = -36
.-. y = 9.
Substitute value of y in (4), x — 9 = — 2.
.-. a; = 7.
174 ALGEBRA.
26. s-2JLz* = 20-^2* (1)
23 -a; 'J v J
Multiply (1) by 2, 2x - -£=-^ - 40 - 59 + 2x.
23 — a;
Ay O «.
Transpose and combine, — * = 19,
Multiply by 23 - xt
Transpose and combine, 4y + 17 a? = 437 (3)
Multiply both sides of (2) by 3,
Transpose and combine,
Multiply by x - 18,
Transpose and combine, 3y — 17 a; = — 297 (4)
Add (3),
Substitute value of y in (3),
23-a;
4y-2* =
= 437 -19a;.
4y + 17a; =
= 437
"♦Si*-
= 90-73 + 3y.
3y-9_
a; -18
*i7.
3y-9 =
. 17a; -306.
3y-17a; =
4y + 17a; =
= -297
= 437
7y
. 140
20.
80 + 17a; =
17a; =
437,
357.
.*. x =
21.
Exercise LXXII.
1. x \-y = a
0)
x-y=>b
(2)
Add, 2a; = a + b
. x~a+h
2
Subtract (2) from (1),
2y =» a — b.
.:y-aZb-
(2)
2. ax + 6y = c
px + qy = r
Multiply (1) by p and (2) by a,
apx + 6py = cp (3)
apx •+ agy = ar (4)
Subt., y(bp — aq) = cp — ar.
6p — aq
Multiply (1) by q and (2) by b,
aqx + bqy = eg
fcpa; + bqy = br
Subt., (ag — bp) x =cq — br
. «. cq — br
ag — op
teachers' edition.
175
3.
mx + ny
px + qy.
Multiply (1) by p
mpx + npy
mpx + mqy
(1)
-* (2)
and (2) by m,
= ap (3)
= mb (4)
4.
ax+by = e
ax 4- cy— d
(1)
(2)
•y = ;
Sub. , (np — mq) y --= ap—mb
ap — mb
np—mq
and (2) by n,
= aq
= nb
Multiply (1) by q
mqx + nqy --
npx + nqy =
Sub., (mq — np)a; = o^ — nb
■ ■■ ;»-«?-"*.
mq — np
Multiply (1) by m',
Multiply (2) by m,
Subtract,
Multiply (1) by n',
Multiply (2) by n,
Add,
Multiply (1) by d,
Multiply (2) by a,
Subtract,
Multiply (1) by/,
Multiply (2) by b,
Subtract,
Subt., (b—c)y= t — d
b — c
Multiply (1) by c and (2) by bt
acx + bey =» ce
abx + bey = bd
Subt., (ac — ab)x = ce-bd
• x — M — bd
a(c-b)
6. maj — wy = r
m'a; + n'y = r1
mm'x — m'ny = mfr
mm9x + m n'y = m/
(m'n + m n')y = mr' — ra'r
•y-
7?17^ ■
m'n + mn'
mnfx — nn'y = n9r
mrnx + 7in;y = n/
(mw' + m'n) a: = n'r + n/
. ~ w;r + nr1
mnf + m'n
6.
ax + by = c
dx+fy = <?
adx -f 6dy = erf
acfe + g^ = ac2
bdy — afy^cd — ac2
- v-c(<*-«0
"y bd-af
a fa + tyy = cf
bdx + bfy = 6c»
(a/- 6c?) a; =
• cf-bc*
af-bd
(1)
(2)
(3)
(4)
(1)
(2)
176
ALGEBRA.
7.
a o
a)
(2)
6 a
Simplify (1), bx + ay = abc (3)
Simplify (2), ax + bey = 0 (4)
Multiply (3) by a and (4) by 6,
abx + c?y — a*bc
abx + IPcy — 0
Subt., a%y — b*cy = a2bc
(a* — 6*c)y = a26c,
a'-62c
Multiply (3) by 6c and (4) by a,
6*cx + a6cy = a6V
a's + a6cy = 0
Subt., b?cx — a2x — ab2c2
(6»c-a*)a; = a6V.
:x- a6a°2
9.
(1)
(2)
b + y 3a + x
as + 26y = d
Simplify (1),
3 a* + ax = 6* + by.
Transpose and combine,
ax- 6y = 6*-3a* (3)
(2) is ax + 2by = a*
Subt., - 3 by = b%- 3a*- </
"y 36
Multiply (3) by 2,
2aaj-26y = 26*-6a*
(2) is ax + 2by=*d
Add, 3a* = 26*-6a* + d
. 26»-6a» + rf
• • x — ^
3a
6*c-
abx + cay =
ax— cy =
Simplify (2),
a6aa; — 6cay *
Multiply (1) by 6,
a6*sc + 6c<fo =
(3) is abdx — bedy ■-
Add, (a62 + a6a> =
(1) 10.
■(2)
bd
d-6(3)
26 (4^
d-bh]
= 6+o*
b + d
or,a; = — ■
Multiply (1) by d,
a6o*a; + cd2y =
(3) is a6aa; — 6coy =
Add, (cd2 + bcd)y--
•••yj
or, y-
o6(6 + d)
1_
a6'
2d
d-b
= b + d
b + d t
~cd(b + dj
"cd
x
a + 6
x
1
a + 6
+ _2 :
a+6 a— 6 a-
Add (1) and (2),
(1)
(2)
1
a+6 a+6 a-6
Simplify,
2s(a-6) = 2a,
x(a — 6) = a.
a
.-. x = -•
a — 6
Subtract (1) from (2),
_2£_ = _1 L
a—b a—b a + ft
Simplify,
2y(a + 6) = 26,
y(a + 6) = 6.
•••y = aTV
teachers' edition. 177
11.
a(a — x) = b(x + y — a) (1)
a<y-ft-*)-%-6) (2)
Simplify (1), a2 - as = bx + 6y - afc. (3)
Simplify (2), ay — ab — ax = by — b2. (4)
Transpose a2 and for in (3), ax + bx =* a2 + ab — by (6)
Transpose ay — ab in (4), aa; — ay — ab — by + 6*. (6)
Divide (5) by (a + b) and (6) by o, x = ^^-fy,
a+ 6
__ay — ab — by -f 6*
a
Equate values of x, *+«»-fr . ay-ot-iy + y
a + 6 a
Simplify, a8 + a26 — a&y =» aty — a2b — 62y -f J3,
aty + a6y — U*y « os + 2a26 — 6s.
... y = a + 6.
Substitute value of y in (5), ax+bx=*a2 + ab — ab — b2.
.*. jc = a — 6. .
12.
^L±l a (1)
ar-y-1
ar + y-1
Simplify (1), x— y + \ = ax— ay — a.
Simplify (2), x + y + 1 = bx + by — b.
Trans, and combine, (a — 1) x — (a — 1) y = a + 1 (3)
(6-l)a> + (ft-l)y-& + l (4)
Multiply (3) by 6-1 and (4) by (a-1),
(a-l)(6-l)*-(a-l)(6-l)y-(a + l)(ft-l) (5)
(o-l)(6-l)a + (a-l)(6-l)y-(o-l)(6 + l) (6)
Add, 2{a-l)(b-l)x = 2(ab- 1)
aJ-1
(2)
(a-l)(ft-l)
Subtract (5) from (6), 2 (a - 1)(6 - l)y = 2(a - 6).
... y «z^
y (a-l)(*-l)
178 ALGEBRA.
13. o* = ty + ^±£ (i)
{a-b)x = (a + b)y (2)
Simplify (1), 2ox-2&y = a» + 6* (3)
Simplify (2), ax - bx - ay - by = 0 (4)
m (3). .-£±*±«Sr.
In (4), -1*rfL
Equate valaes of*, °!+*±2& _ 2JLt|jf.
Simplify,
a8 + ab* + 2a&y - a*b - ft8 - 2b2y = 2a*y + 2a6y.
Transpose and combine, 2a'y + 26*y = a8 — a*6 + a62 — 6s,
a-6
Substitute value of y in (1), ax * ^^ + £±£
14.
ax + by ~c*
a)
a
y a+ x
(2)
Simplify (2),
Add (1) and (3),
ax — by = - o* 4- 0*
ax + by = <?
ax — by = — a* + b*
(3)
2aa; = c2-a, + 6*
Subtract (3) from (1),
c^-a8*©*
•'•— 2a *
aa? + 6y = c*
oar — by = ~a* + o8
2&y = c8 + aa-68
* * y ox
teachers' edition. 179
16.
-4 + -*. =2a (1)
a + 6 a-b '
x — y^ x + y
2a6 a2 + 62
Clear (1) of fractions, ax — bx + ay + Ay = 2 a8 — 2 a62 (3)
Clear (2) of fractions, a2* +62sc — a2y — 6^ = 2 a6z + 2 aby (4)
2a8 — 2a62 — as + 6a?
(2)
In (3), y-
In(4), y-
Hence,
a+6
a'g— 2a6a; + b2x
a2 + 2a6 + 62
2as - 2a62 — ax + bx a?x - 2abx + b2x
a + b a2 + 2a6 + 62
2a4 - 2a262 - a*x + 2a86 - 206s + b*x = a2*- 2abx + 62x.
Transpose and combine, 2a?x — 2 abx = 2 a4 — 2 a262 4- 2 a86 - 2 a68.
Divide by 2 a, asc — bx = a8 — a62 + a26 — 6s
a8 - ao2 + a*b - ft8
a — b
or, a = a2 + 2a6 + 62.
Substitute value of a; in (3),
o8 + 2a26 + a62-a26-2a62-68+ay + 6y = 2a8-2a62.
Transpose and combine, ay -f by = a8 — a26 — 062 + 6s.
a8-a26-a62+68
a + 6
or, y = a* — 2ab + 62.
16.
bx
-6(5 =
-ay-
-ac
(1)
X
-y-
= a —
6
(2)
Transpose (1),
bx-
-«y =
.(&-
-a)c
Multiply (2) by a,
ax-
-«y =
.(«-
-b)a
(3)
Subtract,
(6-
a)a? =
■ c(b
-a)+a(6-
-a)
.•. a? =
= c + a.
bx-
-<*y =
= (b~
■a)c
(1)
Multiply (2) by 6,
bx -
-*y-
.(a-
-6)6
(4)
Subtract,
(b-
a)y =
-. y =
= c(6
= c +
-a) +
b.
6(6-
-a)
180 ALGEBRA.
17.
(1)
y-b
a{x - a) + 6(y - 6) + abc = 0 (2)
Simplify (1), x — cy = a — bc (3)
(2) is ax + by = a* + 6* — a6e (4)
Multiply (3) by a, ax — acy = a* — abc
Subtract, by + acy = 6*
...y— £_
b + ac
Multiply (3) by b and (4) by c, 6s — icy = ab — c6*
aca; + bey = a*c + 6*c — a6c*
Add, bx + acx = ab + a*c — abc1
abc1
,\ x = a r
ac+b
18.
(a + b)x -(a - b)y = 4a6 (I)
(a - 6)x + (a + 6)y - 2a* - 2b1 (2)
Multiply (1) by (a -6),
(a» _ b*) x - (a - b)1 y - 4 a*6 - 4 a6* (3)
Multiply (2) by (a + 6),
(a*-6*)x + (a + 6)*y = 2as-2a6* + 2a*6-26*(4)
Subtract (3) from (4), (2a* + 26*)y = 2a* - 2a*6 + 2a6* - 26s.
.-. y = a-6.
Multiply (1) by (a + 6) and (2) by (a - 6),
(a + 6)*x - (a* - 6*)y - 4a*6 + 4a6*
(a - ft^a, + (gi _ y) y = 2as - 2a*6 - 2a6» + 26s
Add, (2a* + 26*)* = 2a1 + 2a*6 + 2a6* + 26*
.\ * = a + 6.
19.
(» + o)(y + 6)-(x - a)(y - 6) - 2(a - 6)* m
Simplify (1) and (2), *~ y + 2(a-A) = 0 ®
ay 4- 6a; + ay 4- a6 — ay + ay 4- 6a? — ab = 2(a — 6)* (3)
x-y + 2a-26 = 0 (4)
Transpose and combine, 2 ay + 26a? = 2 a* — 4 a6 + 26*
x — y = 26 — 2a
Divide (5) by 2, ay + 6x = a* — 2a6 + 62
Multiply (6) by a, — ay + ax = 2a6 — 2a*
Add, (6 + a)x=6*-a*
. \ x = 6 — a.
Substitute value of x in (6), 6 — a — y = 26 — 2a.
.-. y = a-6.
TEACHERS EDITION.
181
20.
(a + ft)fa> + y)-(a-6)(*-y).
(a-b)(x + y) + {a+b){x-y)~-
Simplify (1), 2 bx + 2 ay ■-
Simplify (2), 2ax-2by-
Multiply (3) by a, 2abx + 2a*y-
Multiply (4) by 6, 2abx-2b2y-
Subtract, (2a2+2b*)y:
Multiply (3) by 6,
Multiply (4) by -
Subtract,
a,
62
a*
b\
a3
b>
•. y =
2b*x+2aby =
— 2a2a? + 2aby --
(2a3 + 26*)* =
a8 -6s
a8 -ft8
2(a* + J8)'
a86
-ab*
a86 + a&*
a6 (a + 6)
2(a* + 68)'
(3)
(4)
Exercise LXXIII.
l + 5-io
x y
4 3 on
- + - =* 20
x y
Multiply (1) by 4,
4 8 Art
- + - = 40
or y
l + ? = 20
* y
5-20
y
•••y=i-
Multiply (1) by 3,
? + ?- 30
x y
(2) by 2, ? + ?= 40
(2) is
Subtract,
Subtract, — -
= -10
or, 10a> = 5.
(i)
(2)
(3)
(5)
2.
1 2
_ + _ _ a (1)
x y
M-5 (2)
x y v
Multiply (1) by 3,
? + 5 = 3a (3)
x y w
3 4 .
- + - = 6
* y
(2)i8
Subtract,
2_
y
.-. y =
Multiply (1) by 2,
3a-6
2
Za-b
(2)is
2 4 o
- + - = 2a
a; y
» + *-»
a? y
1
Subtract, - =b — 2a.
x
. _ 1
b-2a
182
ALGEBRA.
3. 2__5 = 4. (1)
z 3y 27 W
-1 + 1=11 (2).
ix y 72 W
(1)i. ?_5. = A
K) x Zy 27
8X(2)i8 M-£ 0)
Subtract,
29 29
3y 27
,-.y = 9.
Substitute value of y in (1),
2 5 4
* 27 27'
2 9
a? 27
.-. * = 6.
! + ?-4
(1)
* y
?_2_4
(2)
* y
Multiply (1) by 3,
5 + 5-12
a? y
(2) is
3_2«4
x y
Subtract,
5.8
y
.-. y=l.
(l)is
1 + 2 = 4
.» y
(2) is
!_2 = 4
x y
Add,
= 8
1-5 = 6 (2)
* y
Multiply (1) by (4),
* y
(2)by3, ^-15 = 18
* y
Subtract, — i= 2
y
.-.y = -J.
Substitute value of y in (1),
- + 8 = 5.
x
.•.* = -!.
6. ? + * = ?? (1)
x y b
* + ? = ^ (2)
x y a
Multiply (1) by b and (2) by a,
<*> P „.
1- — = ac
* y
Subtract,
I- — = be
x y
,y= —
ac — bc
Multiply (1) by a and (2) by k
a? ah a*c
Subtract,
x y b
6* ab_¥c
x y a
a* -J2 a?c-Pc
x ab
ab(a + b)
ciat + ab + V*)
teachers' edition. 183
7.
A+3 5
ax by
A_JL = 3
ax by
(i)
(2)
Multiply (1) by 5,
10 + 16 = 25
ax by
Multiply (2) by 2,
l°-± = 6
ax by
Subtract,
W-19
i
Multiply (1) by 2,
*+i.-io
Multiply (2) by 3,
15 _ «.- 9
aa; by
Subtract
1?-19.
1
. •. x -= —
a
a
— + — =»m-f n
na; my
ti 771 » •
_ + _ = m2 + n*
a? y
(1)
(2)
Multiply (1) by n,
77171 71 / v
1 =•- n(m + 7i)
Tia; my '
Multiply (2) by m
mn t m* m(m* + na)
nx ny n
Subtract,
7is — 7M8 __ n^m + n) —
m(m2
m*
+ n«)
mny n
7lS-
m27i2 + mws -
1
••• y —
-m4-
-mhi*
Substitute value of y in (2), x =- —
184
ALGEBRA.
n a b
9. - + - = m
x y
b a
x y
(1)
(2)
Multiply (1) by b,
h— =bm
x y
ab a*
= an
x y
Multiply (2) by a,
Subtract,
*=bm — an
V a* + 6*
* ' * ** 6m — an
Multiply (1) by a,
a* t ab ntm
— h = ar»
a? y
* y
Multiply (2) by b,
Add,
°, + J, = «ro+6»
a* + 6»
, + 6n
Exercise LXXTV.
1.
5* + 3y-6z = 4 (1)
3x-y+2z = 8 (2)
x-2y + 2z = 2 (3)
(1) is 5x + 3y-6z = 4 (1)
3 x (2) is 9s-3y + 6z = 24
Add, 14 s =28
.-. x-2.
(l)is 5s + 3y-6z = 4 (1)
3x(3)is 3s-6y + 6z= 6
Add, 8s-3y =10 (4)
Substitute value of x in (4),
16-3y = 10,
-3y = -6.
.-.y = 2.
Substitute values of x and y
in (3),
2-4 + 22 = 2,
2z = 4.
.-.« = 2.
2. 4x-5y + 2z = 6
2a? + 3y-z = 20
7a-4y + 3z = 35
Multiply (1) by 3 and (3) by 2,
12s-15y+6z= 18 (4)
14a?-8y+6z = 70
Subt., -2x- ly =-52 (5)
Multiply (2) by 3 and (3) by 1,
6a;+9y-3z = 60
7a?-4y+3z = 35
Add, 13s+5y =95 (6)
Multiply (5) by 5 and (6) by 7,
10a; + 35y = 260
91s + 35y= 665
Subt., -81* =-405
.*. x = 5.
Substitute value of x in (6),
...y = 6.
2 = 8.
TEACHERS ' EDITION.
185
3. x + y + 2 =
5a; + 4y + 32 =
15ar + 10y + 62 =
(3) is 15aH-10y+62:
6x(l)is 6s+ 6y+62=
Subtract,
(2) is
3x(l)is
Subtract,
(4) is
4x(5)is
Subtract,
9a?+ 4y
5a?+4y+32 =
3a?+3y+32 =
2x+ y
9a; + 4y
8a; + 4y
= 6 (1
= 22 (2)
= 53 (3)
= 53
= 36
= 17 (4)
= 22
= 18
= 4
= 17
= 16
= 1
(5),
= 4.
= 2.
and
= 6.
= 3.
(5)
Substitute value of x in
2 + y =
.\y =
Substitute values of x
in (1), 1 + 2 + 2 =
.'.2 =
L 4a? — 3y + 2 =
9a; + y-52 =
x — 4y + 32 =
(l)i8 4 a;— 3y+ 2 =
3x(2)is 27a;+3y-152 =
Add, 31s -142 =
Multiply (2) by 4,
36a:+4y-202 =
(3) is s-4y+ 32 =
Add, 37a? -172 = 66 (7)
Multiply (5) by 37,
1147ar-527z = 2046
31 X (7) is 1147a;-518z = 2107
Subtract, -92 = -63
.'.2=7.
(7),
= 66,
= 185.
= 5.
and y
9,
-18.
9 (I
16 (2"
2 \Z)
■■ 9
^8 (4)
= 57 (5)
64 (6)
2
Substitute value of 2 in
37* -119 =
37s =
.*. x =
Substitute values of x
in(l), 20-3y + 7 =
-3y =
.-.y =
5. 8a; + 4y-32 =
x + 3y — 2 =
4x — 5y + 42»
(l)is 8a:+4y-32 =
3x(2)is 3a:+9y-32 =
Subt., bx—by
Multiply (2) by 4,
4s+12y-42 =
(3) is 4a;— 5y+42 =
Add, 8a?+ 7y
Multiply (4) by 7 and
35 a; — 35y =
40a; + 35y =
Add,
-6 m
>8 (3)
6
■■ 21
>-15(i)
:28
= 36 (5)
(5) by 5,
= -105
= 180
75a;
Substitute value of x
b-by-
-bl
•••y-
Substitute values of
in (2), 1 + 12-2 =
.*. 2 =
6. 12a; + 5y-42 = 29
13a;- 2y + 5y = 58
17a; — y -2= 15
(l)is 12x+5y-42 = 29
4x(3) is 68a;-4y-42 = 60 (4)
Subt., 56a;-9y =31 (5)
(2) is 13a;-2y+52== 58
5x(3)is85a?-5y-52= 75 (6)
Add, 98a;-7y =133 (7)
Multiply (7) by 9 and (5) by 7,
882a; -63y = 1197(8)
392a; -63y = 217(9)
Subt., 490 a; = 980
.-. x = 2.
Substitute value of x in (7),
196 - 7y = 133.
.-.y = 9.
Substitute values of x and y
in(l), 24+45-42 = 29.
.'.2=10.
18*
ALGEBRA.
x-y~z = 5 (1)
aj + y_2 = 25 (2)
a. + y + 2=,35 (3)
x-y-z=* 5
x + y + g = 35
2* =40
.% x = 20.
Substitute value of a; in (2) and (3),
y-2= 5
y + 2 = 15
(l)is
(3) is
Add,
Add,
Subtract,
2y =20
.-.y-10.
- 2z = - 10.
.-.2 = 5.
8. x + y + 2 = 30 (1)
8x + 4y + 22 = 50 (2)
27* + 9y + 32«64 (3)
Multiply (1) by 2,
2s + 2y + 22 = 60 (4)
(2) is 8a; + 4y + 22 = 50
Sub., -6a;- 2y -10 (5)
Multiply (1) by 3,
3s + 3y + 32 = 90 (6)
(3) is 27a + 9y + 32 = 64
Sub.,-24a-6y -=26(7)
Multiply (5) by 3,
-18a;-6y = 30 (8)
(7) is -24s-6y = 26
Subtract, 6 a; =4
.•-*-!■
Substitute value of x in (8),
-12-6y = 30.
Substitute values of x and y
in(l), 1-7 + ^ = 30,
•\ 2 = 36J.
Multiply (1) by 2,
Multiply (2) by 3,
Add,
Multiply (2) by 7,
Multiply (3) by 16,
Add,
Multiply (6) by 154,
Multiply (9) by 3,
Subtract,
9. 15y = 242-10a; + 41 (1)
15a; = 12y-162 + 10 (2)
18x-(72-13)=14y
20a; + 30y-482= 82
45a-36y + 482= 30
65a;- 6y =112
105a;- 84y + 1122 = 70
288a; - 224y - 112z = - 208
393a; -308y =-138
10,010a; -924y = 17,248
1,179a; -924y = -414
8,831a;
- 17,662
. x-2.
Substitute value of x in (6),
130-6y = 112.
-•-y-3.
Substitute values of x and y in (1),
20 + 45-242 = 41.
(3)
(4)
(5)
(6)
(7)
(8)
(9)
teachers' edition.
187
10.
>.
3*-y
5a; + 3y-
7a;+4y-
+ z
>2z
• bz
= 17
= 10
-3
8
(3)
Multiply (1) by 2,
(2) is
6a?-2y +
5a? + 3y-
2z
2z>
= 34
= 10
(4)
Add,
lla; + y
-44
(5)
Multiply (1) by 5,
(3) is
15a? — by +
7a? + 4y —
bz-
bz-
= 85
= 3
(6)
Add,
(5) is
22 a;- y
11a; + y
= 88
= 44
$
Add,
33a;
. x-
= 132
= 4.
Substitute value of x in (5),
44 + y-
= 44,
From
(1),
z-
= 0.
= 5.
(3)
12. a?+2y + 3« = 6
2a: + 4y+22 = 8
3a; -f 2y + 82 = 101
Multiply (1) by 2,
2a; + 4y + 6*=12
(2) is 2a; -f 4y + 22 = 8
Subtract, 42= 4
.'.2=1.
(2) is 2a;+4y+ 22= 8
2x(3) is 6a;+4y+162 = 202 (7)
Subt.,-4a; -142 = -194 (8)
Substitute value of 2 in (8)f
-4a; -14 = -194,
-4a; = -180.
.\x = 45.
Substitute values of a; and z in (1),
45 + 2y + 3 = 6, '
2y = -42.
.-.y = -21.
11. x + y + 2 = 5 (1)
3s-5y + 72=75 (2)
9x -112+10 = 0 (3)
Multiply (1) by 5,
5a; + 5y + 52= 25(4)
(2) is 3a;-5y + 72= 75
Add, 8a; +122-100(5)
Multiply (5) by 9 and (3) by 8,
72a; + 1082= 900
72a;- 88 z = -80
Subtract, 1962 - 980
.-.2 = 5.
Substitute value of z in (3),
9a?-55 = -10,
9a; = 45.
.*. a; = 5.
Substitute values of x and 2
5 + y f 5 = 5.
.-. y = -5.
13. a;-3y-22=l
2a?-3y+52 = -19
5a? + 2y-2 = 12
Multiply (3) by 2,
10a; + 4y-22 = 24
(1) is x-3y-22= 1
Subt., 9a;+7y =23
Multiply (3) by 5,
25a;+10y-52= 60
(2) is 2a;- 3y+52 = -19
= 41
= 23
Add, 27x+ 7y
(5) is 9a;+ 7y
Sub., 18 a;
18
, x=l.
Substitute value of x in (5),
9 + 7y = 23.
.-.y = 2.
Substitute values of x and y in (1),
1-6-22 = 1.
.•.2 = -3.
188
ALGEBRA.
14. 3x-2y = 5 (1)
4a-3y + 2z=ll (2)
ar_2y-5z = -7 (3)
Multiply (2) by 5 and (3) by 2,
20ar-15y+102 = 55(4)
2s- 4y-102 = -14(5)
Add, 22x-l9y = 41(6)
Multiply (1) by 19 and (6) by 2,
57z-38y = 95 (7)
44s-38y = 82 (8)
16.
Subtract, 13 s
= 13
.*-l.
Substitute value of x in (1),
3-2y = 5,
-2y = 2.
.-.y 1.
Substitute values of x and y
in (2),
4 + 3 + 22 = 11,
22 = 4.
.\z = 2.
15. x+ y =1 (1)
y+ 2= 9 (2)
x + 2= 5 (3)
Add, 2a; + 2y + 22 = 15
x + y + 2 = 7} (4)
Subtract (1) from (4),
2 = 6}.
Subtract (2) from (4),
s = -l£.
Subtract (3) from (4),
y = 2j.
(lHs
(2) is
Add,
(3)is
2s-3y = 3
3y-4z = 7
-5* + 4z = 2
2.r-3y = 3
3y-42= 7
= 10
. 2
2)
(3)
2x
-5x
-42 =
-t-42 =
Add, -3a; =12
.-.* = - 4.
Substitute value of x in (1),
-8-3y = 3,
-3y=ll.
.-.y = -3f.
Substitute value of x in (3),
20 + 42 = 2,
42 = -18.
.-.« — 4J.
17. 3s-4y + 62=l
2sc + 2y — 2 = 1
*x — §y + 72 = 2
(l)is 3s- 4y+62 = l
6x(2) isl2aj+12y-6z = 6
Add, 15 s+ 8y =7
Multiply (2) by 7,
14.r+14y— 72 = 7
(3) is Ix- 6y+7z = 2
Add,
9
2)
(3)
(5)
(6)
(7)
21x+ 8y
Subtract (7) from (5),
-6x = -2.
Substitute value of x in (7),
7 + 8y = 9,
8y = 2.
Substitute values of z and y
in (2),
l + J-«-l,
-*=l_i-J.
TEACHERS EDITION.
189
18.
7»-3y
9y--5z
x + y -f z
Multiply (3) by 7,
7x + 7y + 7z
(1) is 7s-3y
= 30 (1)
= 34 (2)
- 33 (3)
= 231
= 30
Subtract, 10y+7z
Multiply (2) by 10 and
90y- 50« =
90y + 63z =
Subtract, -113z
= 201 (5)
(5) by 9,
= 340
- 1809
n
Substitute value of z in (5),
lOy + 91 = 201,
lOy = 110.
.-.y = ll.
Substitute values of y and z in (3),
a+ll + 13 = 33.
.-. c = 9.
= - 1469
-13.
19.
x + * + z-
2 3
z x
y 2 3
6,
= 17.
Simplify,
6a; + 3y + 2z = 36 (1)
2x + 6y + 3z = -6 (2)
3<c + 2y + 6z=102 (3)
20.
1 2
- + -
x y
3_4
y z
34
z x
= 5
(1)
_- = 5
= -6 (2)
3x(l)is 18aj+9y+6z = 108
(3) is 3aH-2y+6z = 102
Sub., 15x+7y = 6 (4)
2x(2)is4aH-12y+6z = - 12
(3) is 3s+ 2y+Qz = 102
Sub., z+lOy =-114 (5)
Subtract 7 X (5) from 10 X (4),
143x = 858
.\ re = 6.
Substitute value of x in (5),
y = -12.
Substitute values of a? and y in ( 1 ),
z=18.
Multiply (6) by 4 and (3) by 3,
2! + 2? -los
X z
12 9
+-= 15
X z
(3)
Add
41
= 123
Multiply (1) by 3 and (2) by 2,
= 15 (4)
= -12 (5)
3 6
- + - =
x y
_8 6
z y"
Subtract,
3 8
- + _ =
x z
27 (6)
Substitute value of z in (3),
9-i-B.
X
.-. a? = l.
Substitute value of x in (1),
190 ALGEBRA.
21.
i + l_I=a (1) (4)18 i+i+1-a+ft+c
x y z w x y z
x y + 2 W * y 2
i + l-l = c (3) Subtract, - =a + c
y z x y
Add, l + i + I = a+6+c(4) "y~^+7
x y z ill
(l)is l + i-l-a • V ■
* y * (3)is -I+I+l-c
o % y z
Subtract, - = 6 + c
2 Subt., - =a + &
- * 2
6 + c
a + 6
&s + cy =
a
(1)
<K + cx =
-b
(2)
ay + bx =
*c
(3)
abz + acy =
■a*
(4)
abz + 6cx =
-V
©
acy + bcx =
= c*
(6)
2a6z + 2acy + 2bcx =
*a* +
6*
+ C3
(7)
2bcx =
*&* + (*
-a2
(8)
2acy =
= a»-
&»
+ C2
(9)
2abz =
-<*' +
6»
-c*
(10)
Multiply (1) by a,
Multiply (2) by 6,
Multiply (3) by c
Add (4), (5), and (6),
Subtract twice (4) from (7),
Subtract twice (5) from (7),
Subtract twice (6) from (7),
In(8), — * + *-*
In (9),
In (10),
2 6c
a»-6a + c2
2ac
x a3 -f ft2 -
2a6
TEACHERS EDITION.
191
3. 2 4+l = 7| (1)
x by z
l + l+a.ioj (2)
3a; 2y z
l_J_+i=16A (3)
5a; 2y z
Multiply (1) by 60 and (2) by 30,
180 48 60 AKa ...
+ — = 456 (4)
x y z w
10 15 60 OAC /KN
— + —+— = 305 (5)
x y z
24. ?_? + !=2.9 (1)
x y z
5_6_7
x y z
= -10.4(2)
Sub.,™L?3 =151 (6)
x y
Multiply (2) by 60 and (3) by 30, (2) i8 £ _ 5 - 1 . -10.4(6)
y z x w
Add, _I + 2 = 7.4 (4)
Multiply (1) by 2,
1-2 + 5-M (5)
x y z w
5 6 7
5+S + ^.mo (7)
*J$+^-«S .(8)
x y z
4 45
Sub.,--+— =127 (9)
x y
Multiply (6) by 2 and (9) by 85,
^-^ = 302 (10)
_*» + ?«*_ 10795 (11)
x y
Add, 2®?- 11097
y
•••y = i-
Substitute value of y in (9),
-1 + 135 = 127,
x
8a; = 4.
Substitute values of x and y
in (5),
RO
20+45+— = 305,
z
240z = 60.
•\ z = J.
x y z
Subt., -1 + — =16.2 (7)
a; z v '
(4) is _! + != 7.4 (8)
a? z v '
Subtract, £= 8.8
z
Substitute value of z in (4),
-1 + 7.7 = 7.4.
x
Simplify, — 1 + 7.7 a; = 7.4 a;,
3a; =10.
Substitute values of a; and z in (1),
0.6 -- + 4.4 = 2.9.
y
Simplify,
0.6y-3 + 4.4y = 2.9y.
.-.y-lf.
192
ALGEBRA.
25
?+l-5-o
x y z
(1)
(5) is *_* = 2
x z
?_2 = 2
2 y
(2)
(3)
Mul.(3)by3, - + ? = 4
2 Z
X z
Multiply (1) by 2,
Add, 1 =6
X
.•.*=ii.
Substitute value of x in (5),
x y z
(2)is -- + i=2
(*)
24^5 = 2
7 z
Substitute values of x and y in (1),
y 2
(5)
12 1 10 yv
Add, * _? = 2
— + - — — = 0.
... y=_3J.
ax + by + cz *- a (1)
ax — by — cz = b (2)
ax + cy + 62 = c (3)
Add (1) and (2), 2 ox = a + b (4)
a + 6
.\ x = -- — •
2a
Multiply (2) by b and (3) by c,
afo _&*y -&cz = &2 (5)
acx + chj + bcz=*<? (6)
Add, afcx + acx-&2y + chf = 62 + c2 P)
Substitute value of x in (7),
g26 + a2c + a&2 + abc ^ -(*)y = & + <*.
ab + ac + bc — ft2 -2c2
"y~ 2(62-c2)
Substitute values of x and y in (3),
a + b abc + ae* + be2 — 62c — 2c8 , =
2 2(&a-c2)
, 352c-a62-a6c-52
2 2(62-c2)
36c— aft — ac — h*
'■' * 2(&2-c2)
teachers' edition. 193
81
X— I
a a + b + c
y ~ z = x — a — b
b a + b + c
x + z x—a— b
0)
27.
%x~ U _ 3y -f 2z_ x — y — z^ .
3 4 5
Simplify, 40s -20y-45i/ + 30 2 = 12a? - 12y - 12z = 240
40a;-20y = 240 (1)
45y + 30z = 240 (2)
12a?-12y-12z = 240 (3)
Divide (1) by 20, 2x - y = 12 (4)
Divide (2) by 15, 3 y + 2 z = 16 (5)
Divide (3) by 12, x - y - z - 20 (6)
Multiply (4) by 3, 6a? -3y* =36 (7)
(5) is 3y + 2z = 16
Add, 6a? + 2z = 52 (8)
(4) is 2a; -y = 12
[6) is x-y-z=a 20
Subtract, x + z = -8 (9)
(8)-*-2is 3a; + z = 26
(9) is x+z = - 8
Subtract, 2 a; = 34
.-.a; = 17.
Substitute value of x in (4V y = 22.
Substitute value of y in (5), z = — 25.
(1)
(2)
(3)
c a + 6 + c
x(a + b + c) — y(a + b +c) = ax — a2 — ab (4)
-z(a + b + c)+y(a + b + c) = bx — b* — ab (5)
x(a + b + c) + z(a + b + c) = ex — ac — be (6)
Add, 2 a; (a + b + c) = x(a+b+c)-at-b2-2ab-ac-bc
x(a + b + c) = — (a* + b2 + 2ab + ac + be).
.-. x=> — (a + 6).
From(4), -(a+6)(a+6+c)-yfa+6+c} = - 2a2 - 2a&,
or, — y{a+b+c) = — a2+b2 + ac + be.
_(a+6)(tt-6-c)
a + 6 + c
From(6), -(a+6)(a+6+c)+z(a+6+c) = - 2ac - 26c,
- z(a+6+c) = a2 + 2 aft + 62 — ac — 6c.
... z=3(a + 6-c)(a + 6)t
a + 6 + c
194 ALGEBRA.
Exercise LXXV.
1. The sum of two numbers divided by 2 gives as a quotient
24, and the difference between them divided by 2 gives as a quo-
dent 17. What are the numbers?
Let x = first number,
id y = second number.
Then
*±2 = 24
2
id
2
Add (1) and (2),
Subtract (2) from (1),
3 = 41
y-7.
0)
(2)
2. The number 144 is divided into three numbers. When the
first is divided by the second, the quotient is 3 and the remainder
2 ; and when the third is divided by the sum of the other two
numbers, the quotient is 2 and the remainder 6. Find the num-
bers.
Let x = first number,
y = second number,
and z = third number.
Then
x + y + 2=144
(1)
£n2 = 3
(2)
and
x+y
(3)
Simplify (2),
x-3y = 2
W
Simplify (3),
«-2y-2x = 6
(5)
Multiply (1) by 2,
2x + 2y + 22 = 288
(6)
Add (5) and (6),
32 = 294
.-.2 = 98.
Substitute value of z in (1), x + y + 98 = 144,
s + y = 46 (7)
(4) is a:-3y= 2
Subtract, 4y = 44
.-.y-U.
Substitute value of y in (7), x + 11 — 46.
.-.a? = 35.
teachers' edition. 195
3. Three times the greater of two numbers exceeds twice the
less by 10 ; and twice the greater together with three times the
less is 24. Find the numbers.
Let x = greater number,
and y = less number.
Then
3z-2y = 10
(1)
and
2s + 3y = 24
(2)
Multiply (1) by 2,
6x~ 4y = 20
Multiply (2) by 3,
6x+ 9y = 72
Subtract,
-13y = -52
•••y-4.
Substitute value of y
in(l), 3s-8 = 10.
.\x -6.
4. If the smaller of two numbers is divided by the greater,
the quotient is 0.21 and the remainder 0.0057 ; but if the greater
be divided by the smaller, the quotient is 4 and the remainder
0.742. What are the numbers ?
Let x = larger number,
and y = smaller number.
Then - = smaller divided by larger.
x
- = larger divided by smaller.
Hence
2^021 .0.0057
(1)
X X
and
x_ 0.742
y y
(2)
Simplify (1),
y = 0.21 x + 0.0057
y - 0.21 x = 0.0057
(3)
Simplify (2),
x = 4y + 0.742
x— 4y = 0.742
W
Multiply (3) by 4,
4y-0.84x = 0.0228
(4) is
-4y+ x = 0.742
Add,
0. 16 X- 0.7648
.-.3 = 4.78.
Substitute value of x in (4),
-4y= -4.038
.-.y = 1.0095.
1
196 ALGEBRA.
5. Seven years ago the age of a father was four times that of
his son ; seven years hence the age of the father will be double
that of the son. What are their ages?
Let x = number of years in father's age.
Then x + 7 = number of years in father's ago 7 years hence
x — 7 = number of years in father's age 7 years ago.
Let y = number of years in son's age.
Then y + 7 — number of years in son's age 7 years hence,
y — 7 = number of years in son's age 7 years ago.
*-7 = 4(y-7) (1)
s + 7 = 2(y + 7) (2)
<r-4y = -21 (3)
s-2y = 7 (4)
Subtract, -2y 28
.-.y = i4.
Substitute value of y in (4),
ar-28 = 7.
.\ x = 35.
6. The sum of the ages of a father and son is one-half what
it will be in 25 years ; the difference between their ages is one-
third of what the sum will be in 20 years. What are their ages?
Let x = number of years in father's age,
and y =- number of years in son's age.
Then x + y — sum of ages,
x + y + 50 — sum of ages in twenty-five years.
t x 4- y + 50
2
;-y-* + y + 40 (2)
* + y = -l^ (i)
Simplify (1), x -f y =■ 50
Simplify (2), 2 x - 4 y = 40
(3) is x + y = 50
(4) -*- 2 is a;-2y = 20
Subtract, 3y = 30
.-.y = 10.
Substitute value of y in (3),
x + 10 - 50.
.-. x-40.
teachers' edition. 197
7. If B give A $25, they will have equal sums of money ; but
if A give B $22, B's money will be double that of A. How
much has each ?
Let or = number of dollars B has,
and y = number of dollars A has.
Then x — 25 = number of dollars B has after giving $25 to 4,
y + 25 = number of dollars A has after receiving $ 25.
z-25 = y + 25 (1)
y — 22 = number of dollars A has after giving $25 to B.
x + 22 = number of dollars B has after receiving $22.
z + 22 = 2(y-22) (2)
Transpose and combine,
x- y = 50 (3)
x _ 2y = - 66 (4)
Subtract, y = 116
Substitute value of y in (3),
x- 116 = 50.
.-.a =166.
8. A farmer sold to one person 80 bushels of wheat and 40
bushels of barley for $67.50; to another person he sold 50
bushels of wheat and 30 bushels of barley for $85. What was
the price of the wheat and of the barley per bushel?
Let x = number of dollars received per bushel of wheat,
and y = number of dollars received per bushel of barley.
Then 30* + 40y = 671 (1)
50s + 30y = 85 (2)
Simplify (1), 60s + 80y = 135 (3)
Multiply (2) by & 60s + 36y = 102 (4)
ract,
44y
= 33
.-. y
-i
bitute value of
yin(3),
60* + 60 =
135,
60x =
75.
••. X-
=-H.
198 ALGEBRA.
9. If A give B $5, he will then have $ 6 less than B ; but if he
receive $5 from B, three times his money will be #20 more than
four times B*s. How much has each?
Let x = number of dollars A has,
and y = number of dollars B has.
Then x — 5 = number of dollars A has after giving B $ 5,
and y + 5 = number of dollars B has after receiving $ 5.
Hence, x — 5 = y -f 5 — 6,
and 3(x + 5) = 4(y - 5) + 20.
Transpose, x — y = 4 (1)
3*-4y = -15 (2)
Multiply(l)by3, 3s-3y = 12 (3)
(2) is 3s-4y = -15
.-.y= 27
Substitute value of y in (1),
*-27 = 4.
.\*-31.
10. The cost of 12 horses and 14 cows is $1900; the cost of
5 horses and 3 cows is $650. What is the cost of a horse and a
cow respectively?
Let x = number of dollars a hors
and y = number of dollars a cow
e costs,
costs.
Then 12a; + 14y - 1900
and 5a; + 3y = 650
a)
(2)
Multiply (1) by 3, 36* + 42y = 5700
Multiply (2) by 14, 70* + 42y = 9100
(3)
Subtract, 34* = 3400
.-. * = 100.
Substitute value of * in (2),
500 + 3y-650,
3y = 150.
.-.y-60.
teachers' edition. 199
11. A certain fraction becomes equal to 2 when 7 is added to
its numerator, and equal to 1 when 1 is subtracted from its
denominator. Determine the fraction.
Let - = required fraction.
V
By conditions, ^±_? = 2 (1)
and -2— = 1 (2)
Simplify (1), x + 7 = 2y (3)
Simplify (2), x = y - 1 (4)
Transpose (3), x - 2 y = - 7 (5)
Transpose (4), x — y = — 1 (6)
Subtract, y = 6
Substitute value of y in (5),
rr - 12 = - 7.
.*. x = 5.
.•. fraction = 4.
12. A certain fraction becomes equal to \ when 7 is added to
its denominator, and equal to 2 when 13 is added to its numera-
tor. Determine the fraction.
Let - = required fraction.
y
By conditions, — ^— = - (1)
y + 7 2
and ^±13 = 2 (2)
y
Simplify (1), 2s-y = 7 (3)
Simplify (2), x - 2y = - 13 (4)
Multiply (3) by 2,
4a;-2y= 14
(4) is x -2y = - 13
Subtract, 3 a; =27
.-. a; = 9.
Substitute value of x in (3),
18-y = 7.
.-.y = ll.
.•. fraction = -ft.
200 ALGEBRA.
13. A certain fraction becomes equal to J when the denomi-
nator is increased by 4, and equal to ff when the numerator is
diminished by 15. Determine the fraction.
Let _ b fraction.
y
Then -*_ = * (l)
y + 4 9 W
-X — 15 20 /gv
"« ()
Simplify (1), 9 x = ly + 28.
Simplify (2), 41 a? - 615 = 20y.
Transpose, 9 a - 7y = 28 (3)
41s-20y = 615 (4)
Multiply (3) by 20, 180a;-140y = 560 (5)
Multiply (4) by 7, 287s-140y = 4305 (6)
Subtract, - 107 x = - 3845
.-.a; = 35.
Substitute value of x in (3), 315 - 7y = 28.
.\ y = 41.
.\ fraction = ff.
14. A certain fraction becomes equal to J if 7 is added to the
numerator, and equal to § if 7 is subtracted from the denomina-
tor. Determine the fraction.
Let - = fraction.
y
Then £Jl2=2 m
V 3
and -?- - ? (2)
y-7 8 K>
Simplify (1), 3a; + 21 = 2y.
Transpose, 3 x - 2y = - 21 (3)
Simplify (2), 8a? = 3y-21.
Transpose, 8 x - 3 y = - 21 (4)
Multiply (3) by 3, 9a? - 6y 63
Multiply (4) by 2, 16* - 6y = - 42
Subtract, -7x =-21
.\ x = 3.
Substitute value of x in (3), 9 — 2y = — 21.
.\ y = 15.
.\ fraction = -j^.
2 5,3
x + y~2
(1)
? + 5 = 2
y *
(2)
x y
(3)
* + !?.. 10
a; y
(4)
» =7
.•. x = 3.
teachers' editiow. 201
15. Find two fractions with numerators 2 and 5 respectively,
such that their sum is \\ ; and if their denominators are inter-
changed their sum is 2.
Let x = denominator of first fraction,
and y = denominator of second fraction.
Then
and
Multiply (lfby 2,
Multiply (2) by 5,
Subtract,
Substitute value of x in (2), y = 6.
. \ first fraction = $, second fraction = f .
16. A fraction which is equal to J is increased to -ft when a
certain number is added to both its numerator and denominator,
and is diminished to g when one more than the same number is
subtracted from each. Determine the fraction.
Let x equal numerator, y the denominator, and z the number
to be added.
Then 2=§ (l)
y 3o
y + z 11 li
and "(, + 1)-g (3)
Clear of fractions and transpose,
Sx-2y = 0 (4)
llz-8y + 3z = 0 (5)
9s-5y-4z = 4 (6)
Multiply (5) by 4, 44a- 32y + 12*=- 0
Multiply (6) by 3, 27s- 15y- 12z = 12
Add, 71a-47y =12 (7)
Multiply (7) by 3, 213 x - 141 y = 36
Multiply (4) by 71, 213s-142y~ 0
Subtract, y = 36
Substitute value of y in (1), x = 24.
•\ fraction = ||.
202 ALGEBRA.
17. The sum of the two digits of a number is 10, and if 54 be
added to the number the digits will be interchanged. What is
the number?
Let x = digit in tens' place,
and y = digit in units' place.
Then 10 a; + y = number.
By conditions, x + y = 10 (1)
and 10a; + y -f 54 = lOy + xt
9a;-9y = -54.
Divide by 9, x - y = - 6 (2)
Add (1) and (2), 2a; = 4.
r.x = 2.
Subtract (2) from (1 ), 2y - 16.
.-.y = 8.
number = 10 a; + y.
.\ number = 28. •
18. The sum of the two digits of a number is 6, and if the
number be divided by the sum of the digits the quotient is 4.
What is the number?
Let
x = digit in
tens'
place,
and
y -= digit in
units
' place.
Then
10 a? + y =» number,
and
* + y = 6
(i)
But
10a? + y^1
6
(2)
Clear of fractions,
10a; + y = 24
x + y= 6
Subtract,
9x -18
.-. a;«2.
Substitute value of
x in (1),
2 + y = 6.
.-.y-4.
.*. number -» 24.
teachers' edition. 203
19. A certain number is expressed by two digits, of which
the first is the greater. If the number is divided by the sum of
its digits the quotient is 7; if the digits are interchanged, and
the resulting number diminished by 12 is divided by the differ-
ence between the two digits, the quotient is 9. What is the
number?
Let x = digit in tens' place,
and y = digit in units' place.
Then 10 jc + y = number.
By conditions, * = 7 (1)
J x +y x '
10y + a.-12 = &
x — y v '
Simplify (1), 3s-6y = 0 (3)
Simplify (2), -8a; + 19y = 12 (4)
Multiply (3) by f 8s-16y = 0
Add, 3y = 12.
Substitute in (3), x = 8.
.\ number = 84.
20. If a certain number is divided by the sum of its two
digits, the quotient is 6 and the remainder 3 ; if the digits are in-
terchanged, and the resulting number is divided by the sum of
the digits, the quotient is 4 and the remainder 9. What is the
number?
Let x =* digit in tens' place,
and y = digit in units' place.
Then 10 a? + y = number.
By conditions, Wx + y-3 = 6 (1)
J s + y
10y + a?-9 = 4
x + y
Clear (1) and (2) of fractions, transpose and combine,
4s-5y = 3 (3)
-3a + 6y = 9
Divide (4) by 3, - x + 2y = 3
Add 4 x (5) and (3), -4s + 8y = 12
4s-5y = 3
3y = 15
'.y = 5.
Substitute value of y in (3), x =• 7.
•\ number = 75.
204 ALGEBRA.
2L If a certain number is divided by the sum of its two digits
diminished by 2, the quotient is 5 and the remainder 1 ; if the
digits are interchanged, and the resulting number is divided by
the sum of the digits increased by 2, the quotient is 5 and the
remainder 8. Find the number.
Let x = digit in tens' place,
and y = digit in units place.
Then 10 x + y = number.
By conditions, I0x + y = 5 + 1 ^
x+y—2 x+y—2
and My + S-5+— 3
x + y+2 a? + y + 2
Clear of fractions, 10 a: + y = 5 x + by — 10 4- 1.
lOy + x = bx + by + 10 4- 8.
Transpose and combine, bx — 4y = — 9 (1)
5y-4x = 18 (2)
Multiply (1) by 5, 25 x - 20y - - 45
Multiply (2) by 4, - 16 a; + 20y - 72
Add, 9x - 27
.-.a? = 3.
Substitute value of x in (1), y = 6.
.\ number = 36.
22. The first of the two digits of a number is, when doubled,
3 more than the second, and the number itself is less by 6 than
five times the sum of the digits. What is the number?
Let x = digit in tens' place,
and y = digit in units' place.
Then 10 a; + y = number.
By conditions, 2x — y + 3
and 10x + y + 6 = 5x + 5y
Transpose and combine, 2 a; — y = 3
5a;-4y = -6
Multiply (3) by 4, 8 x - 4y - 12
(4) is 5a;-4y = -6
Subtract, 3 a; =18
.-. x -6.
Substitute value of x in (3), y = 9,
•\ number = 69.
(2)
TEACIIKRS* EDITION. 205
23. A number is expressed by three digits, of which the first
and last are alike. By interchanging the digits in the units'
and tens' places, the number is increased by 54 ; but if the digits
in the teus' and hundreds' places are interchanged, 9 must be
added to four times the resulting number to make it equal to the
original number. What is the number?
Let x = digit in hundreds' and units' place,
and y = digit in tens' place.
Then 101 x + lOy = number.
By conditions, 110a + y = 101 x + lOy + 54 (1)
4(lla + 100y) + 9 = 101a;+10y (2)
Transpose and combine (1), 9r — 9y = 54 (3)
Divide (3) by 9, x - y = 6 (4)
Transpose and combine (2),
-57a; + 390y = -9
Multiply (4) by 57, 57 a;- 57y=342
Add,
333 y = 333
.-.y-i.
Substitute value of y in (4),
*-7.
.*. number = 717.
24. A number is expressed by three digits. The sum of the
digits is 21 ; the sum of the first and second exceeds the third
by 3 ; and if 198 be added to the number, the digits in the units'
and hundreds' places will be interchanged. Find the number.
Let x = digit in hundreds' place,
y = digit in tens' place,
and z = digit in units' place.
Then lOOx + lOy + z = number.
By conditions, x 4- y + z — 21 (1)
and x + y - z = 3 (2)
100a + 10y + z + 198 = 1003 + lOy + x (3)
Subtract (2) from (1), 2z = 18.
.'.2 = 9.
Divide (3) by 99, x - z - - 2 (4)
Substitute value of z in (3), x — 9 = — 2.
.-. s = 7.
Substitute values of x and 2 in (2), y = 5.
. \ number = 759.
206 ALGEBRA.
25. A number is expressed by three digits. The sura of the
digits is 9 ; the number is equal to forty-two times the sum of
the first and second digits ; and the third digit is twice the sum
of the other two. Find the number.
Let x = digit in hundreds' place,
y = digit in tens' place,
and z = digit in units' place.
Then x + y + z = 9
100a; + lOy + z = 42(3 + v)
2 = 2(a; + y)
From (2), 58s - 32y + z = 0
From (3), -2s- 2y + z = 0
Subtract, 60a; - 30y =0
Divide by 30, 2»-y=0 (4)
Subtract (3) from (1), 3 a; + 3y = 9
Divide by 3, x + y = 3 (5)
(4) is 2a;-y = 0
Add, 3 a; =3
.-. a;=l.
Substitute value of x in (5), y = 2.
Substitute values of x and y in (1), z = 6.
.*. number = 126.
26. A certain number, expressed by three digits, is equal to
forty-eight times the sum of its digits. If 198 be subtracted from
the number, the digits in the units' and hundreds' places will be
interchanged; and the sum of the extreme digits is equal to
twice the middle digit. Find the number.
Let x = digit in hundreds' place,
y = digit in tens' place,
and z = digit in units' place.
Then 100a; + lOy + z = 48(a; + y + 2) (1)
100a; + lOy + 2 - 198 = IOO2 + lOy + x (2)
and x + z = 2y (3)
From(l), 52a; -38y- 472 = 0 (4)
From (2), 99 a; - 992= 198.
Divide by 99, x - 2 = 2 (5)
From (3), a;-2y + 2 = 0 (6)
Subtract 19 X (6)from (4), 33 x - 66z = 0.
Divide by 33, a;-22 = 0 (7)
Subtract (7) from (5), 2 = 2.
Substitute value of 2 in (5), x = 4.
Substitute values of x ana 2 in (6), y = 3.
.*. number = 432.
teachers' edition. 207
27. A waterman rows 30 miles and back in 12 hours. He
finds that he can row 5 miles with the stream in the same time
as 3 against it. Find the time he was rowing up and down
respectively.
Let x = number of hours he rowed down,
and y = number of hours he rowed up.
By conditions,
x + y = 12
(1)
and
5a; = 3y
(2)
Transpose (2),
5a?-3y= 0
Multiply (1) by 3,
3x + 3y = 36
Add,
8* =36
.-. x = 4J.
Substitute value of x in
(1), 4J+y = 12.
28. A crew, which can poll at the rate of 12 miles an hoar
down the stream, finds that it takes twice as long to come up the
river as to go down. At what rate does the stream flow?
Let x — rate of pulling,
and y = rate of stream.
x + y = rate down stream,
x — y = rate up stream.
Then
a; + y = 12
a;-y= 6
(1)
(2)
Subtract, 2y= 6
. \ y = 3 = rate stream flows.
Substitute value of y in (1),
x + 3 - 12,
x = 12-3.
.-.a = 9.
29. A man sculls down a stream, which runs at the rate of 4
miles an hour, for a certain distance in 1 hour and 40 minutes.
In returning it takes him 4 hours and 15 minutes to arrive at a
point 3 miles short of his starting-place. Find the distance he
palled down the stream and the rate of his pulling.
208 ALGEBRA.
Let x = rate the man sculls,
and y = number of miles he goes.
Then x + 4 = rate going down the stream,
and x — 4 = rate going up the stream.
(x + 4) | = number of miles be goes.
.-.(* + 4)t-y (1)
and (a;_4)V=y-3 (2)
4x(l)is 20a + 80 = 12y (3)
3 X (2) is 51 x - 204 =U2y - 36 (4)
Subtract, - 31 x + 284 = 36
. \ x — 8, rate of pulling.
Substitute value of a? in (1), y = 20.
30. A person rows down a stream a distance of 20 miles and
back again in 10 hours. He finds he can row 2 miles against the
stream in the same time he can row 3 miles with it. Find the
time of his rowing down and of his rowing up the stream ; and
also the rate of the stream.
Let
x = rate of rowing,
and
y = rate of stream.
Then
2 3
X-y X+y
(1)
x+y x—y
(2)
Simplify (1),
i = 5y
(3)
Substitute this value of x in (2),
2^+20 = 10.
6y 4y
•••y = i
From (3), x = 4&.
°0
Therefore, -- — ■ = 4 (time of running down),
20
and — — - = 6 (time of rowing up).
4y — f
teachers' edition. 209
31. A grocer mixed tea that cost him 42 cents a pound with
tea that cost him 54 cents a pound. He had .30 pounds of the
mixture, and by selling it at the rate of 60 cents a pound, he
gained as much as 10 pounds of the cheaper tea cost him. How
many pounds of each did he put into the mixture?
Let x — number of pounds of tea at 42 cents,
and y = number of pounds of tea at 54 cents.
Then
Multiply (1) by 42,
x + y = 30
(1)
42a; -t-54y = 1800-
-420
(2)
42a + 42 y = 1260
42 s + 54y = 1380
12y = 120
.-.y = 10.
x + 10 - 30.
.\ x = 20.
32. A grocer mixes tea that cost him 90 cents a pound with
tea that cost him 28 cents a pound. The cost of the mixture is
$61.20. He sells the mixture at 50 cents a pound, and gains
$3.80. How many pounds of each did he put into the mixture?
Let x = number of pounds of tea at 90 cents,
and y => number of pounds of tea at 28 cents.
28 y = number of cents second kind cost.
Then 90 a; + 28 y = number of cents whole cost,
x + y = number of pounds in whole mixture,
and 50 (a: + y) = number of cents received.
Hence, 50s + 50y = 6500
(i)
90 a? + 28y = 6120
(2)
Multiply (1) by Jf , 28 x + 28 y = 3640
(3)
Subtract, 62 a; = 2480
.-.a; = 40.
Substitute value of x in (1), y = 90.
210 ALGEBRA.
33. A farmer has 28 bushels of barley worth 84 cents a bushel.
With his barley he wishes to mix rye worth $ 1.08 a bushel, and
wheat worth $1.44 a bushel, so that the mixture may be 100
bushels, and be worth $1.20 a bushel. How many bushels of
rye and of wheat must he take?
Let x = number of bushels of wheat,
and y = number of bushels of rye.
Then 2352 = cost in cents of barley,
144 a; = cost in cents of wheat,
108 y = cost in cents of rye,
and 12,000 ~ cost in cents of mixture.
x + y + 28 = 100,
z + y = 72 (1)
144a; + 108y + 2352 - 12000,
144a; + 108y = 9648 (2)
Divide (2) by 36, 4x + 3y = 268 (3)
Multiply (1) by 3, 3a + 3y = 216 (4)
Subtract, x = 52
Substitute value of a; in (1), y = 20.
34. A and B together earn #40 in 6 days; A and C together
earn ft 54 in 9 days ; B and C together earn $80 in 15 days. What
does each earn a day?
Let x =
= number of dollars A earns in one
» day,
y-
: number of dollars B earns in one
day,
and z =
= number of dollars C earns in one
day.
Then
x + y-Y
a; + « = 6
a)
(2)
and
y + * = tt
(3)
Simplify (1),
3a; + 3y -20
(*)
Simplify (3),
3y + 3z-16
(5)
Subtract,
3a; -3z= 4
(6)
Multiply (2) by 3,
3a; +3« = 18
CO
Add,
6a; =22
.-. s = 3f.
Substitute value of a; in (1), y = 3.
Substitute value of ar in (2), z = 2J.
teachers' edition. 21 1
35. A cistern has three pipes, A, B, and C. A and B will fill
it in 1 hour and 10 minutes ; A and C in one hour and 24 min-
utes ; B and C in 2 hours and 20 minutes. How long will It take
each to fill it?
Let x = number of minutes it takes A to fill it,
y = number of minutes it takes B to fill it,
and z = number of minutes it takes 0 to fill it.
-» -» -, = parts A, B, C can fill in one minute.
x y z
Hence, l + I-£ (1)
i 2 84
1=_1
' z 140
h\-± (3)
Add, and divide by 2, I + I + 1 - .1
J x y z 60
Subtract (1), z = 420.
Subtract (2), y - 210.
Subtract (3), x _ 105.
36. A warehouse will hold 24 boxes and 20 bales ; 6 boxes
and 14 bales will fill half of it. How many of each alone will it
hold?
Let x = number of boxes it will hold,
and y -* number of bales it will hold.
Then -, - =» parts one box, one bale, can fill.
x y *
xj 24 20 t /tx
Hence, — + — =1 (1)
x y '
A 6 14 1 ,ox
and _ + _ = _ (2)
4x(2)-(l)
Substitute value of y in (2),
24
20
—
+ — =
1
X
y
6
14
1
—
+ — =
—
X
y
2
36
1.
y
.-. y =
■36.
X-
54.
212 ALGEBRA.
37. Two workmen together complete some work in 20 days ;
but if the first had worked twice as fast, and the second half as
fast, they would have finished it in 15 days. How long would it
take each alone to do the work?
Let x = number of days it would take the first alone,
and y = number of days it would take the second alone.
Then -, - = parts they can do in one day,
- + _ = part both could do in one day,
x y
2 1
and - + — = part they could do if first worked twice as fast,
x 2y an(j Becond worked half as fast.
.•.i + i-i- (1)
and 2 + J_ 1 (2)
(1)-} of (2) is
x 2y 15
4y 60'
.-. y = 45.
Substitute value of y in (1), x = 36.
38. A purse holds 19 crowns and 6 guineas ; 4 crowns and 5
guineas fill J J of it. How many of each alone will it hold?
Let x = number of crowns bag holds,
y = number of guineas bag holds.
Then - = part of bag 1 crown occupies,
- = part of bag 1 guinea occupies.
and
5 x (1) - 6 X (2) is
Substitute value of x in (1),
L9 + ? = i
x y
(1)
4 5_17
x y 63
(2)
71 213
x 63
.\z = 21.
21 y
.-.y-63.
teachers' edition. 213
39. A piece of work can be completed by A, B, and C together
in 10 days ; by A and B together in 12 days ; by B and C, if B
work 15 days and C 30 days. How long will it take each alone
to do the work?
Let x = number of days it takes A,
y = number of days it takes B,
and z = number of days it takes C.
Then -, -, and -, respectively, = part each can do in one day.
x y 12
a 15 30 - /QX
and h — = 1 , (3)
Subtract (2) from (1),
V z
1 60*
\ z - 60.
15 1
Substitute value of z in (3), — + - = 1.
y 2
.-.y = 30.
Substitute value of y in (2), - + — ■ = — -»
x 60 12
1 = i-
x 20*
.\ x = 20.
40. A cistern has three pipes, A, B, and C. A and B will fill
it in a minutes ; A and C in h minutes ; B and C in c minutes.
How long will it take each alone to fill it?
Let x = number of minutes it takes A,
y = number of minutes it takes B,
and z = number of minutes it takes C.
Then -, -, -, respectively, = part each fills in one minute,
x y z x
and - + - = part A and B fill in one minute.
x y x
But - *= part A and B fill in one minute.
214 ALGEBRA.
.-.1+1-1 (i)
x y a
\+\'\ (2)
X Z 0
1+1-1 (3)
y z c
Add, and divide by 2, 1 + 1 + 1 - hc +a M + ab (4)
1
Subtract (1) from (4),
Subtract (2) from (4),
Subtract (3) from (4),
z
. z--
1
y~
•y=
i
a;
be
+ OC +
a6
2abc
ac + ab —
6c
2abc
2abc
ac + ab-
aft—ac+
6c
6c
2abc
2abc
ab
ac
— ac +
— ab +
6c
6c
2abc
2abc
ac — a6 + 6c
41. A man has $10,000 invested. For a part of this sum he
receives 5 per cent interest, and for the rest 4 per cent; the in-
come from his 5 per cent investment is $50 more than from his
4 per cent. How much has he in each investment?
Let x = number of dollars invested at £%,
and y = number of dollars invested at 4%.
Then x + y = total number of dollars invested.
.-. x + y = 10,000 (1)
5a;
As he receives 5% on x dollars, — = interest at 5%.
10 100
As he receives 4% on y dollars, —^ = interest at 4%.
But interest at 5% is $50 more than that at 4%.
... l£_i£ = 50 (2)
100 100 W
Simplify (2), 5s - 4y * 5000
Multiply (1) by 5, 5s + 5y=* 50000
Subtract, - 9 y = - 45000
.-. y = 5000.
Substitute value of y in (1), x = 5000.
teachers' edition. 215
42. A sum of money at simple interest amounted in 6 years
to $26,000, and in 10 years to #30,000. Find the sum and the
rate of interest.
Jet x = number of dollars at interest,
and y = rate of interest.
Then -^- = interest on x dollars for one year,
100
-— 2£ = interest on x dollars for six years,
100 *
and — ?&. = interest on x dollars for ten years.
100
...|^ + x = 26,000 (1)
and ^+ x = 30,000 (2)
Multiply (1) by 5, y^ + 5 x = 130,000
Multiply (2^ by 3, ^- + 3 x = 90,000
Subtract, 2x = 40,000
.-. x = 20,000.
Substitute value of x in (1), y = 5.
43. A sum of money at simple interest amounted in 10
months to $26,250, and in 18 months to $27,250. Find the sum
and the rate of interest.
Let x = sum,
and y = rate of interest.
Then -^- = interest for one year.
100 J
Since 10 months equals J of a year,
- of -^- = interest for 10 months,
6 100
and - of ^- = interest for 18 months.
2 100
But 26,250 - x = interest for 10 months,
and 27,250 — x = interest for eighteen months.
216 ALGEBRA.
.-. 1^=26,250 -x (1)
600 v '
and ^ = 27,250-* (2)
200
Multiply (1) by*, |g, 2**0-9,
Subtract, 0^100,000-4*
4a; =100,000.
.-.a; = 25,000.
Substitute value of x in (2), y = 6.
44. A sum of money at simple interest amounted in m years
to a dollars, and in n years to b dollars. Find the sum and the
rate of interest.
Let x = sum,
and y — rate of interest.
Then —^ = interest on sum for m years,
and ^2£ = interest on sum for n years.
.-.^+a; = a (1)
100 W
and ™% + x = b (2)
100 W
Multiply (1) and (2) by 100,
mxy + 100.t = 100a (3)
nxy +100* =1006 (4)
Multiply (3) by n, mnxy + 100 nx = 100 an
Multiply (4) by m, mnxy + lOOmx = 1006m
Subtract, lOOnx - lOOma; = 100an - 1006m
Divide by 100, nx — mx = an — 6m,
an — bm
.*. x =
n — m
Substitute value of x in (3),
many — m26y 100an— 1006m
t = lUUa.
7i — m 7i — m
Multiply by n — m,
many - m%by + 100 cm — 100 6m = 100 an — 100 am,
many — m26y = 1006m — 100 am,
any - m6y = 1006 - 100a.
100(6 -a)
y an — bm
teachers' edition. 217
45. A sum of money at simple interest amounted in a months
to c dollars, and in & months to d dollars. Find the sum and the
rate of interest.
Let x =
= sum,
and y =
= rate of interest.
Then -3U
100
= interest of $x for one year,
1200
= amount of $# for a months,
1200
= amount of $rc for 6 months.
™!L + x = c
1200
(i)
*%L + x = d
(2)
1200
Simplify (1),
axy + 1200 x = 1200c
(3)
Simplify (2),
% + 1200 a = 1200 d
(4)
Multiply (3) by
6 and (4) by a,
abxy + 12006a? «= 12006c
(5)
abxy + 1200oa; = 1200ac?
(6)
(6) -(5) is
1200s(a-&) = 1200 (ad-
ad — be
.*. x = — •
-be)
a — b
Substitute value of x in (3),
ay/«^|£\ + 12(X)(^^U 1200c.
Simplify, ay (ad- be) + 1200 (ad - be) = 1200(ac - be).
Transpose and unite, ay (ad — be) = 1200 a(c — d).
, 1200(e-rf)
ad— 6c
46. A person has a certain capital invested at a certain rate
per cent. Another person has $ 1000 more capital, and his capi-
tal invested at one per cent better than the first, and receives an
income $ 80 greater. A third person has $ 1500 more capital, and
his capital invested at two per cent better than the first, and re-
ceives an income $ 150 greater. Find the capital of each, and
the rate at which it is invested.
218 ALGEBRA.
Let x = capital,
and y — rat§.
Then (* + 1000)(y + fi - interest on capital $1000 greater, at 1%
100 greater,
and (x + 1500) (y + 2) = interest on capital $150o greater, at 2%
100 greater.
12L + 80 = interest on first capital, increased by $80,
^- + 150 = interest on first capital, increased by $150.
(s-HOOO)(y + l) = i^ 8Q (1)
100 100 W
(x-M500)(y + 2) = ^_ 15Q (2)
100 100 W
Simplify, xy + lOOOy + x + 1000 = xy + 8000,
ay + 1500y + 2x + 3000 = xy + 15000
Combining, 1000 y + a; - 7000 (3)
1500y + 2x= 12000 (4)
Multiply (3) by 2, 2000y + 2x = 14000 (5)
(4) is 1500y + 2s =12000
Subtract, 500y = 2000
.-. y = 4.
Substitute value of y in (5), x = 3000.
.% the capitals are $3000, $4000, $4500; and the rates 4%, 5%, 6%.
47. A person has $12,750 to invest. He can buy three per
cent bonds at 81, and five per cents at 120. Find the amount
of money he must invest in each in order to have the same
income from each investment.
Let x = number of dollara in three per cent bonds,
and y = number of dollars in five per cent bonds.
Then = interest of money invested in three per cents.
81
But J**i# =, interest of money invested in five per cents.
120
300a = 500y ,^
'"" 81 120
x + y = 12750 (2)
Reduce (1), 8a;-9y = 0
Multiply (2) by 8, 8a; + 8y=* 102000
Subtract, 17y- 102000
.-. y = 6000.
Substitute value of y in (2), x = 6750.
teachers' edition. 219
48. A and B each invested $1500 in bonds; A in three per
cents and B in four per cents. The bonds were bought at such
prices that B received $5 interest more than A. Both classes of
bonds rose ten points, and they sold out, A receiving $50 more
than B. What price was paid for each class of bonds?
Let x = amount paid for $1 three per cents,
and y = amount paid for $ 1 four per cents.
= face value of three per cents.
x r
1500
= face value of four per cents.
y
X — = income from three per cents.
x 100 r •
- X — income from four per cents.
y 100
Then »XA\_«X 3 N = 5
\ y lOOJ \ x lOOJ
V * loo) V y iwy
Simplify,
60 45&5
y z"
(1)
5-2-1
x y
(2)
Multiply (2) by 20,
_«2 + 60_20
y x
(3)
Add (1) and (3),
X
.-. 3 = 0.60.
Substitute value of x in
(2), 6-2-L
.-.y-0.75.
That is, the three per cents were bought at 60 and the four
per cents at 75.
220 ALGEBRA.
49. A fferson invests $10,000 in three per cent bonds, $16,500
in three and one-half per cents, and has an income from both in-
vestments of $ 1056.25. If his investments had been § 2750 more
in the three per cents, and less in the three and one-half per
cents, his income would have been G2\ cents greater. What
price was paid for each class of bonds?
Let x = amount paid on $ 1 three per cent bonds,
and y = amount paid on $ 1 three and one-half j>< r
cent bonds.
10000 3
Then X = number of dollars income from first invest-
* 10° ment,
and x -^- = number of dollars income from second in-
y 100 vestment.
.30000 115500 _ 105625 (1)
lOOx 200y
12*750 3
— — X — = number of dollars income of 3 per cents if
x 100 the stated addition in the amount invest-
1 ohca oi ed had been made,
— - — x — *- = number of dollars income of 3J per cents if
V 100 the stated deduction in the amount invest-
<iA9*n QA9*n e<* na(* Deen made.
Then ^^ + ^^ - number of dollars income on both.
lOOx 200y
... S + W"?1056'87} ^
Multiply (1) by 5, ±^- + ^2 = 5281.25 f (3)
Multiply (2) by 6, Sgg+fgf- 6341.25 (4)
Subtract, 1^2=1060,
100a;
106000 a; = 79500.
.-. x=0.75.
That is, the 3 per cent bonds were bought at 75.
Substitute value of x in (1),
30000. + 115500 = 105625>
75 200y
115500
656.25.
200y
.-. y = 0.88.
That is, the 3£ per cent bonds were bought at 88.
teachers' edition. 221
. ' - c _.
50. The sum of $2500 was divided into two unequal parts
and invested, the smaller part at two per cent more than the
larger. The rate of interest on the larger sum was afterwards
increased by 1, and that of the smaller sum diminished by 1 ; and
thus 'the interest of the whole was increased by one-fourth of its
value. If the interest of the larger sum had been so increased,
and no change been made in the interest of the smaller sum, the
interest of the whole would have been increased one-third of its
value. Find the sums invested, and the rate per cent of each.
Let x = number of dollars in larger part,
and y = number of dollars in smaller part.
Then x + y = 2500 (1)
Let z = rate per cent on larger part,
and z 4- 2 = rate per cent on smaller part.
Then xz + y(z + 2) — interest on whole amount.
Changing rate per cent,
2 + 1 = rate per cent on larger part,
and 2 + 1 = rate per cent on smaller part.
Then x (z + 1) -f y (z + 1) = interest on whole after change.
Then z(2 + l)+y(2 + l) = $ [xz+y(z + 2)] (2)
Changing rate per cent again,
2 + 1 = rate of larger part,
2 + 2 = rate of smaller part.
Then x(z+ 1) + y (2 + 2)= %[xz + y (2 + 2)] (3)
Simplify (2), 4 x — 6 y — xz — yz = 0.
Simplify (3), 3 x - 2y - xz - yz = 0. (4)
Subtract, x — 4y = 0 (5)
Subtract (5) from (1), by = 2500.
.-. y = 500.
Substitute value of y in (4), x = 2000.
Substitute values of x and y in (3),
6000 - 1000 - 20002 - 5002 = 0,
~2500z = -5000.
.-. 2 = 2.
222 ALGEBRA.
51. If the sides of a rectangular field were each increased by
2 yards, the area would be increased by 220 square yards ; if the
length were increased and the breadth were diminished each by
5 yards, the area would be diminished by 185 square yards.
What is its area?
Let x = number of yards in length,
and y = number of yards in width.
Then xy — number of yards in area.
(* + 2)(y-f2) = sy + 220 (1)
(x + 5)(.y-5) = :ry~185 (2)
Simplify (1), xy + 2x + 2y + 4 = xy + 220, .
2x + 2y = 216,
x + y = 108 (3)
Simplify (2), xy - bx + by - 25 = xy - 185,
5a-5y = 160
x-y = 32 (4)
Add (4) and (3), 2 x =140
.-.3 = 70.
Subtract (4) from (3), 2 y = 76.
.-.y = 38.
.*. xy = 2660 square yards.
52. If a given rectangular floor had been 3 feet longer and 2
feet broader it would have contained 64 square feet more ; but if
it had been 2 feet longer and 3 feet broader it would have con-
tained 68 square feet more. Find the length and breadth of the
floor.
Let x = number of feet in length,
and y = number of feet in breadth.
Then xy = number of feet in surface.
(s + 3)(y + 2) = zy + 64 (1)
(z + 2)(y + 3) = zy + 68 (2)
Simplify (1), xy + Zy + 2x + 6 = xy + 64.
3y + 2a = 58 (3)
Simplify (2), xy + 2y + 3a; + 6 = xy + 68.
2y + 3x = 62 (4)
Multiply (3) by 2,
Multiply (4) by 3,
Subtract,
Substitute value of x in (3),
6V
+ 4z =
116
Oy
+ 9z =
186
~bx =
-70
.-. x =
14.
32/
+ 28 =
3y =
.-. y =
58,
30.
10.
TEACHERS EDITION.
223
53. In a certain rectangular garden there is a strawberry-bed
whose sides are one-third of the lengths of the corresponding
sides of the garden. The perimeter of the garden exceeds that
of the bed by 200 yards ; and if the greater side of the garden
be increased by 3, and the other by 5 yards, the garden will be
enlarged by 645 square yards. Find the length and breadth of
the garden.
I
*
Let x = number of yds. in length of garden,
and y = number of yds. in width of garden.
Then 2x + 2y = perimeter of garden,
and xy = area of garden.
Also, | = number of yds. in length of bed,
and
Then
Add 3 to one side of garden, x + 3,
| = number of yds. in width of bed.
o
— + -^ = perimeter of bed.
Add 5 to other side of garden, y + 5.
Then (s + 3)(2/ + 5) = area.
Simplify,
Simplify,
Multiply (1) by 5,
Subtract,
+ =^ = 200.
3 3 )
x + y = 150
(a> + 3)(y + 5) = zy + 645.
5z + 3y = 630
5a; + 5y = 750
Substitute value of y in (1),
2y = 120
.-. y = 60.
x 4- 60 = 150.
.-.a; = 90.
(1)
(2)
224 ALGEBRA.
64. In a mile race A gives B a start of 100 yards, and beats
him by 15 seconds. In the second trial A gives B a start of 45
seconds, and is beaten by 22 yards. Find the rate of each in
miles per hour.
Let x = number of yards A runs in one second,
and y = number of yards B runs in one second.
Since there are 1760 yards in one mile,
and = number of seconds A ran in first and second
x x trials respectively,
and = number of seconds B ran in first and second
y y trials respectively.
Then 1^_1^=15 0)
and I?k_i™ = _45 (2)
x y
Multiply (1) by 88, 146080 _ 154880 = ^
y x
™ u- i /ox k Qo 146080 144254 0„0K
Multiply (2) by 83, + = — 3735
Add, -^^ = -2415
x
Therefore, A runB 4^*^ yards, or ^fa of a mile, in one second,
and in one hour (= 3600 seconds), 9 miles.
Substitute value of a? in (1), y = 4.
Therefore, B runs 4 yards in one second, or 8-j^- miles in one
hour.
55. In a mile race A gives B a start of 44 yards, and beats him
by 51 seconds. In the second trial A gives B a start of 1 minute
and 15 seconds, and is beaten by 88 yards. Find the rate of each
in miles per hour.
Let x = number of yards A ran in one second,
and y = number of yards B ran in one second.
, = number of seconds A ran in first and second
x x trials, respectively.
1 71 6 1 7fi0
-- — , -— — =« number of seconds B ran in first and second
V y trials, respectively.
teachers' edition. 225
Then
1716 1760 _
y x
(1)
and
1672 1760 _
x y
(2)
Multiply (1) by 19,
33440 +32&04_ ^
x y
Multiply (2) by 20,
33440 35200 1rAA
== — 1500
• x y
Add,
_?5?6 = _ 531
y
Therefore, B runs 4f yards per second, or 10 miles per hour.
Substitute value of y in (1), x = 5|$.
Therefore, A runs 5|| yards per second, or 12 miles per hour.
56. The time which an express-train takes to go 120 miles is
T9T of the time taken by an accommodation-train. The slower
train loses as much time in stopping at different stations as it
would take to travel 20 miles without stopping; the express-
train loses only half as much time by stopping as the accommo-
dation-train, and travels 15 miles an hour faster. Find the rate
of each train in miles per hour.
Let x = the rate of the accommodation-train,
and y = the rate of the express-train.
120
= number of hours accommodation-train goes 120
x miles without stopping,
120
= number of hours it takes express-train to go 120
V miles without stopping,
20
— = number of hours accommodation-train loses in
x stopping,
— = number of hours express- train loses in stopping,
120 20
1 = number of hours accommodation -train goes 120
x x miles including stops,
1 = number of hours express-train goes 120 miles
V x including stops.
226 ALGEBRA.
120 10 9/120 20 \ m
— + — = tt — + — )' 0)
y x 14 \ x x J
y-x=15 (2)
Simplify (1), 120a; - 80y - 0
Multiply (2) by 80, ♦ -80a + 80 y = 1200
Add, 40s = 1200
.\x = 30.
Substitute value of x in (2), y — 45.
57. A train moves from P towards Q, and an hour later a
second train starts from Q and moves towards P at a rate of 10
miles an hour more than the first train ; the trains meet half-way
between P and Q. If the train from P had started an hour after
the train from Q, its rate must have been increased by 28 miles in
order that the trains should meet at the half-way point. Find
the distance from P to Q.
Let x = number of hours first goes half the distance.
Then x — 1 = number of hours second goes half the distance.
Let y = rate of first.
Then y + 10 = rate of second.
Hence, xy = one-half of the whole distance.
But (x— l)(y -l- 10) = one-half the whole distance.
.•.(*-l)(y + 10)-ay.
Simplify, 10a;-y = 10 (1)
In Becond statement,
if x — 2 = number of hours "first goes half distance,
and y + 28 = rate of first,
then (x — 2)(y + 28) = one-half of the whole distance.
But, from (1), xy = one-half of the whole distance.
.-.(a-2)(y + 28)-*y (-)
Simplify (2), 28z-2y = 56
2x(l)is 20z-~2y = 20
Subtract, 8x =36
.\*-4J.
Substitute value of x in (1), y = 35.
Therefore, one-half the distance, xy, is 157J miles, and the
whole distance is 315 miles.
teachers' edition. 227
58. A passenger-train, after travelling an hour, meets with an
accident which detains it one-half an hour ; after which it pro-
ceeds at four-fifths of its usual rate, and arrives an hour and a
quarter late. If the accident had happened 30 miles farther on,
the train would have been only an hour late. Determine the
usual rate of the train.
Let x = number of miles train usually goes per hour,
and y = number of miles train travels.
^ = number of hours usually required,
x
and 2L — = number of hours actually required after accident.
y
Since the detention is J hour, and the train is 1 J hours late, the
loss in running-time is } of an hour.
. y-x y-x 3 (1,
4z x = 4 K '
5
If the accident had occurred 30 miles farther on, the loss in
running-time would have been J an hour.
y-g-30 y - a: - 30 _ 1 ,™
\x - -o v;
5
Simplify (1),
Simplify (2),
Subtract, x = 30
59. A passenger-train, after travelling an hour, is detained 15
minutes ; after which it proceeds at three-fourths of its former
rate, and arrives 24 minutes late. If the detention had taken
place 5 miles farther on, the train would have been only 21 min-
utes late. Determine the usual rate of the train.
Let x = usual rate of train per hour,
and y = number of miles train has to run.
y — x = number of miles train has to run after detention,
= number of hours usually required to run y — x
x miles,
and r = number of hours actually required to run the
~* y — x miles.
4 ^
x
2
y-
-4ar
= 0
y-
-3*
= 30
228 ALGEBRA.
Since the detention was 15 minutes, and the train is 24 minutes
late, the loss in running- time is 9 minutes = fa of an hour.
. y~x y~x=S (I)
" 'Sx a; 20 w
4
If the detention had occurred 5 miles farther on, the loss in
running-time would have been 6 minutes = ^ of an hour.
• .V~-T~5 y—x—h^ 1 ,2)
3j x 10 K '
4
Simplify (1), 20y-29s = 0
Simplify (2), 20y - 26 x = 100
Subtract, 3 x =100
.-. s = 33J.
60. A man bought 10 oxen, 120 sheep, and 46 lambs. The
cost of 3 sheep was equal to that of 5 lambs ; an ox, a sheep, and
a lamb together cost a number of dollars less by 57 than the
whole number of animals bought; and the whole sum spent was
$2341.50. Find the price of an ox, a sheep, and a lamb, respec-
tively.
Let x -= number of dollars paid for an ox,
y = number of dollars paid for a sheep,
and z = number of dollars paid for a lamb.
lOx + 120y + 46z = number of dollars paid for all.
.-. lOar + 120y + 48z = 2341.50 (1)
x + y + z = 119 (2)
3y = 5z (3)
(1) is lOx + 120y + 46z = 2341.50
Multiply (2) by 10, 10 x + lOy + lOz = 1190
Subtract, 110y + 36z = 1151.50 (4)
Multiply (4) by 3, 330y + 108 2 = 3454.50.
Multiply (3) by 110, 330y - 550z = 0.
Subtract, 658 z = 3454.50.
.-. z = 5.25.
Substitute value of z in (3), y = 8.75.
Substitute values 01 y and 2 in (2), x = 105.
teachers' edition. 229
61. A farmer sold 100 head of stock, consisting of horses,
oxen, and sheep, so that the whole realized $ 11.75 a head ; while
a horse, an ox, and a sheep were sold for $110, $62.50, and
$7.50, respectively. Had he sold one-fourth of the number of
oxen that he did, and 25 more sheep, he would have received the
same sura. Find the number of horses, oxen, and sheep, respec-
tively, which were sold.
Let x = number of horses,
y = number of oxen,
and z = number of Bheep.
Then a; + y + 2 = 100 (1)
110* + 62Jy + 7i2 = 1175 (2)
and no* + (lx^)+lf+^=1175 (3)
Multiply (3) by 8, 880a; + 125y + 602= 7900
Multiply (2) by 8, 880 x + 500 y + 60 z = 9400
Subtract,
-375y
= -1500
-y-
= 4.
Substitute value of y in (1), x + z = 96
Substitute value of y in (2),
110* + 250 + 7*2 = 1175.
110a: + 7*2 = 925.
Multiply by 2,
Multiply (4) by 15,
220 a; + 152 =
15a; + 152 =
= 1850
= 1440
Subtract,
205 a;
- 410
,:x = 2.
* Substitute values of x and y in (1), 2 — 94.
(4)
230 ALGEBRA.
62. A, B, and C together subscribed $100. If A's subscrip-
tion had been one-tenth less, and B's one-tenth more, C's must
have been increased by $ 2 to make up the sum ; but if A's had
been one-eighth more, and B's one-eighth less, C's subscription
would have been $ 17.50. What did each subscribe?
Let x — number of dollars A subscribed,
y = number of dollars B subscribed,
and z = number of dollars C subscribed.
-^ = -j^ of A's subscription,
-— ^ = ii of B's subscription,
z + 2 = $ 2 more than C's subscription,
9x
— = { of A's subscription,
8
-% — J of B's subscription,
8
100 = number of dollars all subscribed,
..* -|- — V- + z + 2 = number of dollars all subscribed,
10 10
— + — %- + 17.5 = number of dollars all subscribed.
8 8
.-. x + y + z = 100
(1)
?* + litf + 1 + 2-100
10 10
(2)
^ + Iy + 17.5 = 100
(3)
Multiply (1) by 10,
Simplify (2),
10a; + 10y + 10z= 1000
9a; + lly + 10a= 980
Subtract,
x - y =20
(4)
Simplify (3),
Multiply (4) by 7,
9a; + 7y = 660
.7z-7y = 140
Add,
lGa; = 800
.-. a; = 50.
Substitute value of x in (4), y — 30.
Substitute values of x and y in (1), z = 20.
TEACHERS* EDITION. 231
63. A gives to B and C as much as each of them has ; B gives
to A and C as much as each of them then has ; and C gives to A
and B as much as each of them then has. In the end each of
them has $6. How much had each at first?
Let - x = number of dollars A had at first,
y = number of dollars B had at first,
and z — number of dollars C had at first.
x — y — z = number of dollars A had at 1st distribution,
2y — number of dollars B had at 1st distribution,
2z = number of dollars C had at 1st distribution,
2y— {(%— y— z)+2z} = number of dollars B had at 2d distribution,
or 3y — x — z = number of dollars B had at 2d distribution,
2x — 2y — 2z = number of dollars A had at 2d distribution,
4z = number of dollars C had at 2d distribution,
4z-{(2x— 2y-2z)+(3 y-x-z)}} or 7z - x - y
= number of dollars C had at 3d distribution,
4 a; — 4y — 4z = number of dollars A had at 3d distribution,
Qy — 2x — 2z = number of dollars B had at 3d distribution.
.\ 7z-x-y = 6 (1)
4a;-4y-4z = 6 (2)
Gy-2x-2z = 6 (3)
Multiply (1) by 4, 28z - 4s - 4y = 24
(2) is -4z+4a;-43/= 6
Add, 24z -8y = 30 (4)
Multiply (1) by 2,
(3) is
Subtract, 16z -83/= 6 (5)
(4) is
Subtract, 8z =24
.-. z = 3.
Substitute value of z in (5), y — 5J.
Substitute values of y and z in (1), x = 9}.
24z
-8y = 30
14z-
-2a;-2y=12
-2z-
-2x + 6y= 6
16z
-83/= 6
24z
-8y = 30
232 ALGEBRA.
64. A pays to B aud C as much as each of them has ; B pays
to A and C one-half as much as each of them then has ; and C
pays to A and B one-third of what each of them then has. In
the end A finds that he has $1.50, B $4.16f C $0.58*. How
much had each at first?
Let x = number of dollars A had at first,
y = number of dollars B had at first,
and z = number of dollars C had at first.
x — y — z = number of dollars A has left after giving to B
andC,
2y — number of dollars B has after A pays him,
2z = number of dollars C has after A pays him,
x~ y ~ — - = number of dollars A has after B pays him,
y ~g~J „ number of dollars B has left after paying A and C,
3z = number of dollars C has after receiving B's money,
x~ — y^ — - = number of dollars A has after C pays him,
y ~ z = number of dollars B has after C pays him,
— Z~~X~~V = number of dollars C has after paying A and B.
4<r-4y-4«_1J0 {1)
2 v '
10y-2x-2z = 416? (2)
O
ll*-*-y==0.58t (3)
Simplify (1), x-y-z= 0.75 (4)
Simplify (2), 10y - 2x - 2z = 12.50 (5)
Simplify (3), 11 z - x - y = 1.75 (6)
(4) is x-y-z= 0.75
Add (4) and (6), 10z - 2y = 2.50 (7)
(5) is 10y-2x-2z = 12.50
Multiply (4) by 2, -2.y + 2x-2g- 1.50
Add, 8y -4z = 14.00 (8)
Multiply (7) by 4, - Sy + 40 2 = 10.00
(8) is 8y- 4z = 14.00
Add, 36 z = 24.00
.\ z = 0.66f.
y = 2.08}
Substitute values of y and z in (4), x « 3.50.
Substitute value of 2 in (8), y = 2.08J.
ad z '
teachers' edition. 233
Exercise LXXVI.
5.
V2as&V
1. (a*)2 2. (z5)3 3. (afy3)8
=*a8xS = i5x3 =a;2xay3xa
= a6. -a;15. = z*y«.
a3x*&ax4 ^ 35a;axV
" 24 ""^a8"5^5
^ala68 _ 243 x10 y5
16 ' 32 a15 610'
6. (s + 2)8
= (^ + 3(^(2) + 3(s)(2)' + (2)»
= a* + 6a?+12a; + 8.
7. (a: -2)*
= a*- 4(a;)8(2) + 6(af(2)a- 4(a;)(2)8 + 2*
= z*-8a*+24a*-32ar + 16.
8. (x + Sf
= (af + 5(a;)4(3) + 10(^(3)' + 10(a;)*(3)8 + 5(a;)(3)4 + (3)5
= a* + 15a* + 90a* + 270ar» + 405a; + 243.
q /I i 9<r\5
' = l5 + 5(2a;) + 10(2a;)a + I0(2xf + 5(2a;)4 + (2a;)5
= 1 + 10a; + 40 a? + 80 x9 + 80 a* + 32a£
10. (2m -l)8 13. (-7m8naV)3
= (2m)8-3(2m)2+3(2m)-(l)8 = - 7am3x*naa*xVx2
= 8m8 - 12ma + Sm -1. = 49m«n2 »V-
11. (2aa&c»)4 1A / 2a?yV
= 2*a2x46icSx4 14, '
= 16a86icls.
5
(-
_ 25a?x*y5
12. (-bax*y*? = S^^c5
= -58a8a^xSy2x3 32a;15v5
= -125a8a*y«. . = ~2l3^W
15. (3 a; + 1)*
= (3a?)4 + 4(3a;)8 + 6(3a;)a + 4(3a;) + 1
- 81a* + 108ar* + 54a? + 12a; + 1.
16. (2a; -a)*
= (2a;)4 - 4(2a;)3(a) + 6(2a;)a(a)a - 4(2a?)(a)8 + (a)4
= 16a* - 32aa* + 24a2a? - 8 a3a; + a4.
17. (3x + 2af
- (3a;)5 + 5(3a?)4(2a) + 10(3a;)8(2a)2
-f 10(3a;)a(2a)s + 5(3a;)(2a)4 + (2a)5
- 243a* + 810aa* + 1080a2*3 + 720a8a:2 + 240a4a; + 32a5
234 ALGEBRA.
18. (2a; -y)*
= (2*)*-
= 16a?-
19. (x*y-2ry*f
(2a; -y)*
= (2xf-A(2xf(y) + e(2xyW-^(2x)(yf^(yY
= 16a?- 32a?y + 24a?y* - 8xy* + y*.
(xty-2xy1f
= (^yy-G(^yf(2xy^)+l5(^y)\2xy^-20(x*vf(2xy^
+ I5(x*yn2xy')*-6(x*y)(2xyfy+(2xy*j
- a?V - 12a?»y7 + 60a?°y« - 160a?y» + 240a?y*°
if
- 192a? y11^- 64a?y»
20. (ab-Sy
= (oft)7 - 7(o5)«(3) + 21(oft)5(3)» - 35(oft)*(3)s
+ 35(a&)*(3)* - 21 (aftf (3)* + 7 (a6)(3)« - (3)'
= aT67 - 21 a6 66 + 189a5 6* - 945 a* 6*
+ 2835 a« 6s - 5103 a* 6s + 5103 ab - 2187.
21.
(-3a'6'C)* 05 / a?yVY
= -35a2*5&2*5c5 *°" I 2~"J
243a»ft»A V ^xTyixy^ixr
22. (-3 ay3)* 27
= -3*a?yS*« a;1**'1**
= 729a?y» ^3—
23. (-5 a2 6a?)5
= -55a2x*ft5a^x*
- S12WW. 26. (l-a-a«)*
0>( / 3a6»V = {l-(a+a*)}*
24. -— ) -l»-2(a + tf) + (<i + «fr
V **"/ =l-2a-2as+a2 + 2as + a4
= 81 a4 y = 1 - 2a-a2 + 2os + 0*.
256 <?*
27. (2-3* + 4x»f
-[2-(3x-4^)]8
= (2)» - 3(2)3(3jc - 4 a?) + 3(2)(3x -4a?)» - (3a; - 4 a?)8
= S - 36a; + 48a? + 54a?2 - 144a?
+ 96 a? - 27a? + 108a? - 144a? + 64a*
- 8 - 36ar + 102a? - 171 a? + 204a?- 144a? + 64a?.
28. (l-2x + a?)»
= {(l-2z) + a?}»
= (1 - 2xf + 3 (1 - 2x? a? + 3(1 - 2a;)(a?)a + (a?)3
= 1 - 6x + 12a? - 8a? + 3 a? - 12a? + 12a? + 3a? - 6a? + a?
= 1 - 6a; + 15a? - 20a? + 15a? - 6a? + a?.
29. (l-a;+a?)s
= U-(z-a?)}*
« I _ 3 (x - a?) + 3 (x - a;*)* - (x - a?)*
= 1 - 3a; + 3a? + 3a? - 6a? + 3a? - a? + 3a? - 3a? + a?
- 1 - 3 x + 6a? - 7a? + 6a? - 3a? + a?.
TEACHERS EDITION.
235
(1 + X + 3*)*
= {l+(* + 3*)}*
= 1* + 4(l)3(a; + a?) + 6(l)2(ar + a*)' + 4(1) (a? +a?)3 + (a; + a?)*
= 1 + 4a; + 4a;2 + 6a;2 + 12a? + 6a? + 4a? + 12a* + 12a?
+ 4a;6 + x* + 4a? + 6a? + 4a;7 + a?
= 1 + 4a; + 10a;2 + 16a? + 19a? + 16a? + 10a? + 4a?7 + a?.
Exercise LXXVII.
1. Va* = ±a*.
v^? = ±a?,
^64 = 4,
v^l6a12W = ±2a36c2,
v'-32a* = -2a3.
2. -fr-1728 <?d"ay= -12cWgy»t
^33756"?*= 15 6V,
Mfcv^X 3* x 74,
v^2*x3*x7W= ± 42 c4*.
3. V53361 6* (*y»«»
32
7
7
ll2
53361
5929
847
121
V53361=V32x72xll2.
• ^63361 6*C8yi2zi«= ± 231 Pfety«cP.
▼ 343 22*
2a?
66<?
i
64 a^8
729 z30*
3s5
4. V25a2Mc2+v/8a36V-^81a*y<?-v^32as610(?
= V52a26V+^23a36«c3-v/3*a*68c*-v/26a56l0c5
- 5a£2c + 2a62c - 3a62c - 2a62c
= 2a62c.
5. v^7^6x^243^xVl6^V
= 3ay2x3yzx4a;2z
= 36a?y322.
6. 4V2x -y/abxy + 5y/a?b*xy !
= 4 V2 x 2 - Vl x3_x2x6+5Vl2x33x2x6
= 4\/4-V36 + 5V324
= 4x2-6 + 5x18
= 92.
7. 2ay/Sax + by/\2by + 4a5a;Vkcy
= 2xlV8xlx2+3v^l2x3x6 + 4xlx3x2V3x2x6
= 2Vl6 + 3^216 4- 24V36
= 2x4 + 3x6 + 24 XG
= 170,
236
ALGEBRA.
8. Vas + 2ab + » X v'a8 + 3a2 6 + 3a&2 + 6s
= (a + 6) x (a + 6)
= (l + 3)x(l + 3)
= 16.
9. vfo - 362a + 36a* - a8 -s- V62 + a2- 2ab
= ^(5_a)8_5_V(6-a)a
= tf-a)-*-(6-a)
Exercise LXXVIII.
1.
a* + 4a8 + 2a* - 4a + lla2 + 2a-l
a*
2a* + 2a
4a8
4a8
+ 2a2
+ 4a2
2a8 + 4a
-1
-2a2-4a + l
-2a2-4a + l
x*-2x*
2.
y + 33y-2ay + y4|32-
- ay -f y3
2^-^-2^ + 3^^
y|-2r»y + s2y2
2a2 — 2xy + ya
2a?y2- 2ay +y*
2s2y2-2a;y3 + y*
3.
4a*-12q5a; + 5a*j2 + 6a33r» + ayi2q8-3a2j-q32
4a6
4a8- 30*71 -12a5a? + 5aV
h9aV
to*
30^]-12a5a: +
l-12a5a?4-
4a8 — 6a2jc — oar*
-4aV + 6aV + aV
-4aV + 6a8ar»4-aV
9a*-24a;V-12^v3+16a;V+16xv5+4v613^-4a^-2y3
9a*
6a*-4:cya
-24a;y-12^y8+16xy
24a74y2+16ar>y*+16g2y4
6arl-8a;y2-2y3|-12ic3y8
I -12 ay
+16.Ty5+4y6
+16 ay5 +4 y*
teachers' edition.
237
5.
4a8 4- 16a6 c2 - 32a2 c6 4- 16c8! 2a4 4- 4a2 c2 - 4c*
4a8
4a4 4- 4a2^21 16a'c2 ~ 32a2 c6
| 16 a6
c^iea4^
4a44-8a2c2-4c4
-16a4c4~32a2d64-16c8
lGa4c4-32a2c64-l(>c8
4ar4 - 20ar> 4- 37a;2 - 30a; 4- 9 |2s2-5.r4-3
4a*
±x*-bx
-20^ + 37^
-203s 4- 25 s2
4s2 -10a; 4- 3
12a* -30a; 4- 9
12a;2 -30a; 4- 9
16a*- 16a6a* 4- 1662a* 4- 4a262 - 8a63 4- 464|4a;2- 2a6 4- 262
16a*
8a*-2^-16a&a*4-16&2a* + 4a2&2
|-16a6a;2 4-4a262
1662a* -8a63H-464
1662a* -8a634-464
8a*-4a&4-262
8.
16 - 24a; + 25a* - 20a* 4- 10a* -4a* 4- a*[4- 3s 4- 2a* -a*
16
8-3a;
-24 a; 4- 25 a;3
-24a; 4- 9xa
-20 a* 4- 10a*
-12a* 4- 4a*
8-(
\x + 2xl
16a^
16 a*
8-6a?4-4a*
-a*
- 8a* 4- 6a*-4a*4-a*
- 8a* 4- 6a*-4a* 4- a*
9.
a*-4a*y4-8aV-10a^y34-8a;2y4-4^v54-/lar>-2a^.v4-2a;.v2-y3
a*
2x*-2x*y
2a*-4x2y + 2xy2
-4a*y+8a*y2
- 4 a*y 4- 4a* ya
2a*-4a*y4-4a^2 — if
4a*y2-10a*ys4-8a*y*
4a*y2- 8ar>y34-4ar'y4
— 2a*y34-4a*y4— 4a:y54-y6
2 a* y8 4- 4 x2 1/4— 4 x?f 4- y6
238
ALGEBRA.
10.
4a« _ 4a5_ iia* + i4as + 5aa_ 12a + 4l2a8-c^-3a + 2
4a«
4a8 -"a8"
— 4a{
-4a!
>-lla4
> + a*
14a8 + 5a*
6a8 + 9a8
4 a8- 2a'
-3a
-12a4 +
-12a4 +
4a8 -2a1
-6a + 2
8a8-4a8-12a + 4
8a8-4a2-12a+4
11.
1 3a-b+ 5c+d
9 a8 -6 a&+68+30 ac-10 6c+25c8+6aa^-26d+10 cd+d8
9a8
6a-6
-6a&+68
-6a&+68
6a-26 + 5c
30OC-106C+25C8
3000-1060+25^
6a-2b + 10c + d
12.
Gad^&d+lOcd+d8
6ad-26aT+10cd+d2
|5a?8-3a8y-4gyl + y8
25 a* - 30a*y - 31a*y8 + 34*V + 10a"y* - 8xy5 + y6
25 a*
10a* -3*^
10a*-6a*y-4zy8
-30a^y-31a*y"
-300*7+ 9g*.Va
10a*- 6a*y- 80^ + 3*
40a?*y8 + 343*^ + lOa^y4
- 40a*y8 + 243*3* + 16 a* y4
103*3*- ea^^-Sa^ + y6
10aV- 6a?V- Sgy5 + y»
13.
| m4— 2 m8+3 ro8— 4 m+5
m8-4m7+10m6-20m5+35m4-44m8+46m2-40wH-25
2m4-2ma
-4m7+10m6
-4mT+ 4m6
2m4-4m8+3m8
6m«-20m5+35m4
6m6-12m5+ 9m4
2 m4— 4 m3+6 m8— 4 m
- 8ra5+26m4-44m8+46m8
- 8m5+16m4-24m8+16m8
2m4-4m8+6m8-8m+5
10m4-20m8-f30m8-40m+25
10m4-20m8+30m8-40m4-25
TEACHERS EDITION.
239
14.
x* — x*y — Jafy2 + xy* + y4 \x* — jxy — y2
x*
2x*-ixy
2x*-xy-y2
• a?y + j&y*
- 2a?y + xy* + y*
2s2y2 + sy3+y4
15.
a4-4aV+6aV-6a^, + By4-— + ^l
a4 xx
x2 — 2xy + y* — *
2xi-2xy
— 4a?y + 6afy2
— 4afy + 4a2y2
2a? — 4a;y + ya
2afy2-6;Fy8 + 5y4
2sy-4gy8+ y*
2a?-4a;y + 2y2 -
y8
-2^.+ 4^-3£ + g
-W + 4/-tf + S
16.
^_^j?4- 43 g2a;2 3 as3 a^la2 3aa? . a?
9 2 48 4 4 3,42
_9
2a2 Sax
3 4
asa? 43a2a;2
2 48
a8a? 27 oV
2 48
.)/^_3aa2\ of
-U 4 y 2
aV_3ar» ^
3 4 4
aV_3aa? a£
3 4 4
240
ALGEBRA.
17.
10 20,25 24.
X2 X* X4, X6
1+- + --5 + -3 + -Z -f — +
f 2 3 4
9
2 + -
X
4
- +
4
- +
X
10
4
X*
20 25
12 9
X* + X*
"+1
X
♦s
4 6
X X2
4
8 16 24 16
x3 x* b6 je6
8 16 24 16
X3 X* if Xs
18.
is 2o , o 26^6*|a , ^b
>* o a ar\b a
2a
-1
-^ + 3
b
b
6
?,b
2a
« 6
^V
z + -
2-
+ —
o a
a
a1
o
2b
+ *'
a
a«
19.
^ + ^ - 3 + 9 +2 3
12
2*2 + f
2xJ + a;-i - —
X8
_5r*
12
^-T"
4
1
2z*
3 9
3
3
2s2
-2 + 1
3 9
3
teachers' edition.
241
Exercise LXXIX.
(1)
(2)
(3)
(4)
(5)
120409(347
9
(1)
16803.9369(129.63
1
64)304
22J68
256
44
687)4809
249)2403
4809
2241
2586)16293
15516
4816.36(69.4
25923)77769
77769
36
129)1216
(2)
4.5449976i (2.1319
1161
4
1384) 5536
4lj54
5536
41
423)1349
1269
1867.104i (43.21
4261)8097
16
4261
83)267
42629)383661
249
383661
862)1810
1724
(3)
0.24373969(0.4937
8641) 8641
16
89)837
8641
' 801
983)3639
1435.652i (37.89
9
2949 »
9867)69069
69069
67)535
469
(4)
0.5687573056(0.75416
748)6665
49
5984
145J787
7569)68121
725
68121
1504)6257
6016
64.128064(8.008
15081)24130
64
15081
16008)128064
150826)904956
128064
904956
242
AU1KBRA.
3.
(1) 0.9000000000(0.94868
81
184)900
736
1888)16400
15104
(5)
18966) 129600
113796
189728)1580400
1517824
17.00(4.1231
16
8ljl00
81
822)1900
1644
8243)25600
24729
82461)87100
82461
(2)
(3)
W
6.21(2.4919
4
44J221
176
489)4500
4401
4981)9900
4981
49829)491900
448461
0.43(0.6557
36
125)700
625
1305)7500
6525
13107)97500
91749
0.008620(0.0923
81
182)420
364
1843)5600
5529
(6)
(7)
i29.O0060606(lL3578
2l}29
21
223)800
2265)13100
11325
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158949
227148)1855100
1817184
347.2590(18.6348
1
28 J247
224
366)2325
2196
3723)12990
11169
37264)182100
149056
372688)3304400
2981504
TEACHERS* EDITION.
Ui
(1)
(2)
(3)
14295.3870(119.5633
1 '
21)42
21
229)2195
2061
2385) 13438
11925
(4)
23906) 151370
143436
239123)793400
717369
2391263) 7603100
7173789
2.50000(1.5811
1
25)150
125
308)2500
2464
3161)3600
3161
31621)43900
31621
2000(44.7213
16
84)400
336
887)640Q
6209
8942)19100
17884
89441)121600
89441
0.3006o6o6 (0.5477
25
104J500
416
1087)8400
7609
10947)79100
76629
(5)
(6)
(1)
(2)
(3)
0.03000000(0.1732
1
27)200
189
343)1100
1029
3462)7100
6924
111(10.5356
205)1100
1025
2103)7500
6309
21065)119100
105325
210706)1377500
1264236
O.OOlil (0.0333
9
63)210
189
663)2100
1989
O.06460606 (0.0632
36
123)400
369
1262)3100
2524
0.0050(0.07071
49
1407)10000
9849
14141)15100
14141
244
ALGEBRA.
(5)
(6)
(<)
iOOOOOOOO (1.4142
1
24jl00
96
281)400
281
2824)11900
11296
28282)60400
56564
5.00(2.2360
4
42)100
84
443)1600
1329
4466)27100
26796
3.25(1.8027
1
28)225
224
3602) 10000
7204
36047)279600
252329
8.600000(2.9325
4
G.
(1)
vT-J.
(2)-
Vii = f
(3)
v7ji-h=j
(4)
vib-h.
(5)
V|!| = «.
(6)
V«{ = H=!
7.
(1)
J = 0.5.
0.50)0.7071
49
(2)
(3)
49)460
441
583)1900
1749
5862)15100
11724
1407) 10000
9849
14141)15100
14141
}= o,6mm.
0.666666(0.8164
64
16lj266
161
1626)10566
9756
f = 0.75.
0. 750000 (0.86(50
64
166)1100
996
1726)10400
10356
1732)4400
(4) ^ = 0.03125.
0.031250(0.1767
1
27)212
189
346)2350
2076
3527)27400
teachers' edition.
245
(5) ^ = 0.0546876.
m
f = 0.857142.
0.05468760(0.2338
0.857142(0.9258
4
81
43)146
182)471
129
364
463)1787
1845)10742
1389
9225
4668)39850
18508)151700
37344
148064
(8)
^ = 0.08333333.
0.08333333(0.2886
(6) xt 3 =0.048.
4 V
0.0480(0.2190
48)433
4
384
41)80
568)4933
41
4544
429)3900
5766)38933
3861
34596
Exercise
LXXX.
1.
3a*
(3a; + 2y)2,y = 6a;y + 4ya
3a;2 + 6a;y + 4ya
a* + 6x*y + 12a-y2 + 8y*\x + 2y
^6a?y + 12a^2 + 8y8
6a;2y + 12a?y2 + 8y8
2.
g8-9a2 + 27a-27|a-3
3a2
-3(3a-3)= -9a + 9
3a2-9a + 9
- 9a2 + 27a -27
- 9a2 + 27a -27
(3<c + 4)4 = _
3a;2
12* + 16
3.
a" + 12a? + 48a; + 64[a; + 4
~*12a;a + 48a; + 64
12a;2 + 48 a; + 64
3a;2 + 12a; + 16 1
246
ALGEBRA.
3**
(3x*-ax)(-ax) = -SaaP+atx2
x* —3a3?+ba*j!*-3a?x—a* \x*—ax-a2
3*
-3aa-5+5a8z3-3a5a;
-Zax*+Za2x*-a*x*
3ar*-3aa^+aV
3(z*-aar)2 = 3^-60^+3^^
(3xJ-3ax-a*X-<»,) = -3aV+3a8z +a*
$x*Sax*
+3a8a?+a*
-3a2ar*+6aV -3a5a?-a«
-3aV+6a8a*-3a5x-a6
i6 +335+6rf*+7a*+6a?*+3a:+l la^+ar+l
a*
(3a*+a:)c=
3ar*
+3a8+3f
3ar*+3a*-Kr*
3(ar* + a?)*=3ar*+6a*+3a*
(3a"+3ap+l)(l)- 3s*+3a:-H
3a*+6a?*+7a*
3a*+3a>*+ Xs
3ar*+6a*+6ar»+3a!+l
Sa^+ea^+ea^+Sar+l
3o?4+6ar,+6xa+3a;+l
6.
[l-3g + 4s»
1 - 9x + 39 x1 - 99a* + 156a?* - 144a* + 64a*
1
3
-9a? + 9x2
-9x+39a?*-
-9a? + 27a?»-
-99ar»
3-9a; + 9a^
-27 x*
3~18x + 27aa
12x2-36a!,
+ 16ar*
12s2-
12a*-
- 72ar» + 156a* - 144 a* 4- 64 a*
3 - 18s + 39a?- 36 a^
* + 16ar*
- 723? + 156a* - 144s6 + 64a*
7.
|of-2o-l
a*-6a*+ 9a* + 4a8-9a2-6a-l
3 a*
(3o*-2aX-2a)= -6a84-4a2
3a*-6as + 4a2
3(a2-2a)2 = 3a*-12a8+12a2
(3a2-6a-l)(-l)
-6a6+ 9a*+4a8
-6a6 + 12a*-8a8
3a2+6a+l
3 a* -12 a8 + 9a2+6a+l
-3a* + 12a8-9a2-6a-l
-3a* + 12a8-9a2-6a-l
TEACHERS EDITION.
247
48**
(12*3+4*) 4s 48a8+16*2
8. |4*2+4*-l
64*«+ 192**+144**- 32*8-36*2+ 12*- 1
64a*
192*6+144**-32orl
48**+48*8+16*2
3(4x*+4aj)2 =48**+96*8+48*2
(12x«4-12*-lX-l)=
192.U5 +192** f64*»
12*2-12*+1
48**+96s8+36*a-12*+l
-48**-96*«-36*a+12*- 1
-48**-96*8-36*J+12*-l
9. li-g+s*-*8
1 -3*+6*»-10*8+12ar*-12*6+10a*-6*7+3*8-*»
1
3
-3*+**
Sx+Gx'-lOar1
-3*+3*2- x9
3-3*+**
3-6*+3**
*»+**
3*2- 9*3+12**-
3**- 6*»+ 6s*-
-12s8+10s8
- 3**+ a*
3-6*+6*2-3*3+**
3-6*+9*J-6*s+3**
-3*s+3**-3*5+*6
- 3*»+ 6**-
- 3*8+ 6s*-
- 9**+ g^-e^+Sa8-*8
3-6*+9*a-9s8+6a!*
-3s*H
-s6
- 9**+ 9s8-6s'+3s8-s9
10. |a2+3a6-962
a6 +9a56-135as6»+729a66-7296«
3a*
9a86+9a262
3a*+9a86+9a2&2
3a*+18a86+27a»6a
-27a262-81oy+81M
3a*+18a86
-&la&+m4'
9a*6-135a8&8+729a&8
9a66+27a*62+27a8&3
-27a*62-162a363+729a68-729&*
-27a*62-162a868+729a68-72966
11. |c2-46c+4&2
<J8-126c8+6062c*-160&8c8+240Mc2-19268c+6468
3c* I -126c5+6062c*-160J8c8
-126c8+166Vl-126cs+48yc*- 64W
S^-^W+ieto2
3c*-246c8+4862c2
12i2c2-4868c+166*
3c*-246c8+6062c2-4868c+166*
1262c*- 9668c3+2406*c2-19265c+646*
126V- 96&8c8+2406*c2-19266c+646*
248
ALGEBRA.
12.
|2a*+4o&-3y
8a«+48a56+60aW- 80a368-90a264+108a^-276«
8a«
12a4 48a5&+60a*&*- SOaW
24a86+16a*&2
12a4+24a8&+16a26*
12a4+48a3&+48a*&»
48a56+96a4&*+ 64a868
-18aW-36aP+96*
12a4+48as&+30a2&*-36a68+9&4
36a462-144a8i8-90a264+108a55-27is
-36a4y-144a86a-90a»64+108a65-27y
Exercise LXXXI.
68 =
274625165
216
63 =
262144164
216
3(60)* = 10800
3(60x5)= 900
5s = 25
58625
3(60)* = 10800
3(60x4)= 720
4*= 16
46144
11725
58625
11536
46144
2.
4.
110592148
48 = 64
884.73619,6
9s = 729
3(40)*= 4800
3(40x8)= 960
8*= 64
46592
46592
3(90)» = 24300
3(90x6)= 1620
6*= 36
155736
6824
25956
155736
4s -
5.
1092153521478
64
3(40)*= 4800
3(40x7)= 840
7* = 49
5689
45215
39823
3(470)* = 662700
3(470x8)= 11280
82= 64
5392352
674044
5392352
TEACHERS EDITION.
249
Is =
3(10)2 =
3(10x1) =
12 =
6.
300
30
1
i48i544|
1
481
331
114
9.
Is =
3(10)2= 300
3(10x4)= 120
42= 16
2.80322J |1.41
1
1803
331
436
1744
3(110)2 =
3(110x4) =
42 =
36300
1320
16
150544
150544
3(140)2 = 58800
3(140x1)= 420
P= 1
59221
37636
59221
59221
10.
18=
1601.613J11.7
1
3(10)2= 300
3(10x1)= 30
12= 1
601
331
331
3(110)2 = 36300
3(110x7)= 2310
(7)2 = 49
270613
38659
270613
Is-
3(10)2 =
300
3(100)2 = 30000
3(100x8)= 2400
&2= 64
32464
12597121108
1
"259712
259712
1» =
3(10)2 =
3(10x9) =
92;
7077888|192
3(190)2 =
3(190x2) =
300
270
81
6077
651
108300
1140
4
5859
218888
109444
218888
11.
2» =
3(20)2 =
3(20x3) =
32 =
3(20x3) =
2(3)2.
3(230)2 =
3(230x4) =
42 =
12.81290412.34
1200 -
180
9
8
4812
1389
4167
180
18
645904
158700
2760
16
161476
645904
250
ALGEBRA.
12.
38 =
3(30)a = 2700
3(30x8)= 720
8*= 64
3484
3 (380)» = 433200
3(380x4)= 4560
4*= 16
437776
56.623104|3.84
_27
29623
27872
1751104
1751104
13.
38 =
33076.16J |32.1
27
3(30)* = 2700
3(30x2)= 180
(2)*= 4
6076
2884
3(30x2)= 180
2(2)a = 8
5768
308161
3(320)* = 307200
3(320x1)= 960
1*= 1
308161
308161
14.
48 =
102503.232146.8
64
3(40)*= 4800
3(40x6)= 720
6*= 36
38503
5556
33336
3 (460)* = 634800
3(460x8)= 11040
8* = 64
5167232
045904
5167232
TEACHERS EDITION.
251
15.
9» =
3 (90)* = 24300
3(90x3)= 810
3*= 9
820.02585619.36
729
91025
25119
75357
810
18
15668856
3(930)2 = 2594700
3(930x6)= 16740
62 = -36
2611476
15668856
16.
2» =
3(200)*= 120000
5(200x5)= 3000
(of = 25
8653.002877120.53
8
653002
123025
615125
3(2050)2 = 12607500
3(2050X3)= 18450
(3)2 = 9
37877877
12625959
37877877
17.
Is -
3(10)2 = 300
3(10X1)= 30
1»- 1
i.37i33063i 11.111
~ 371
331
3(110)*= 36300
3(110 x 1) = 330
(1)2= 1
331
40330
36631
36631
3(1110)2 = 3696300
3(1110x1)= 3330
12= 1
3699631
3699631
3699631
252
ALGEBRA.
18.
3(20)* =
3(20x7) =
7* =
3(270)* =
3(270x5) =
5* =
3(2750)a =
3(2750 x 5) =
5* =
20910.518875127.55
8
1200
420
49
12910
1669
11683
420
98
1227518
218700
4050
25-
222775
1113875
4050
50
113643875
22687500
41250
25
22728775
113643875
19.
4s =
3(40)* = 4800
3(40x5)= 600
(5f = 25
9J.398648466125 14.5045
64
27398
5425
27125
600
50
273648466
60750000
3(4500x4)= 54000
42= 16
243216064
60804016
54000
32
30432402125
6085804800
3(45040x5)= 675600
52 = 25
6086480425
30432402125
teachers' edition.
253
20.
18 =
3(10)* =
3(10 X 7) =
(7J»»
3(170)* =
3(170x4)
5.340104393239|1.7479
1
X4) =
3(1740)* =
3(1740x7) =
7* =
3(17470)* =
3(17470 x9)-
= 300
= 210
- 49
4340
559
210
98
3913
427104
=86700
= 2040
■ 16
88756
355024
2040
32
72080393
- 9082800
. 36540
49
9119389
63835723
36540
98
82446702
= 915602700
471690
81
916074471
82446702,
21.
(1)
3(10)* =
3(10x3) =
3* =
3(130)* =
3(130x5) =
5* =
3(1350)* =
3(1350x7) =
7* =
2.500|1.3572
1
300
1500
90
9
1197
399
303000
90
18
50700
1950
25
263375
52675
39625000
1950
50
5467500
28350
49
38471293
5495899
1153707000
254
ALGEBRA.
00
5* =
>.206006006006|0.5848
125
3(50)*= 7500
3(60x8)= 1200
8*= 64
75000
8764
70112
1200
128
4888000
8(580)* = 1009200
3(680x4)= 6960
4* = 16
1016176
4064704
6960
32
823296000
3(5840)* = 102316800
3(5840x8)= 140160
8* = , 64
102457024
819656192
(3)
0.(
2" =
3(20)* = 1200
3(20x1)- 60
1*= 1
)10000000000 10.2164
8
2000
1261
1261
60
2
739000
3(210)'= 132300
3(210 X 5) = 3150
5* = 25
135475
677375
3150
50
61625000
3(2150)* =13867500
3(2150x4)= 25800
4*= 16
13893316
55573264
TEACHERS EDITION.
255
w
Is =
3(10)* =
3(10x5^ =
3(150)* =
. 3(150x8) =
8* =
3(1580)* =
3(1580x7) =
V-
3(15870)* =
3(15870x4) =
4* =
4.00000000000011.5874
1
300
150
25
3000
475
2375
150
50
. 625000
67500
3600
64
71164
569312
3600
128
55688000
7489200
33180
49
7522429
52657003
33180
98
3030997000
755570700
190440
16
755761156
3023044624
(5)
7s-
0.40000000000010.7368
343
3(70)* = 14700
3(70x3)- 630
3*= 9
57000
15339
46017
630
18
10983000
3(730)* = 1598700
3(730 X 6) = 13140
6*= 36
1611876
9671256
13140
72
1611744000
3(7360)"=. 162508800
3(7360x8)= 196640
8*= 64
162705504
1301644032
256
ALGEBRA.
Exercise LXXXII.
1.
81a4 -540a86+1350aa5a-1500a6a*+625&4l 9 a*-30ab +25b>
81a4
18aa-30a&
-540a8&+1350oa&*
-540a8&+ 900aa62
18tt*-60ai+2562
450aa6a-1500a&,+625&4
450a2y-1500qy+6256*
9«'-30a6+256a13a-56
9a»
6a-^56]-30a6+25&1
-30a&+25&»
2x*
a;8 -4o;7+10a!«-16a*+19«4-16a!»+10«2-4a;+l 1 s4-2s8+3a;a-2g+l
x8 "
2a7|-4a;7+10a*
|-4a;7+ 4a*
2a74-4a*+3ar'
6a*-16ar>+19z*
6a*-12a*+ 9s*
2a?4-4a*+6z*-2a>
-4s»+10a*-16z8+103s
-42*+ 8^-12^+ 4x*
2a!*-4a8+6<r1-4z+l
2a?4- 4a*+ 6a*-4aM 1
2s4- 4a»+ 6s»-4s+l
a* -2ss+3za-2aH-l |g«— a?+l
a?4
2aj»-a7Ufcr»+332
2aa-2*+l
2a*-2a;+l
2xa-2a;+l
3.
64- 192s + 240 a?- 160 a* + 60 a;4- 12a? + a*18-12s + 6a?-s'
64
16-12*
- 192a + 240 a*
-192* + 144 a3
16 - 24a: + 6s2
16-24a?+12xJ-x»
96xa-160aj»+p0aJ*
96*a-144a8 + 36&4
- 16a* + 24a4- 12a* + a*
- 16a* + 24 s4 -12a* + a*
teachers' edition.
257
2* =
3(2)2=12
(6 -*)(-*)
8 - 12x +$x2 - a?\2-x
8
12a; + 6^-3^
-6x4- x2
12-6»+V
• 12a; 4- 6a;2 -s8
4.
729a* -1458g5-hl215g*-540a!a+135g>-l8g4-l|9a*-6x-fl
(9a?)8** 729a*
243ar* -1458a*+1215a*-540a*
-162a8
4-36a?
243a^l62*8+36<ra
-14583*4- 972a^-216a*
243a^324ar»4-108a*
27^-18 x
4-1
243 a*-324 ar*4-135 a*-18 s+1
243a*-324ar»4-135a*-18a;4-l
243^-324^4-1 35a*-18a;4-l .
9a?-6a; + l|3a? — 1
9a?
6»-l
-6»4-l
-6x 4-1
1 -8t/4-28t/>-56vg4-70v4-56v54-28v6-8y74-v8 1 l-4y4-6y2-4y84-y4
2-4y
-8y4-28y2
-8y4-16y*
12a;g-48y84-36y4
2-8y+6y2
2-8y+12y2-4y8 |_ 8y84-34y*-W4-28y*
|- 8y84-32y*-48y»4-16y«
2-8y+12y»-8y»+y*
2y*_ 8i/54-12y6-8v74-y8
2y*~ 8y54-12yg-8yT4-y8
1 -4y4-6y*-4ys4-y4ll--2y4-y*
2-2y
-^y+Sy1
— 4y4-4ya
l-2y4-y2|l-y
2-y
2-4y4<^|2y2-4y84-y*
1 2y2— 4y*4-y*
-2y4-y2
-2y4-y*
258
ALGEBRA.
Exercise LXXXIII.
1.
s3 - 3 = 46,
x> = 49.
.-. x = ±7.
2. 2(x*-l)-3(x»+l)+14 - 0.
Simplify,
2x*-2-3x*-3+14 = 0,
x* = 9.
.-. x = ±3.
3.
s*-5 2x» + l
6
3
Simplify,
2xa-10 + 2x* + l = 3,
4x»«12,
a? = 3.
.-. x = ±V3.
-*- + -*- -8.
1 + x 1 — x
Simplify,
3-3x + 3 + 3x = 8-8x»,
8x* = 2,
.-.x = ±}.
5. J— -L- 1
4x» 6x* 3
Simplify, 9-2 = 28x»,
-28x* = -7,
x» = J.
.\ x = ± J.
6. 5x*-9 = 2x*+24,
3x» = 33,
a* = 11.
.-. x = ±Vll.
7. (x + 2)a = 4x+5.
Simplify,
3* + 4a: + 4 = 4s + 5.
Transpose and combine,
x* = l.
.\ x = ±l.
8 ^_?1-10
■ 7-
50+^
25 '
5 15
Simplify,
15a*-W+50 = 525-150-32*.
Transpose and combine,
13 a2 = 325,
x» = 25.
.*. a; = ±5.
9 3x^-27 90 + 4xV
x* + 3
x* + 9
■7.
Simplify, (3x*-27)(x2 + 9) + (904-4x»)(x* + 3) = 7(xa + 3)(*« + 9).
3x4 - 243 + 4x* + 402x» + 270 = 7x* + 84x* +189,
18 a? = 162,
x* = 9.
.-. x=±3.
10.
8x+^ =
65x
x 7 '
Simplify, 56 x2 + 49 = 65 x*.
Transpose and combine,
-9x* = -49,
x2 = ^.
.-.x = ±J
11.
4x*+5 2x»-5 7x*-25
15
20
10
Simplify,
24 x*+30-8 x»+20 - 21 x*-75,
24x*-8x*-21 x* - -75-30-2
-5x» = -125,
x» = 25.
.\ x«=±5.
teachers' edition.
259
12.
10** + 17 12s2 + 2 5a;2 -4
18 11^-8 9
Simplify, 110s* + 107s2 - 136 - 216s2- 36 - 110s*-168s2 + 64.
Transpose and combine, 59 s2 = 236,
s2 = 4.
.-. s = ±2.
13.
14s2 + 16 2s2 + 8
2tf
' 3 '
21 8s2 -11
Simplify, 112s4 - 26s2 - 176 - 42s2 - 168 - 112s* - 154s2.
Transpose and combine, 86 s2 = 344,
s2 = 4.
.-. x~±2.
14.
s2 + bx + a = 6s(l — bx),
Xs + bx + a = bx — 62s2,
s2+62s2 = — a,
a
s2 = --
1 + 62
> 1 + ft2
15. wis2 -t- n = g,
mx2 = q—n.
16. s2 — ax + b = as(s — 1),
s2 — as + b = as2 — axt
x*-ax* = -bt
x*(l -a) = -6,
*2 = -
a-1
Exercise LXXXIV.
1. s2 + 4s = 12,
Complete tbe square,
s2 + 4s + 4=16.
Extract the root,
x + 2 = ± 4.
.-. s = 2or-6.
2. x*-6x = 16.
Complete the square,
s2-6s + 9 = 25.
Extract the root,
x - 3 = ± 5.
, s «= 8 or - 2.
260
ALGEBRA.
3. a*- 12* + 6 -J.
x»-12x=-V.
Complete the square,
x* - 12x + 36 = ifi.
Extract the root,
x-6 = ±V-
.\ s = llJor}.
4. s3-7s = 8.
Multiply by 4,
4**- 28a? = 32.
Complete the square,
4x2-() + 49 = 81.
Extract the root,
2x-7 = ±9,
2x = 7 ± 9.
.\x = 8or-l.
5. 3s»-4s=7.
Multiply by 3,
9x*-12s = 21.
Complete the square,
9s*- 12s + 4 = 25.
Extract the root,
3x-2 = ±5,
3s = 7 or -3.
.\ x = 2§or-l.
7. x2 - x = 6.
Complete the square,
4 4
Extract the root,
x-l = ±5,
2 2
1±5
, x = 3or-2
8. 5s»-3x = 2.
Multiply by 5,
25x*-15x = 10,
Complete the square,
25*-() + |-i?.
Extract the root,
5—|-±7
5x
2 2
3±7
, a? = lor-§.
6. 12x8+s-l = 0.
12s2 + x=l.
Multiply by 3,
36x* + 3x = 3.
Complete the square,
36x* + () + JL-^.
W 16 16
Extract the root,
6x + £ = ± J,
6x = f or -2.
.*. x = J or — J.
9. 2x2-27x = 14.
Multiply by 8,
16x2-216x = 112.
Complete the square,
16^-0 + 729 = 841.
Extract the root,
4x-27 = ±29,
4x = 56or-2.
. •. x = 14 or - J.
teachers' edition.
261
10.
12
3
a2-
2£ = _Jl
' 3 12*
Complete the square,
rf-^ + i-l.
3 9 36
Extract the root,
3 6
-14
3 6
2 6
12.
3a , _£ .
4 3a =
13
6*
Simplify,
9a2 -26 a = -16.
Complete the square,
169 = 25
9 9*
Extract the root,
13
9r>_26* + ^ = ^.
-a-4
3s-6orf.
3
.-. a = 2or-
9
11.
12* + 24,
24.
2 3
Simplify, 3a2-2a
3a2 -14a
Multiply by 3,
9a* -42a =72.
Complete the square,
9^-0 + 49=121.
Extract the root,
3a-7 = ±ll,
3a « 18 or
=2(a + 2). 13,
-4.
, a? ~ 6 or — 1 \.
a + 1
a; + 4
Simplify,
(a + l)(a + 6) =
2a-l
a + 6 '
= (2a-l)(a + 4>
a2 + 7a + 6 = 2a2 + 7a- 4,
a2-2a2+7a-7a = - 4 - 6,
-a2 = -10,
a2 = 10._
.\a = ±Vl0.
14.
Simplify, 18 a2* 72 a
Transpose and combine,
Multiply by 10,
Complete the square,
Extract the root,
x x + S ^ 1
a + 1 2(a + 4) = 18*
- 9a? - 36a - 27 = - a2 - 5a
10^ + 41a = 23.
100a2 + 410a *- 230.
lOa + L1^,
2 2
10a =
41±51
" 2 2'
, a =■ J or — 4|.
262 ALGEBRA.
15. 2 - 3 + 2 •
x— 1 x— 2 x — \
Simplify, 2 a? -
\2x + 16 = 3a* - 15x + 12 + 2a? - 6x- + 4.
Transpose and combine,
-3a? + 9x = 0.
Divide by —3,
a?-3a; = 0,
orx(x-3) = 0.
,\ a; = 3or0.
16.
5<c(x - 3) - 2 (a? - 6) = (x + 3)(x + 4).
Simplify,
5 a? - 15s - 2a? + 12 = a? + 7x + 12.
Transpose and combine,
2a?-22*«0,
a?-llx = 0,
or x(x-ll) = 0.
.-.a; =11 orO.
17.
3a; 5 3a? 23
2(x + l) 8 a?-l 4(x-l)
Simplify,
12a2 - 12a; - 5a? + 5 = 24x2 - 46a? - 46.
Transpose and combine,
17a? -34 a; = 51,
Divide by 17,
a? -2a; = 3.
Complete the square,
a?-() + l = 4.
Extract the root,
a-l = ±2.
.-. x = 3 or - 1.
18.
(*-2)(
x-4)-2(x-l)(x-3) = 0.
Simplify, a? -
-6x + 8-2a? + 8x-6 = 0.
Transpose and combine,
a?-2x = 2.
Complete the square,
a?-2x+l = 3.
Extract the root,
s-l-i V3.
.% x = l±V3.
19.
i(a-4)-|(x-2) = i(2x + 3).
Simplify,
5a? - 20a; - 14a;2 + 28a; = 70a; + 105.
Transpose and combine,
9a? + 62a; = -105.
Complete the square,
324x2 + () + (62)2 = 64.
Extract the root,
18x + 62 = ±8,
18* = -54 or -70.
... x = -3or-3f
Simplify,
Multiply by 17,
Complete the square,
Extract the root,
TEACHERS EDITION.
263
20.
5)-I(x"-l)-2(*-2>>.
18a;2 - 6s - 30 - 5s2 + 5 = 30s2 - 120s + 120,
17s2 -114s = -145.
289s2 - 1938s = - 2465. ( (. 7
289 s2 -0 + 3249 = 784.
17s -57 = ±28.
,\ s = 5 or l|f.
21.
-50
Simplify,
2x 3s-
15 '3(10 + s) = ... ^
760s + 71s2 + 5^0s - 9500 - Sfs^TBlOs + 2100,
40s2* 760s = 11600. \
Divide by 10, 4 s2 + 76 s = 1160.
Complete the square, 4 s2 + ( ) + 361 = 1521.
Extract the root, 2 s + 19 = ± 39.
.\s=10or-29,
(7)
Simplify,
Multiply by 7,
Complete the square,
Extract the root,
_15-7s
-1 8(l-s)'
7s -15
(s + l)(s-l) 8(s-l)
8s = 7s2 -8s -15,
7s2 -16s = 15.
49s2 -112s =105.
49^-0 +64 = 169.
7s- 8 =±13.
.*. s = 3 or — ^.
23.
2s-l 1 = 2s -3
s-1 6 s-2*
Simplify, 12s2 -30s + 12 + s2-3s + 2 = 12s2- 30s + 18.
Transpose and combine, s2 — 3 s = 4.
Complete the square, 4^ — 0 + 9 = 25.
Extract the root, 2 s — 3 = ± 5.
.*. s = 4or — 1.
264 ALGEBRA.
24.
x + 2 _ 4-x ^ 7
x-l 2x ** 3*
Simplify, 6x* +
Transpose and combine,
Multiply by 5,
12a? + 3x» - 15* + 12 - Hi1 - 14x.
5*»-llx=12.
25V -55* = 60.
Complete the square,
25^-() + l|i = ?61.
4 4
Extract the root,
K 11 ^19
2 2
5x = 15 or —4.
.-. x = 3or-£
25.
14x-9 *»-3
X — ■ '■' "» - •
8x-3 x + 1
Simplify, 8x* + 5x* - 3x - Ux3 - 5a; + 9 = 8*«- 24x- 3s2 + 9.
Transpose and combine, — 6x* + 16x = 0.
Divide by - 2, 3*1 - 8x = 0,
x(3*-8) = 0.
.\ x = 0or 2}.
26.
1 J? -1-5 = x — 6
2x + l~x-2*
Simplify, 2x* - 3x - 2 - x» - 3* + 10 = 2** - 11* - 6.
Transpose and combine,
*»-5x = 14.
Complete the square,
*-o+f-fi.
Extract the root,
* 5 4-9
x = ± —
2 2
.-. x = 7or~2.
27.
7 — x x
Simplify, 10** + 490 - 140* + lOx2 = 203* - 29c2.
Transpose and combine, 49 x* — 343 x = — 490.
Divide by 49, x* - 7x = - 10.
49 9
Complete the square, ^-(1 + 7*7
4 4
7 3
Extract the root, * — - = ± —
2 2
. •. x = 5 or 2.
teachers' edition. 265
28.
2a + 3 7-a; 7-3a;
2(2a?-l) 2(as + l) 4-3a;
Simplify, -14a;2-12ar»+22a;+24-12a?+106a;2-162a;+56
- 44a;2-10a;-24a?-28.
Transpose and combine, 48 x2 — 180 a; = — 108.
Divide by 12, 4 x* - 15 x = - 9.
Multiply by 16 and complete the square,
64^-0 + 225 = 81.
Extract th« root, 8a; — 15=<±9,
8a; =24 or 6.
. \ x = 3 or #.
29.
12a?-lla;2 + 10a;-78 -. _1
8a?-7a; + 6 "" * X 2*
Simplify, 24ar»-22a;2+20a;-156 = 24a?-21a?+18a;-8a;2+ 7a;-&
Transpose and combine, 7 a? — 5 a? =■ 150.
Multiply by 28, 196a? - 140a; - 4200.
Complete the square, 196 a? - ( ) + 25 - 4225.
Extract the root, 14 x — 5 = ± 65.
.-. a? = 5or-4f
30.
6 18 = 7 8
a; — 1 x + 5 x + 1 a? — 5
6 7 18 8
Combine,
or ' -. n c e
a; — 1 a? + l a; +5 a; — 5
13 -a 10a; -130
a?-l a?-25
13 -x _ 10(13 -a;)
OT a?-l a?- 25
.-. a; = 13.%
Hence, if 13 — a? «= 0, the equation is satisfied.
Otherwise we may divide by 13 — x.
1 -10
a?-! a?-25
Simplify, a? - 25 = 10 - 10a;2,
11a? = 35,
* 35
155
266
ALGEBRA.
Exercise LXXXV.
1. a* + 2ax = a*. ^ ^ bnx 3n2^Q
Complete the square, 2 2
x* + 2ox + a* = 2a2. 2a?- 5 tut- 3 w2 = 0,
Extract the root, 4 a* — 10 nx = 6 w2,
x + a = ±a\/2. ,. ,,,2571* 49n2
. a; = — a±a
V2. «-() + ^£-ffi!
a? = 4aa; + 7a2. 2.-5? -±1?,
2 2
2a; = 6n or -n.
Transpose,
a^-4aa? = 7a2.
Complete the square, . '. x = 3 n or
3*-4aa; + 4a2 = lla2.
Extract the root,
a;-2a = ±aVTl. 5. —*— *L_.
.-.*-2a±aVlL (* + a£ £-«>"
a2(a;-a)2 = 62(a; + a)2,
3. a»-^5?-3ii«. *(*-a)-±b{x + a),
4 ax — a2 =» 6a: + a5,
4a?2 = 7m2 - I2mx, x(a - b) = a2 + a&.
4x2 + 12mx=7m2, a(a + 6)
4a? + () + (3m)2=16m2, ■'•*- a_h >
2 a; + 3 m - ± 4 m, 0r oa; - a2 = - bx - ab,
2a; ~± 4m -3m. a?(a + &) = a2-o&.
2 2 ••*-- jrnr
6* ca; = az2 + 6*2--^-.
a + 6
aca: + Jca; = a2x2 + 2a&c2 + i2*2-
a2a;2 ^ 2a6a;2 + 62a;2 - aca; - Jca; = ac,
x*(a? + 2a& + 62)_a;(ac + 6c) - ac,
a^(a2+2a6 + 62)~() + £2=Bi«l±4
4 4
*(a + &)-g-±V4ac + *
2 2
2
2(a + b)
teachers' edition. 267
7 a2a^ + — = 2ax 8 (a2 + 1) .t = aa? + a.
b2 c2 c a2a; + a; = ax2 + a,
a«Asi4.ft*-9flW«B as2-(a2+ l)a; = -a,
dW-2aW«--^ 4aV-() + (a2+l)2 = a*-2a2+l,
aW - H + * = 0 2aa;-(a2 + 1) = ± (a2 - 1),
aca; = 62, .\ a? = a or -
52 a
.•. » = —
9.
a b = 2c
x — a x—b x—c
a(x - b)(x - c)'+ 5 fa — a)fa - c) - 2c(a; — a)(x- b),
ax*- abx-acx + abc + bx* - abx - bcx + abc = 2 cx2-2 aca-2 &ca:+ 2 ata,
rf(a + &-2c) + «(ae + &c-2a&)-0. 2ab-ac-bc
,\ x=*Q or •
a + b — 2c
10.
1 -M + -1
a + 6 + a; a 6 a;
afoc = afoc + 62a: + 6a?2 + a?x + abx + ax2 4- o26 + o52 + afa,
a;2(a + b) + x(a 4-' 6)2 = - a6(a + b).
Divide by (a + b), x* + (a + b)x = — ab,
4 a;2 + ( ) + (a + bf = a2 - 2 ai + &2,
2a; + (a + 6) = ± (a — 6),
2x = -2&or-2a.
. •. x = — & or — a.
11.
_J. 1 3 + ar2
a— a: a + a; a2 — ar2
a + a;— a + a; = 3 + 0?2,
a:2- 2 a: =-3,
a?-2* + l«-2f
.-. .r=l±V^2.
268
ALGEBRA.
13
(2»-«r aft,
x-a + 2b
4x* - 4ox + a* = 26x-a& + 26',
4x»- 2x (2a + 6) = -a*-a6 + 26»,
16*»-()+(2a+&)2 = 9&*,
4x-(2a + 6) = ±36,
4a? = 2a -26,
or 2a + 46.
. ~ a-6_a + 26
.*. x = or ■
2 2
14. x* + ax = a + x.
a? + ox — x — a,
x* + (a-l)x = a,
4x2+()+(a-l)2 = a2+2a + l,
2x + (a - 1) = ± (a + 1),
2x = -(a-l)±(a+l).
. \ x = 1 or — a.
15. x2 + ax = bx + a6.
x* + (a — 6) x = ab,
4x2 + () + (a-6)2 = (a + 6)2>
2x + (a - b) = ± (a + 6),
2x = -2aor26.
.•. x=« — a or 6.
16.
x a_x 6
a x b x
x*b + a26 = ax* + aft1,
att-a^a-a^-a2*,
x»(6-a) = a6(6-a).
Divide by (b — a),
x* = a&. _
.*. x = ± Va6.
17.
1111
- + r - - + r
x x+b a a+ 6
a2x + a26 + a6x + ati* + a2x + a&x = ax1 + a&x + bx2 + 6*x + ax* + ate,
20X3 + Jx2- 2a*x + 6»x = a*b + a&2,
(2a + tyx2 - (2a* - 6») x = a26 + aft2,
4x*(2a+ 6)2- ( ) + (2a* - &*)' - 4a* + 8as6 + 8a262 + 4a6» + b\
2x(2a + b) - (2a' - 62) - ± (2a8 + 2a6 + b*),
2x(2a + 6) = (4 a2 + 2a6) or -(2a6 + 262).
2a + 6
18.
a 5x_ x2 ^Q
3 4 3a
4a2 + 15ax-4x2 = 0,
4x2 — 15ax = 4a2,
64x2-() + 225 a2 = 289a2,
8x — 15a = ± 17a,
8x = 32aor-2a.
.*. x — 4a or — -•
19
= a +
= a,
x + 3
x + 3
x-3
x+3 x-3
x-3 x+3
x2 + 6x + 9-x2
+ 6x-9 = ax2-9a,
ax»-12x = 9a,
aV-12ax = 9aJ,
a2x2-() + 36 = 9a2 + 36,
ax-6
ax^
, x«= -
±V9(a'+4),
GiSVtf + l
6±3V?+4
TEACHERS EDITION.
269
20.
mx2 — 1 ■■
x(m? — n*)
m2nx2 — mn = x(m? — n2),
m2nx2 — (m8 — n2)x = mn,
4m*n2a? — ( ) + (ra8 — n2)2 = m6 + 2m*n2 + n4,
2 m2 nx — (ra8 — ?i2) = ± (m8 + n2),
2m2nx = 2m8 or — 2n2.
771 71
.\ x = — or »•
n m?
21.
(oa; — b) (bx — a) = c2,
a&r2 — b2x — a?x + ab = <?,
abx2 — (a2 + b2)x = c2 — a6,
4a262ar» - 4a& (a2 + &2)z = 4a&c2 - 4a2 62,
4a2 b*x* - ( ) + (a2 + ft2)2 = (a2 - i2)2 + 4aic2,
2ata-(a2+&2) = ± Vfa2
62)2 + 4a6c2.
a2 + 62±V(a2-62)2 + 4a6c2
2a6
23.
ra + a;
m — a;
m2
-mx + rri
*-J-77ia; =
cm? —
ca;2,
ca;2 =
xVc =
cm? —
■2m2,
'c-2.
.*. a? —
±m-i
/?
22 oa; + 6 _^ ttkp + n
bx + a nx + m
aTia? 4- 6na; + aTna; -f bm
= fois2 + 6na; + amx + an,
aTix2 — bmx2 -=an — bm,
x*(an — bm) = an — bm,
z2=l.
.\ x = ± 1.
24.
(a-l)2x2 + 2(3a-l)a; -.
4a-l
(a-l)2a?-f2(3a-l)a; = 4a-ll
4 (a - l)4 x* + ( ) + (6 a - 2)2 = 16 a3,
2(a - l)2x + (6a - 2) = ± 4aVa\
2(a-l)2a: = 2-6a±4aVS.
. „, 1 — 3 a ± 2 a Va
(a -If
270 ALGEBRA.
26.
(aJ -&»)(*» + 1)
■ 2x.
a2+62
flW-P^ + tf-P-
■ 2a2x +
262x,
a2 x2
-62x2-2a2x-262x =
= 62-a2,
*»(
a2-62)-2x(a2 + 62) =
2-62)*_2x(a*-6*) =
-PP-O + ^ + Pr-
:62-a2,
Xs (a
*-a* + 2
a*P-ft*,
^(a2-
4^^,
x(a2 - 6s) =
.±2a6,
= a2 + 62±2a6
-(a±6y.
a + &,™
a — 6
.% x =
1 01
a-b
•
a + 6
26.
x2 — 4mnx
■ (m — nV
m
(m + n)2
x1 — 4 mnx = m* — 2m2 n2 + n*,
«• — () + 4m,n1 = m4 + 2m,nl + n*,
aj-2mn = ± (m2 + n2),
x = 2mn±(m24-n2)
= (m + n)2 or — (m — n)2.
27.
. a-6r_14a2-5a6-1062 (2a-3b)*
a62 18a262 2a6
18a262x2-(18a26-27a62)x = -4a2+13a6-1062,
144a2 62x2 - (144a2 6 - 216a62)x = - 32a2 + 104 a* - 80 J2,
144a262x2 - ( ) + (6a - 96J2 - 4a2 - 4aA + J2,
12a6x - (6a - 96) = ± (2a - 6),
12a6x = 8a - 106 or 4a - 86.
4a-56 a-26
""'*" 6a6 3a6 "
28.
abx* + h*x =< 6a* + ^ ~ 2ft> 3o*x
c c2 c
06c2 x2 + 62cx =» 6a2 + a6 - 262 - 3a2 ex,
a6c2x2 + (3a2c + 62c)x - 6a2 + a6 - 262,
4a262c2x2 + 4a6cx (3 a2 + 62) - 24 a8 6 + 4a2 62 - 806s,
4a262c2x2 + ( ) + (3a2 4- 62)2 = 9a* + 24a86+10a262-8a62+64,
2a6cx + 3a2 + 62 - ± (3a2 + 4a6 - 6s).
2a-6 3a + 26
.-. x = or =
ac be
teachers' edition. 271
29.
x2 m2 — 4 a2 _ x
3m-2a 4a-6m 2*
2a;2 4a2-m2
= xt
3m — 2a 3m— 2a
2s2 — 4a2 + m2 = 3ma: — 2aar,
2s2 + (2a - 3m) a; = 4a2 - m2,
16a? + ( ) + (2a- 3m)2 = (36a2 - 12am + m2),
ix + (2a - 3m) = ± (6a - m),
4a; = — 8a + 4m or 4a + 2?/t.
.*. x = m — 2a or a + — •
2
30.
6« + feL+#_5(«-6) + ?*2*
£ 6z
36a? + 6(a + 6)2 = 30a?(a- 6) + 25a6,
36a;2 - 30a; (a - 6) = 25a6 - 6 (a + 6)2,
4 4
6*-|(a-6) = ±^±*,
6g==6a-46 4a-^66.
2 2
. m 3a-26 rt„ 2a-36
.*. x= or •
6 6
31.
i (x2 + a2 + ab) = |a;(20a 4- 46).
8a? + 8a2 + 8a6 = 20aa; + 46a;,
fa? - (20a + 46)a; = - 8a2 - 8a6,
16a? - 2(20a + 46)a; = - 16a2 - 16a6,
16a?-() + (5a+6)2 = 9a2-6a6 + 62,
4z-(5a+6) = ±(3a-6),
4a; = (5a+6)±(3a-6).
... a.=s2aor^±-6.
2
32.
a? — (6 — a) c = ax — bx + ex.
x2 + bx—ax — cx = (b — a) c,
x2 + (b — a — c)x = (6 — a)c,
4a? + ( ) + (b - a - c)2 = a2+ 62 + <?-2a&-2ac + 26c,
2a; + (6 — a — c) = ± (a — 6 — c),
2a; = 2a -26 or 2c.
. \ x = a — 6 or c.
272 ALGEBRA.
33.
x2 — 2 mx = (n — p + m) (n — p — m) .
a2 — 2mx = n2— 2?ip +p?—m*t
s2 - ( ) + m2 = n2 — 2np +|)2,
a? — m = ±(n — />).
.•. x = m±(n — p).
34.
z1 (m + n)x = J(l> + S' + m + n)(P+5r-m-"n)-
4 a2 — 4(m + n)x = (p + a+m + n)(p + a— m— n),
4x2-( ) + (m + n)2 = jo2 + 2pq + y2,
2a?-(m+n) = ±(p + a),
2x = m + n ± (j? + j).
. wi + n±(p + y)
2
35.
mnx*— (m+n)(mn+l)x+ (m+rif = 0.
mnx2 —{m*n -f mn2 + m + n)x = — ?ra2 — 2mn — n2,
4 mWx*— 4 win (m2n -|- mn1+ m + n) x = — 4 rasn — 8 m2n2 — 4 mns,
4mW-( ) 4- (wi2n + mn2 + m-f n)2 = m¥+2mV- 2m3n— 4 mV
+ra2n4— 2 mns+m,+2mn+n8,
2mnx — (nfn + ??in2 + ra + n) = ± (m2n + ran2 — w — n),
2mna; = 2m2n + 2mn2 or 2m + 2n.
.-. x = m+n or ^±7i.
mn
36.
26-a:-2a 46-7a = a?-4g
6.r oa: — 6a: o6 — 62
4a6-oa;-2a2-262 + 6a; + 462-7a6 = 32-4aa;,
a*-3aa>-6a; = 262-3a6-2a2,
x2 - (3a + b)x = 2 62 - 3 a6 - 2a2,
4z2 - 4(3a + b)x - 862 - 12a6 - 8a2,
4s2 - ( ) + (3a + bf = a2 - 6a6 + 962,
2a:-(3a + 6)-±(a-36),
2a; -4a -26 or 2a + 46.
.\ »— 2a - 6 or a + 26.
teachers' edition. 273
37.
2x*(a* - 62) - (3a2 + 62)(a? - 1) = (362 + a2)(.r + 1).
2a*(a*- 62)- 3a2x - to; + 3a2 + 62 =- 362x + a2z + 362 + a',
2z2(a2 - 62) - 4a2z - 462a; - 262 - 2a2.
Divide by 2, z2 (a2 - 62) - 2 (a2 + 62) a: = 62 - a2,
x2 (a2 - 62)2 - 2(a* - A*) a - (62 - a*)(a* - J2),
ar^a2 - ft2)2 - ( ) + (a2 + 62)2 = 4a262,
s(a2-62)-(a2 + 62) = ±2a6,
a;(a2 - 62) - (a + ft)2 or (a - 6)2.
a + 6 a — b
.«. a; = ^-£-f or- — £.
a— o a + 6
38.
a-26-a; 56-a; 2a-a;-196 Q
a2 — 462 as + 26a: 2bx — ax
a-2b-x 56-a? 2a-a?-196_ Q
(a-26)(a + 26) (a + 26)x (a-26)«
ax - 2bx-x*- bob + ax + 1062 - 26a? - 2a2
+ as+ 15a6 + 26a? + 3862 = 0,
3aa? - 2bx - a?2 + lOab + 48 62 - 2a2 _ 0.
«• - (3a - 26)a? - - 2a2 + 10a6 + 48 62,
4a?2 - ( ) + (3a - 26)2 = a2 + 28a6 + 19662,
2a?-(3a-26) = ±(a+146),
2x = 4a + 126 or 2a - 166.
.% a? = 2a + 66 or a — 86.
s + 13a + 36 1 = a — 26
5a — 36 — x x + 26
s? + 13az + 56a; + 26a6 + 662
— box + 5bx + x2 + 662 - 10a6 = 5a2 - 13a6 - ax + 66* + 26a?,
2a2 + (9a + 86)a? = 5a2 - 29 a6 - 66*.
16 t2 + 8(9a + 8 b)x = 40a2 - 232a6 - 48 62,
16s2 +( ) + (9a + 86)2 = 121a2 - 88a6 + 1662,
4a? + (9a + 86) = ±(lla-46),
4a; = 2a - 126 or - (20a + 46).
... iC-a.-36or-(5a + 6).
A
274 ALGEBRA..
40.
x + 3b 36 a+36
= 0.
8a2-12a6"96s-4a2 (2a + 3b)(x-3b) =
s + 36 3j> a + 36 Q
4a(2a-36) (2a-3&)(2a + 36) (2a+36)(x-36) '
(a*-9&*)(2a + 3&)+12ata-36a&*-4a(2aa + 3a&-962) = 0.
(2a + 3 ty's2 + I2abx = 8as + 12a26 + 18a^ + 276s,
4(2a + 3J)2 x* + ( ) + (12a&)2 - 64a* + 192as5 + 432a262
+ 432a&3 + 3246*,
2(2a + 36)* + 12a& = ± (8a2 + 12a6 + 18&2),
(2a + Sb)x = 4a* + 96* or -(4a2 + 12aft+9&2).
2a+36
41.
nx* +px —px* — mx+ m — n = 0.
tub2 — jbc2 + px — mx = n — w,
^(n — j>) + & (j> - m) = n - m,
4z*(n - j»y + ( ) + (p - m)2 = 4na-4mn-4pn+4pm+p«-2pm-Hn,1
2s(n -j>) + (p - m) = ± (2n -p - m),
2s(n -jj) = m -p + 2w -p - m,
or m-j3 — 2n+^> + TO«
...«-l or^LZl^.
42.
(a + 6 + c)x* - (2a + 6 + c)x + a - 0.
(a + b + c)x* -(2a + b + c)x = -a,
4a?(a+& + c)2-() + (2a + & + c)2 = &2 + 2&c + c2,
2x(a + 6 + c) - (2a + b + c) = ± (6 + c),
2z(a + 6 + c) - (2a + 6 + c) ± (6 + c),
2s (a + 6 + c) « 2a + 2 6 + 2c or 2a.
.-. x=l or 2
a + b + c
teachers' edition.
275
43.
(ax — b)(c — d) = (a — b)(cx — d) x.
acx — be — adx + bd=> acx2 — adx — bcx* + bdx,
bcx* — acx* + acx — bdx — be — bd,
(be — ac)x* + (ac — bd)x = be — bdt
4 (6c - acfx* + ( ) + (ac - bd )2 = 4 b V - 4 abc* - 4 62crf
+ 2a6co* + a2c2 + 62d2,
2(6c — ac)a? + (ac — bd)^± (26c — ac - W),
2(6c — ac)x = — (ac — bd) ±(2bc — ac — bd),
— ac + bd + 2bc — ac — bd
2(bc-ac)
— ac+bd — 2bc + ac + bd
2(bc-ac)
c(a — b)
44.
2x
6 x\b a)
3s+l
2x + \ a-
26_3s+l
a
b abx
2ax* + ax — a + 26 = 36s2 + bx,
2axi — 3bxi + ax — 6a; = a — 26,
x*(2a-3b) + x(a-b)=a-2bt
4a*(2a-36)2 + 4a;(a-6)(2a+36) = (4a-86)(2a--36)l
ix*(2a - 36)2 + ( ) + (a - 6)2 - 9a2 - 30a6 4- 25ft2,
2a;(2a-36)+(a-6) = ±(3a-56),
2a; (2a - 36) « - a + 6 ± (3a - 56),
2a;(2a-36) = 2a-46or-4a + 66.
a-26
2a-36
or-1.
276 ALGEBRA.
45.
1 1 = a _ 26x + 6
2x2 + x-l 2x2-3x+l 2bx-b ax*-a
1 1 a 2 6s + 6
(2x-l)(x + l) (2x-l)(x-l) = 6(2x-l) a(x-l)(x + 1)'
L.C.D. = a6(x-l)(x + l)(2x-l).
Simplify, a6x — ab + a6x + a6 = a2x2 — a* — 4 6V + J2,
2a6x = oV - a2 - 46V + 62,
46V - aV + 2a6x = 62 - a2,
x2(462 - a2) + x(2a6) = 62 - a2,
4x2(462- a2)2 + 4x(2a6)(462 - a2) = 166*- 2062a2 + 4a*,
4x2(462-a2)2 + ( ) + (2a6)2 - 166* - 1662a2 + 4a*t
2x(462- a2) + 2a6 - ± (46* - 2a2),
2x(462-a2) = 462-2a6-2a2
or2a2-2a6-462,
262-a6-a2 a2-a6-262
x *= or •
462-a2 462-a2
. « h — a ^ 6 + a
.*. x = or •
26-a 26 + a
Exercise LXXXVI.
1. (x + l)(x-2)(x2 + a;-2) = 0.
(x + l)(x - 2)(x - l)(x + 2) - 0.
.-. x = -1,2, 1,-2.
2. (x2-3x+2)(x2-x-12H0. 4 2x» + 4x2-70x-0
(x-2)(x-l)(x-4)(x+3) = 0. *' J' + Jf^J
"X = ' ' ' ' 2x(x+7)(x-5) = 0;
3. (a: + 1) (x - 2) (x + 3) = - 6. which is satisfied if a? = 0,
x8 + 2x2 - 5 X - 6 = - 6, 3 + 7 = o
*(a» + 2»-5)-0; or lf *-5-C.
which is satisfied if x = 0, .\ x — 0, — 7, 5.
or if x2 + 2x-5 = 0.
By solving x2+2x-5 = 0, 5. (x2-x-6)(x2-x-20) = 0.
x - - 1 ± V6. _ (x-3) (x+2) (x-5) (x+4) = 0.
.-. x = 0, -1 ±V6. .-. x=3, -2, 5, -4.
teachers' edition. 277
6.
x(x + l)(s + 2) = a(a + l)(a + 2).
ar* + 33*+ 2x = a* + 3a? + 2a,
x* + 3a;* + 2x - a8 - 3a2 - 2a = 0,
(a8 - a8) + (3s2 - 3a2) + (2* - 2a) = 0,
(xi + ax+a*)(x-a) + (3x+3a)(x-a) + 2(x-a) = 0,
(x* + ax + a2 + 3a? + 3a + 2) (a; - a) = 0.
• \ a; — a = 0,
and a; = a.
Or, a?2 + as + a2 + 3a; + 3a + 2 = 0.
x2 + aa? + 3x = - a2 - 3a - 2,
a;2 + a;(a + 3) = — a2 — 3a — 2,
4s2 + ( ) + (a + 3)2 = 1 -6a -3a2,
2x + (a + 3) = ± VI- 6a -3a2.
.-. s=--^±£vl-6a-3a2.
7. ars-a^-a: + l = 0. 10. a*-l=0
fr+ig;=l!i::ii:S f ^J^-}!-*
.-.3 = 1 1 -1 (* + l)(*»-a + l)
(a?-l)(a^ + a; + l) = 0,
8. 8ar»-l = 0. and a; = -1,1.
(2a;-l) (4a*+2a;+l) = 0. From x* - x + 1 - 0,
From the first factor, ar2 - a: = - 1,
s = J, 4s2-() + l«=-3,
or 4z2 + 2a: = -l. 2a?-l=±VT3;
16a^ + () + l 3, 1±VZ3
4^+1 = ^x^3. and * = ~ J"2'
••• * = J(-l± V^3). From ar2 + a: + 1 = 0,
&* + a;-=-l,
9. 8s8 + 1=0. 4a^ + () + l = -3,
(2*+l)(4a:2-2.T+l) = 0. 2x + 1 = ± V^
From the first factor, _1 ± y^3
a: = - J, and
or 4ar2-2a? = -l!
4a*-2a: = -l, .-.*-!,
l±V-3
lGx*-() + l = -3t ' ' 2
1 = ±V^3.
ar=J(l±VZ3),
4x-l = ±VT3. and rl*V^
278 ALGEBRA.
11. a:(x-a)(3*-&s) = 0.
x(x-a){x + b)(x-b) = 0.
.-. x = 0, a, ±b.
12. n(x* + 1) + (x + 1) = 0.
(x + l)(nx* - nz + n + 1) = 0.
(n + l)(x + l)(x* -x +1) = 0.
If a; + 1 = 0,
s = -l;
or if nz* — na: + n + 1 = 0. ,
By solving, a; = J ± — V-3n*-4n.
"2w
Exercise LXXXVII.
1. (s-2)(s-l) = 0. c / 2\/ +3\_0
a*-3a; + 2 = 0. V 3y V 2>
^ + ^-1=0,
2. (*-7)(s + 3) = 0. Ma«
^-4s-21-0.
6a* + 5:r-6 = 0.
3. (._IW._J)_a 6 (* + 5)KH
V 2^ 3/ or ^ + 11^ + 5 = 0
(2*-l)(3*-l)-4 °r ^+ 2 +2 '
6js*-5;r + l = 0. or2s^ + ll* + 5 = 0.
I (. + 2)(-|)-o.
(9a+7)(7a;-9) = 0t
63 a*- 32a; -63 = 0.
t.(-3)(. + 8)(.-f)(. + f)-a
, 153a* 81 _0
*~ i6" + 16-0'
16s* -153z» + 81=0.
8. (s-0)(:r-l)(a;-2)(a;-3) = 0.
a*-6a* + lla;2-6a; = 0.
9. (-o)(. + |)(.-|)(. + i)-a
4a*-7&*-3a> = 0.
TEACHERS EDITION.
279
10. {x-(a-2b)}{x-(3a + 2b)}=0.
(x-a + 2b)(x-3a-
x2 — 4a# + 3a2 — 4a& — 4&2
= 0.
11. {a - (2a - b)}{x - {b - 3a)} = 0.
(x - 2 a + 6) (* - 6 + 3 a) = 0,
a2 + as -6a2 + 5a6 - b2 = 0.
12. {s-(a2+a)}{a;-(l-a) = 0. 17. 32 + 4s+l = 0.
(a — a2 — a)(x— 1 + a) = 0,
a2 — a2x — x — a8 + a = 0.
13. rf-7* + 12-0.
In this equation p is 7 and q
is 12.
.-. Vp2-4?=V49-48
-VI.
.•• roots are rational, and both
positive.
14. a?*-7a:-30 = 0.
In this equation p is — 7 and
q is - 30.
v. Vp2-4y=V49 + 120
= 13.
.*. roots are rational, and of
opposite signs.
15. s2+4a-5 = 0.
In this equation p is 4 and q
is — 5.
.-. V^2~4y=V16 + 20 = 6..
.*. roots are rational, and of
opposite signs.
16. 5s2 + 8 = 0.
In this equation p is 0 and q
is 8.
.-. Vp2-4y=V0^32
.*. roots are imaginary.
In this equation p is 4 and
q is 1.
.-. Vp2-4y=\/l6^4
= Vl2.
. \ roots are surds, and neg-
ative.
18. x2~ 2x + 9 = 0.
In this equation p is — 2
and j is 9.
...Vp2-4y=V4^36
= v^32.
.\ roots are imaginary.
19. 3x*-4*-4-0.
* a;2-4a;-- = 0.
* 3
Herep--! ,-- *
>9 3
.•. roots are rational, and of
opposite signs.
20. s* + 4a; + 4 = 0.
In this equation p is 4 and
q is 4.
.-. Vp2-4y=Vl6-16
=V0 = 0.
.•. roots are rational, equal
in value, and both negative.
280 ALGEBRA.
Exercise LXXXVIII.
1.
Let 4+6x — x* = wi.
x2— 6x — 4 = — m,
Xs — 6x = 4 — in,
x*-() + 9 = 13-m,
x-3 = ±Vl3-ro.
, x = 3±Vl3-m.
Since V13 — m cannot be negative, w cannot be greater than 13 ;
that is, the maximum value is 13.
Let (x + a?
m.
x
Then x* + 2ax + a* = mx,
and 4ac* + 4x(2a — m) = — 4a*.
4a* + ( ) + (2a - mf = m» - 4am,
2x + (2a — m) = ±Vm(m— 4 a),
2a: = — (2a— m)±Vwi(m — 4a).
a; = _ 2«Lzi» ± i y/m{m _ 4a)t
x «= — J(2a — m) t Vm(m — 4a).
Since <Jm — 4a cannot be negative, m cannot be less than 4 a.
Hence 4 a is the minimum value.
Let
3.
x* + l
=m.
x
x2 -f 1 = mx,
x* — mx = — 1,
4x*-() + (ro)2 = m»-4,
2 x — m = ± Vm* — 4.
Since ^m* — 4 cannot be negative, m cannot be between + 2
and —2, but may have any other values. Hence +<» is the
maximum and — » is the minimum value.
TEACHERS ' EDITION. 281
4.
Let (a — x)(x — b) = m.
Then ax — ab — x1 + bx = m,
or x2 — x (a + b) = — ab — m,
4 a? - ( ) + (a + J)2 = (a - by - 4 m,
2a?-(a+6) = ±V(a-&)*-4
m.
.-. a; = i (a + 6) ± V(a - bf -4m.
Now, for all possible values of xt (a — 6)a — 4m cannot lie nega-
tive ; that is, m cannot be greater than ^ — — *- ; hence this is the
maximum value.
5.
Let
1 + z2 ""
x= m-h ra2,
mx2 — » x = — m,
4wW-() + l = l-4m2,
2ma>-l = ± Vl-4m2.
...s=^-(l± Vl-4m2).
For all possible values of a:, 1 — 4m2 cannot be negative ; that is,
m cannot be greater than £, and for this value x = 1.
.-. i is the maximum value.
6.
Let a?* + 8a; + 20 = m.
x* + 8a;=m — 20,
a;* + ( ) + 16 = m - 4,
x + 4 = ± Vra — 4.
,\ a? = — 4 ±Vm — 4.
For all possible values of a;, m cannot be negative; that Is, m
cannot be less than 4.
*\ 4 is the minimum value.
ALGEBRA.
7.
Let x,-2a: + 9 = m.
x*-2x = m-9t
rf-O + l-iii-8,
x — 1 = iVm- 8.
.-. x = 1 ±Vm- 8.
For all possible values of x, m — 8 cannot be negative ; that is, m
cannot be less than 8, and for this value x «= 1.
.*. 8 is the minimum value.
8.
Let
{x + a)(x-b)
x9 = mx* + amx — bmx — abm,
x2 — mx* — amx + bmx = — abm,
x*(l — m) — x(am — 6m) = — abm,
4(1 - m)»3* - ( ) + (am - 6m)* = ahn? + 2abm* + 6*m*- 4o6m,
4(1 - m)8^2 — ( ) + (am — 6m)* — (am + 6m)1 — 4a6m,
2a?(l — m) — (am - 6m) = ± V(am + 6m)* — 4 abm,
2x(l — m) — (am — 6m) = ± Vm* (a + 6)* — 4a6m.
.^ = i(?!lz^±_J_Vm*(a + 6)*-4a6m}.
I 1 — m 1 — m J
For all possible values of x, m*(a + 6)* — 4o6m cannot be nega-
tive; that is, m*(a + 6)* cannot be less than — 4o6m, and for thia
value s=am-6m.
2(1 -m)
is the minimum value.
(a + by
Let
9.
a + a*
x = am + mjj*f
mx* — x = — am,
4m»at -0 + 1-1 -4 am*,
2mx— 1 = ±\/l — 4am*.
... x = — (l±Vi-4am*).
2m
For all possible values of a;, 1—4 am* cannot be negative ; that is,
m cannot be greater than + Ja/-» an(* for this value x =
VI. a 2m
- is the maximum value.
a
teachers' edition. 283
10. Divide a line 20 in. long into two parts so that the sum of
the squares on these two parts may be the least possible.
Let x = number of inches in first part,
and 20 — x — number of inches in second part.
x2 + (20 - x)2 = m,
x* + 400 - 40 x + x2 = m,
2a* -40 a: = m- 400,
4z* - ( ) + 400 = 2m -400,
2 x - 20 = ± V2m-400.
Then, as v2m — 400 cannot be a negative expression, 2m cannot
be less than 400.
.•. 200 is the minimum value.
For this value, 2 x2 - 40 x = - 200,
x2- 20 s = -100,
s*-() + 100 = 0,
x - 10 = 0.
.-. x = 10,
20-s = 10.
11. Divide a line 20 in. long into two parts so that the rect-
angle contained by the parts may be the greatest possible.
Let x = one part,
then 20 — x = the other part.
20x — x2 = m,
x2 — 20x = — m,
g«-()+ 100 = 100 -mt
x - 10 = ±V100-m.
.-. a=10±Vl00-m.
For all possible values of x, 100 — m cannot be negative ; that is,
m cannot be greater than 100, and for this value x = 10.
.•. 100 is the maximum value.
Substitute value of m, x = 10 ±V100— 100,
x = 10,
20-3 = 10.
284 ALGEBRA.
12. Find the fraction which has the greatest excess oyer its
square.
Let x = the fraction,
then 2s = the square of the fraction.
x — x* = m,
x* _ x = — m,
4a*-() + l = l-4m,_
2x - 1 = ± Vl -4m.
.\x = J(l±Vl-4m).
For all possible values of x% 1 — 4m cannot be negative ; that is.
m cannot be greater tlian J.
.'. \ is the maximum value.
For this value, x = J.
Exercise LXXXIX.
1. a* + 7x8 = 8. 2. z4-5a?a + 4 = 0.
4a* + ( ) + 49 = 81, a4 -5a? = -4,
2^+7-* 9 4*4-() + 25 = 9,
2ar»=-7±9, 2a* = 5±3,
x» = -8orl. a* = 4orl.
Since a? = — 8, .\a?=±2, ±1.
or (*+2)(z'-2*+4) = 0. * 4af- 37a* = -9,
Whence x + 2 = 0, 64a* - ( ) + (37)a = 1225,
and x = -2% 8x*-37 = ±35,
or ar2-2x + 4 = 0. 8a;2 =72 or 2,
x*-2x=-4, a* = 9 or f
aB» — () + l — — 3, .\a?~±3or±J.
x-l = ±V^3. 4. 16^=17jc*-1.
.-.a: = l±v^3". 16a*- 17a4 = -1,
Since *»=!, 1024^i>i(1^ = ***
. 1 a 32a?4 - 17 = ± 15,
^_1==u' 32a-4 = 32 or 2,
or (*-l)(*»+* +1) = 0. xa - 1 or A-
Whence a; — 1 = 0, Since a;4 = 1,
and a?=l, **-l = 0f
or
a* + x + 1 = 0,' or («I+1X*+1X*-1)- 0.
a* + a; = - 1 .\ * = ^v^, - 1. or 1.
4a» + () + l--3,_ Since ^ fl^'
2z+l = ±V-3 Or(a*HX*+iX*rlH0'
.•.a:=$(-W-3). ...*=±iV-lf-},ori.
. a? =- — 2, 1, 1 ± V- 3, .-. the roots are
andJC-liV11^). ±1. ± V-lf±J, ±iV^l.
teachers' edition.
285
5. 32^-33^ + 1 = 0.
32a:10- 33s5 = -1,
4096z10-() + (33)2 = 961,
64 a* -33 = ±31,
64a^ = 64or 2,
.*. x= 1 or {.
Other roots may be found by
methods given later.
6. (s»-2)' = K*2+12).
x* — 4a:2 + 4 =
a* + 12
4a?*-16x2+16 = s2+12,
4a*-17a* = -4,
643*-() + (17)2 = 225,
8a? -17 = ±15,
8 a? = 2 or 32,
x9 = £ or 4.
.*. 3 = ±£ or ±2.
7. s4*_5^_25 = 0
3 12
12z*»-20s2»-25 = 0,
12a*»-20a2» = 25,
36a?*»-() + 25 = 100,
6x2»-5 = ±10,
6x2* = 15 or -5,
3?w = f or-f
a;.
0,
0,
0,
0.
-1,
-i
-J±jV-3.
i(-l±V=3>
•.x=±*/\ot±*/— j.
(aJ»-9)2 = 3+ll(a^-2).
a?* -18a* +81 = lis2 -19,
3^-293? = -100,
4^_() + (29)2 = 441,
2a;2 - 29 = ± 21,
x2 = 25 or 4.
.\ a; = ±5 or ±2.
10. 19** + 216 a;7 =
216z7+19s*-a; =
*(2163*+19a*-l) =
^(273^-1X8^+1) =
a;(3a:-lX9iCa+3x+l)
(2x+l)(4r»-2x+]) =
.*. x =
From
9a;a + 3x + l =
9a^ + 3a? =
9^ + () + t =
3aJ + J =
3a; =
From
4a*-2a; + l =
4a;2-2a; =
4a?-2ar+ J =
2*-* =
roots are 0, J, -
i(-l±V^3),
11. ^ + 223* + 21 = 0.
a* + 22a^ = - 21,
Xs + ( ) + 121 = 100,
x* + 11 = ± 10,
a:* --lor -21.
.*. a? = ±V— 1
or ivQn.
That is, the roots are imaginary.
a* + 14ar, + 24 = 0.
3* + 143* = -24,
a* + () + 49 = 25,
3* + 7 = ±5,
a? = -2 or
•.*=V^2or #-12.
-12.
12. a^"» + 3a«l-4«0.
4a^»» + 12a^-16 = 0,
4z2"* + () + (3)2 = 25,
2a^ + 3 = ±5,
. 2a^=2or-8.
.-. x = 1 or V - 4.
286
ALGEBRA.
13.
4a?-20a? + 23a? + 5a; = 6.
4a? - 20a? + 23a? + 5x - 6 = 0.
4 a? - 20 x8 + 23 a2 + 5a; - 6 [2^^5^-J
4a?
4a? — 5s
- 20 a? + 23 a?
- 20 a? + 25 a?
4a? -10a; -J
-2a?+5a;-6
-2a;2 + 5a; + }
-¥
If ^ were added to both members the square would be complete,
and the equation would read
4a? - 20a? + 23 x* + 5a; + i = *£.
Extract the root, 2a?-5a;-j = ±f,
2a?-5x = 3or-2,
16a?-() + 25 = 49or9,
4a; — 5 = ± 7 or ± 3,
4a; = 12, or -2, or 8, or 2.
.-. s = 3, -h 2,}.
14.
_L+ 1-20 = 0.
£** x11
1 + 3a^-20a?* = 0,
20a?»-3a?» = 1,
1600a?»-() + 9 = 89i/_
40a^-3 = ±V89.
• :x=V&±&^.
■0.
15.
x*_4a?-10a? + 28a-15 =
Extract root of left side,
x* - 4a* - 10a? + 28a; - 15|a?-2a?-7
x*
2a?-2a;
-4a? -10a?
-4a? + 10a?
2a?-4a; -7
-14a? + 28 a; -15
- 14a? + 28a; 4-49
-64
Add 64 to both sides to complete the square,
x* - 4a? - 10a? + 28 a; + 49 = 64.
Extract the root, a? — 2a; — 7 = ±8,
a?-2a?=15or-l,
43*_() + 4 = 64or0,
2a;-2 = ±8or±0,
2a; = 10, -6, 2, 2.
.-. a; = 5, -3, 1, 1.
teachers' edition.
287
16.
a? - 2s? - 13a;2 + 14a; =- 24.
Extract root of left side,
a?- 2a? - 13a;2 + 14a; + 24|a?-a;-7
2a?-4
-2a?-13a?
-2a? + a?
- 14a? + 14a; + 24
-14a? + 14a; + 49
-25
Add 25 to both sides to complete the square, -
a? - 2a? - 13a? + 14a; + 49 = 25.
Extract the root, a?-a;-7 = ±5,
a? - x = 12 or 2,
4a?-() + l = 49or9,
2a;-l=±7or±3.
.-. a; = 4,-3,2, -1.
17.
108 a? = 20 a; (9 a?-
108a? - 108a? + 51 a? + 20a; - 7.
Multiply by 12, and add 16 to both sides,
1296a?- 2160a? + 612a? + 240a; + 16 = 100.
■l)-51a? + 7.
1296a?- 2160a? + 612a? + 240a; + 16136a;2 - 30a; -
1296 a? '
72a? -30a;
- 2160a? + 612a?
- 2160a? + 900a?
72a? -60 a; -4
- 288a? + 240 a; + 16
- 288 a? + 240 a; + 16
36a?-30a;-4 = ±10,
36a? -30a; =14 or- 6,
144a?-() + 25 = 81orl,
12a;-5 = ±9or ± 1,
12a; =14, -4, 6, 4.
18.
(a? - l)(a? - 2) + (a? - 3)(a? - 4) = a? + 5.
Simplify, a?- 3a? + 2+ a?- 7a? + 12 = a? + 5.
Transpose and combine, a?— 10 a? = — 9.
Complete the square, a?- ( ) + 25 = 16. '
Extract the root, a? — 5 = ± 4,
a? = 9 or 1.
.\ai = ±3or±l.
288 ALGEBRA.
Exercise XC.
1. The sum of the squares of three consecutive numbers is
865. Find the numbers.
Let x = first number,
x + 1 = second number,
and x 4- 2 = third number.
.-. x* + x* + 2x + l+x* + 4x + ± = 365,
3a? + 6a; = 360,
a* + 2x = 120,
a* + () + l = 121,
x '+ 1 = ± 11.
.-. a; = 10 or -12.
Hence, the numbers are 10, 11, 12.
2. Three times the product of two consecutive numbers
exceeds four times their sum by 8. Find the numbers.
Let x = first number,
and x + 1 = second number.
3 a* -f 3 s = three times product,
8 a; + 4 = four times sum.
.-. 3a? + 3a; -(8a; + 4) = 8,
3a?-5a;=12,
36a*-() + 25 = 169,
6x-5 = ±13,
6a;=18or-8.
.-. a; = 3or-f
Hence, the numbers are 3, 4.
3. The product of three consecutive numbers is equal to three
times the middle number. Fiud the numbers.
Let x = first number.
Then x 4- 1 = second number,
and x + 2 = third number.
,\x(x + l)(x + 2) = 3(07 + 1),
x*+Sx*'+2x = 3a; + 3,
a* + 3a?-a;-3 = 0,
(* + l)(a-l)(s + 3)-0.
.•.s = l,~l,-3.
Hence, the numbers are 1, 2, 3.
teachers' edition. 289
4. A boy bought a number of apples for 16 cents. Had he
bought 4 more for the same money he would have paid J of a
cent less for each apple. How many did he buy?
Let x = number of apples bought.
■I n
Then — = number of cents one apple costs,
1 ft
and = number of cents one apple costs when he gets
x + ^ four more.
... 16" 16 =lt
x x + 4 3*
48s + 192 -48 a? = a* + 4a;f
x2 + 4s = 192,
a;' + () + 4 = 196,
a; + 2- ±14.
.\ a; = 12or-16.
Hence, 12 = number of apples bought.
5* For building 108 rods of stone-wall, 6 days less would have
been required if 3 rods more a day had been built. How many
rods a day were built?
Let x = number of rods built in a day,
108
— = number of days in which the whole wall was
* built,
108
= number of days it would have taken to build
x + ^ the whole wall if 3 rods more a day had
been built.
Then 12§_J08_ = 6.
X 3 + 3
108a; + 324 - 108 a; = 6a;2 + 18a?,
6a? + 18a; = 324,
a? + 3a; =54,
4a?+ () + ^=225,
2a; + 3= ±15,
2a; = 12 or -18.
.\ a; = 6 or -9.
Hence, 6 =* number of rods built in a day.
290 ALGEBRA.
6. A merchaut bought some pieces of silk for #900. Had he
bought three pieces more for the same money, he would have
paid $ 15 less for each piece. How many did he buy?
Let x = number of pieces bought.
900
Then — = number of dollars each piece cost,
x
900
and = number of dollars each piece would have cost
as + 3 if be had received three more for $900.
Then *X>-*KL = i5t
x x + 3
900a; 4- 2700 -900 a; = 15a* + 45s,
15a* + 45a; = 2700,
a* + 3x=180,
4a* + () + 9 = 729,
2a? + 3 = ± 27.
.*. x = 12 or -15.
Hence, 12 = number of pieces bought.
7. A merchant bought some pieces of cloth for .$163. 75. He
sold the cloth for $ 12 a piece, and gained as much as 1 piece cost
him. How much did he pay for each piece?
Let x = number of pieces,
12 a; = number of dollars received for all,
'■ — = number of dollars paid for one piece.
Then 12 a; — 168.75 — number of dollars gained.
io icQKK 168.75
.-. 12 a; — 168.75 = ,
x
12a2- 168.75 a; = 168.75.
Multiply by j, 16 x* - 225 a; = 225,
1024 a? - ( ) + (225)* = 65025,
32a; -225 = ±255,
32 a; = 480 or -30.
.-.a; = 15 or -1|,
and 15^5=11.25.
15
Hence, one piece cost $11.25.
16
teachers' edition. 291
8. Find the price of eggs per score when 10 more in 62J cents'
worth lowers the price 3IJ cents per hundred.
Let x = number of eggs at 62J cents.
Then — — = cost of one egg in cents,
x
62 5
and : — = cost of one egg in cents, if he had received ten
x + 10 more.
6250 6250 ,«• v j i
,\ = difference in price per hundred.
x a? + 10 r r
. 6250 6250 _ 125
x x + 10 4 '
Divide by 125, 52 — -t»
J x a; + 10 4
200a; + 2000 - 200a; = x* + 10a;,
a? + 10a; = 2000,
a* + ( ) + 25 = 2025,
x + 5 = ± 45.
.-.a; = 40.
on e on e
Hence, one egg cost — — cents, and 20 eggs cost — — X 20
&e> 40 °° ■ 40
= 31 J cents.
9. The area of a square may be doubled by increasing its
length by 6 inches and its breadth by 4 inches. Determine its
side.
Let x = the side of the square.
(a? + 4)(a? + 6)-2*\
a;a + 10a; + 24 = 2arl,
a;2 - 10s = 24,
a*-() + 25 = 49,
x — 5 = ± 7.
.-. a; =12 or -2.
Hence, the side of the square is 12 inches.
10. The length of a rectangular field exceeds the breadth by 1
yard, and the area is 3 acres. Determine its dimensions.
Let x = number of yards in breadth,
x + 1 = number of yards in length,
and x(x -f 1) = number of square yards in area.
But area is 3 k, or 14,520 square yards.
.-. x2 + x = 14,520.
4a? + ( ) + 1 = 58,081,
2a; + 1 = ± 241.
.-.a; =120 or -121.
Hence, the field is 121 yards long by 120 broad.
292
ALGEBRA.
11. There are three lines of which two are each $ of the third,
and the sum of the squares described on them is equal to a
square yard. Determine the lengths of the lines in inches.
Let x — number of inches in third line,
and
— = number of inches in each of the others.
^+ 49 + 49
1fi T* lftfT*
Then a* + -^-+— — = the sum of the squares.
49 49 *
1 square yard = 1296 square inches.
= 1296,
81 x2
49
9*
7 "
.-. s = ±28.
Hence, the lengths are 16, 16, and 28 inches.
■ - 1296,
= ±36.
12. A grass plot 9 yards long and 6 yards broad has a path
round it. The area of the path is equal to that of the plot.
Determine the width of the path.
Let x = number of yards in width of path.
Then (9 + 2x) 2 + 6 X 2 = entire length of path in yards.
Also, [(9+2a) 2+6x2] x
or (30 + 4 a:) x = area of path in square yards,
and 9x6 = area of grass plot in square yards.
But area of path equals area of grass plot.
.-. (30 + 4*)* = 54.
4s* + 30 a? = 54,
16s* + () + 225 = 441,
4* + 15 -±21.
.-. »=l}or-9.
Hence, the width of the path is 1 J yards.
teachers' edition. 293
13. Find the radius of a circle the area of which would be
doubled by increasing its radius by 1 inch.
Let x = radius of circle,
and x + 1 = radius increased.
The ratio of the circles is the same as the ratio of the squares on
the radii. ... 2a;2 = a* + 2x + 1,
a*-2a;=l,
x*-() + l = 2,
«-l = ±V2,
s«l±V2,
• a: =2.4142.
14. Divide a line 20 inches long into two parts so that the
rectangle contained by the whole and one part may be equal to
the square on the other part.
Let x = one part.
Then 20 — x = the other part.
.-. 20(20-30 = 0?,
400-203 = 0*
a?2 + 20 a? = 400,
a* + () + (10)2 = 500,
.T + 10=±\/oXJb,
x = - 10 ± 22.36,
ar=12.36.
Hence, one part is 12.36 inches, and the other is 7.64 inches.
15. A can do some work in 9 hours less time than B can do it,
and together they can do it in 20 hours. How long will it take
each alone to do it?
Let x = number of hours it takes B.
Then x — 9 = number of hours it takes A,
and - = part B could do in 1 hour.
x
= part A could do in 1 hour.
x — y
x x-9 20
20a; - 180 + 20x = x*- 9x,
a* -49 a? = -180,
4a?-() + (49)2 = 1681,
2a: -49 = ±41,
2a = 90 or 8.
.*. a; = 45 or 4.
Hence, B can do the work in 45 hours and A in 36 hours.
294 ALGEBRA.
16. A vessel which has two pipes can be filled in 2 hours less
time by one than by the other, and by both together in 2 hours
55 minutes. How long will it take each pipe aloue to fill the
vessel?
Let s = number of hours it takes first pipe,
x — 2 = number of hours it takes second pipe.
2 hours 55 minutes equals f f hours.
. 1 1 _12
"x 3-2 35*
35* _ 70 + 35s - 12s* - 24x,
12a2 -94s = -70,
144 s2 -() + (47)2 = 1369.
Extract the root, 12 s - 47 - ± 37,
12s = 84 or 10.
.-. s= 7 or f.
Hence, one pipe will fill it in 7 hours, the other in 5 hours.
17. A vessel which has two pipes can be filled in 2 hours less
time by one than by the other, and by both together in 1 hour
52 minutes 30 seconds. How long will it take each pipe alone to
fill the vessel?
Let s = number of hours it takes first pipe.
Then s + 2 = number of hours it takes second pipe.
- = part first pipe fills in 1 hour,
and = part second pipe fills in 1 hour.
s + 2
1 hour 52 minutes 30 seconds equals -^ hours.
...1 + -1— L
s x + 2 15
15s + 30 + 15s = 8s2 + 16s,
8s2 -14s = 30,
64s2 -112s = 240,
64s2 -0 + 49 = 289,
8s -7- ±17,
8s = 24 or -10.
.-. s = 3 or -1J.
Hence, one pipe will fill it in 3 hours, the other in 5 hours.
18. An iron bar weighs 36 pounds. If it had been 1 foot
longer, each foot would have weighed J a pound less. Find the
length and the weight per foot.
teachers' edition. 295
Let x = number of feet in length.
36
Then — = weight in pounds per foot,
and = weight per foot if it had been 1 foot longer.
But = weight per foot if it had been 1 foot longer.
.36 1 = 36
x 2 x + l'
72a: + 72 -a;2- a: = 72a:,
x* + x = 72,
4s* + () + 1 = 289,
2a: + l=±17.
.-. x = 8 or -9,
25-4J.
% x
Hence, the bar is 8 feet long, and weighs 4i pounds per foot.
19. A number is expressed by two digits, one of which is the
square of the other, and when 54 is added its digits are inter-
changed. Find the number.
Let x = digit in tens' place,
Then a? = digit in units' place,
10 a? + x9 = number,
and 10 x8 + x = number with digits reversed.
.-. 10s + x2 + 54 = 10a?2 + a;,
- 9a2 + 9a: = -54,
x2 - x = 6,
4a*-() + l = 25,
2s-l = ±5.
.'.x = 3.
Hence, the number is 39.
20. Divide 35 into two parts so that the sum of the two frac
tions formed by dividing each part by the other may be 2^.
Let x = one part.
Then 35 — x = the other part.
x 35-a?_25t
'"'35 -a x 12
12a? + 14700 -840 a: + 12a? = 875* -25 a2
49a? -1715a: = -14700,
x2 -35 x = -300,
4a?-() + (35)2 = 25>
2x-36 = ±5.
.-. a: = 20 or 15.
Hence, the parts are 20 and 15.
296 ALGEBRA.
21. A boat's crew row 3} miles down a river and back again
in 1 hour 40 minutes. If the current of the river is 2 miles per
hour, determine their rate of rowing in still water.
Let x = rate in still water,
x + 2 = rate down stream.
1 hour 40 minutes equals J hours.
— *— = number of hours going down stream,
x + 2
2
7,75
•- number of hours going up stream.
2(x + 2) 2(a-2) 3
21x- 42 +21 x + 42 = 10 a? -40,
10a? -42s = 40,
400a? - ( ) + (42)2 = 3364,
20aj-42 = ±58,
20 a = 100 or -16.
.*. x = 5 or — J.
Hence, the rate of rowing in still water is 5 miles an hour.
22. A detachment from an army was marching in regular
column with 5 men more in depth than in front. On approach-
ing the enemy the front was increased by 845 men, and the whole
was thus drawn up in 5 lines. Find the number of men.
Let x = number of men in front,
and x + 5 = number of men in depth.
Then a? + 5x = number of men in all.
But x + 845 = number of men in front,
and 5 = number of men in depth.
Then 5x + 4225 = number of men in all.
.-. ic2 + 5a; = 5a; + 4225,
a:2 = 4225.
.*. x = ±65.
Hence, the whole number of men is 4550.
23. A jockey sold a horse for $144, and gained as much per
cent as the horse cost. What did the horse cost?
teachers' edition. 297
Let x = number of dollars the horse cost.
Then -^-~ = gain per cent,
100 & r
of x = whole gain,
100 6
and x + — = amount received.
100
.... + JL..1*
x* + 100 a; = 14400,
x7 + ( ) + 2500 = 16900,
x + 50 = ± 130.
.\ a; = 80 or -180.
Hence, the horse cost $80.
24. A merchant expended a certain sum of money in goods,
which he sold again for $24, and lost as much per cent as the
goods cost him. How much did he pay for the goods?
Let x = number of dollars paid for goods.
Then — = per cent lost,
100 r
and -^- of x = whole loss.
100
100
100* -x* = 2400,
x2- 100 a; = -2400,
aa_() + (50)» = 100,
a;-50 = ±10,
x = 60 or 40.
Hence, the goods cost either $60 or $40.
25. A broker bought a number of bank shares (f 100 each),
when they were at a certain per cent discount, for $ 7500 ; and
afterwards when they were at the same per cent premium, sold all
but 60 for $5000. How many shares did he buy, and at what
price?
298 ALGEBRA.
Let x = number of shares bought,
7500
— — = number of dollars each share
x cost,
and
Then
100-
7500
X
loo-irrr
X
100*^-7500
100 Ui
Also,
Then
100a;
x-QO
5000
= number of dollars discount on
each share.
■ rate of discount.
■■ number of shares sold.
-- number of dollars received for
5000 x~ eacn Bhare»
and 100 = number of dollars premium on
K/y^ 05 — 60 eacn share.
ouuu -100
x-60 11000 -100a: . , *
— ___ or = rate per cent of premium.
100 100 x -6000 r r
But rate percent discount was equal to rate percent premium.
. 100a? - 7500 _ 11000 - 100a;
100a; 100a;- 6000 '
x* -135 a; + 4500 = 110a? -a*
2a?2 -245 a? = -4500,
16a*-() + (245)2 = 24025.
Extract the root, 4 a; — 245 = ± 155,
4a? = 400or90.
.-. a?=100or22£.
Hence, the broker bought 100 shares at 75.
26. The thickness of a rectangular solid is } of its width, and
its length is equal to the sum of its width and thickness ; also,
the number of cubic yards in its volume added to the number of
linear yards in its edges is $ of the number of square yards in its
surface. Determine its dimensions.
Let 3 a? = number of yards in width,
2 a? = number of yards in thickness,
and 5 x = number of yards in length.
303? + 40 a; = 1(622?*),
90 a?8 - 310 a?2 = - 120 a?.
Divide by 10 x, 9 a?2 - 31 a? = - 12,
9^-() + (W = W.
3a;-V = ±V.
3a? = 9 or*.
.\ a? = 3 or J.
Hence, the dimensions are 15 X 9 X 6 yards,
or 2} X \\ X f yards.
TEACHERS EDITION.
299
27. If a carriage-wheel 16J feet round took 1 second more to
revolve, the rate of the carriage per hour would be 1 J miles less.
At what rate is the carriage travelling?
Let
59400
3600
» number of seconds it takes the wheel to
revolve;
= number of revolutions it makes per hour,
or 16$ X
x
3600
x
59400
= number of feet it goes per hour,
■ number of feet it would go, if it took one
second more to revolve,
Then
x + 1
59400_59400 = 99(X)j number of feet in x , miles>
x x + 1
59400a; + 59400 - 59400a; = 9900x2 + 9900x, "
9900 a* + 9900 x = 59400,
a2 + x = 6,
* + () + *-¥.
re + i = ± J.
.-. x = 2 or -3.
ai £lq. = 29700.
Since 29700 feet equal 5$ miles, the carriage is travelling at the
rate of 5f miles per hour.
Exercise XCI.
3
(3)
(4)
1. x + y = 13
ay = 36
Square (1),
x* + 2xy + y2 = l69
(2)x4is ±xy =144
Subt.,rc2-2icy + y2= 25
Extract root, x — y = ± 5 (5)
Add (1) and (5), 2 s = 18 or 8.
.•. x = 9 or 4.
Subtract (5) from (1),
2y = 8 or 18.
f\ y = 4or 9,
2. a? + y = 29
ay = 100
Square (1),
a*+2a#+y2 = 841
(2)x4is 4 ay =400
(1)
(2)
(2)
Subt., x2 - 2xy + y2 = 441
Extract root, x — y = ± 21 (5)
Add (1) and (5), 2a; = 50 or 8.
. \ x = 25 or 4.
Subtract (5) from (1),
2y = 8 or 50.
•\ y = 4or25.
300
ALGEBRA.
3. s-y = 19
xy=* 66
Square (1),
xi-2xy + y2 = 361
(2)x4is 4zy =264
Add, x2 + 2:ry + y, = 625
Extract root, x + y = ± 25
Add (5) and (1), 2x = 44 or
.-. a; = 22 or
Subtract (1) from (5),
2y = 6 or -
.-. y = 3or-
4. x — y = 45
sy = 250
Square (1),
x*-2xy+yi
(2)x4is 4sy
(1)
(2)
(3)
(4)
(5)
-6.
-3.
-44.
-22.
5. x — y = .
x*+y2 = 178 (2)
Square (1),
a? — 2:cy + y* =
(1)
(2)
Add, x* + 2xy + y* =
Extract root, x + y =
Add (5) and (1), 2x =
.*. x =
Subtract (1) from (5),
2y =
.-. y =
2025 (3)
1000 (4)
3025
±55 (5)
100or-10.
50 or -5.
10or-100.
5 or -50.
= 10 (1)
(2)
(2) is x2 + ya =
Subt.,
100
178
(2) is a*
2xy
-fyV
■ 78
= 178
(3)
Add, x* + 2xy + y* =
Extract root, x + y =
Add (5) and (1), 2x =
.*. x =
Subtract (1) from (5),
2y =
.-. y =
6. x — y =
s* + y» =
Square (1),
x*—2xy+y* =
Subtract (2) from (3),
-2xy =
Subtract (4) from (2),
x1 + 2xy + y* =
Extract root, x + y =
Add (1) and (5), 2x =
Subtract (1) from (5),
2y =
256
±16 (5)
26 or -6.
13 or -3.
6 or -26.
3 or -13.
i1'
(2)
196 (3)
-240 (4)
676.
±2Q (5)
40 or -12.
20 or -6.
12 or -40.
6 or -20.
Square (1),
Subtract (2) from (3),
Subtract (4) from (2),
Extract root,
Add (1) and (5),
Subtract (5) from (1),
s + y = 12 (1)
xa + y2 = 104 (2)
a? + 2&y + y2 = 144 (3)
2sy = 40 (4)
x*-2xy + yi = 64:.
a;-y = ±8 (5)
2x = 20 or 4.
.-. x = 10 or 2.
2y = 4 or 20.
.\ y = 2or 10.
TEACHERS EDITION.
301
8. 1 + 1.
x y
X* V*
Square (1), 9
1 2 1
— + — + — =
3* Xy y*
Subtract (2) from (3),
2_ =
xy
Subtract (4) from (2),
i_A+I=
x2 xy y*
Extract root, =
x y
Add (1) and (5), ? =
x
Subtract (5) from (1),
2
y~
.-. y =
3
4
_5
16
$_
16
4
16
J_
16*
-\
lor}.
2 or 4.
Jorl.
4 or 2.
(1)
(2)
(3)
(4)
(5)
H=5 (1)
Square (1), *
-. + - + -. = 25 (3) n.
xy y
Subtract (2) from (3),
2
Subtract (4) from (2),
L-l- + ±- =
x2 xy y2
Extract root, =
x y
Add (1) and (5), -~
.•. x =
Subtract (5) from (1),
y
.-. y =
^=12 w
1.
±1 (5)
6 or 4.
JorJ.
4 or 6.
J or J.
10. 7x2-8zy = 159 (1)
5a; + 2y=7 (2)
2y = 7-5x.
•' 2
Substitute in (1),
7aa-8s(^i^ = 159,
14^-56a; + 40a;a = 318,
54aJ-56x = 318.
Divide by 6,
9x*-*£x = 53.
Complete the square,
»*-< ) + (W = J4f»-
Extract root,
3a;-V=±V.
3» = ^or-^.
.-. z-3or-lff.
Substitute value of a; in (2).
.-. y=--4or8if
s + y = 49 (1)
a^ + yJ = 1681 (2)
Square (1),
3a + 2ajy+y» = 2401. (3)
Subtract (2) from (3),
2ay = 720 (4)
Subtract (4) from (2),
a?-2zy + ya = 961.
Extract root, x—y = ± 31 (5)
Add(5)and(l),2a> = 80orl8.
.-. x = 40 or 9.
Substitute value of x in (1).
.\y = 9 or 40.
302
ALGEBRA.
12. 7? + f =
x + y =
Divide (1) by (2),
x2- ay+y2 =
Sq. (2), s»+ 2ay+y2 =
Subt., — 3ay
.\-ay =
Add (3) and (5),
a*-2ay + y2 =
Extract root, x — y =
Add (2) and (6), 2a; =
.•. x =
Subtract (6) from (2),
2y =
.-. y =
341 (1)
ii (r
2)
31
121
-90
-30
(5)
1.
±1
12 or 10.
6 or 5.
(6)
10 or 12.
5 or 6.
8!
8
(5)
13. ^+^=1008
s + y = 12
Divide (1) by (2),
x*— xy+y* = 84
Sq.(2), x*+2xy+y*= 144
Subt., - Sxy = - 60
... -xy = - 20
Add (3) and (5),
aa-2»y + y, = 64.
Extract root, re — y = ± 8 (6)
Add (2) and (6), 2 a; = 20 or 4.
.\ x = 10 or 2.
Subtract (6) from (2),
2y = 4or 20.
.\y = 2or 10.
14. x*-tf = 98
x-y = 2
Divide (1) by (2),
x2+ ay+y2 = 49
Sq. (2), a?-2sy-fy2 = 4
Subt., 3ay - 45
,\ xy = 15
Add (3) and (5),
a* + 2sy + y2 = 64.
Extract root, x + y = ± 8
Add (2) and (6), 2 s = 10 or -
. \ x = 5 or -
Subtract (2) from (6),
2y = 6or-
• \ y = 3 or —
(3)
(4)
(5)
(6)
-6.
-3.
10.
5.
15. s8-y3 = 279
a;-y = 3
Divide (1) by (2),
x*+ xy + y2 = 93
Sq.(2), x*- 2 xy + y*= 9
Subt., 3 ay =84
,\ ay = 28
Add (5) and (3),
a* + 2ay + y2 = 121.
Extract root, x -f y = ± 11
Add (6) and (2), 2a; = 14 or
.\ a; = 7 or -
Subtract (2) from (6),
2y = 8 or -
. \ y = 4 or -
8
(6)
-8.
4.
14.
7.
16.
-3y = l
(1)
ay + y2 = 5 (2)
Transpose (1), x = 1 + 3y.
Substitute m (2),
y(l+3y) + y2 = 5,
y + 3y2 + y2 = 5,
4y2 -f y = 5,
2y + i = ±f.
.-. y = lor-l}.
Substitute value of y in (1),
a; = 4or-2f.
17.
4y =
2xy =
y =
5s+l (1)
-** (2)
33
5s + l
4
in (2),
33 -x2,
Substitute value of y
10a? + 2a?_
4
14a2 + 2a; =132.
Divide by 2,
7^ + x = 66,
196 a? + () + 1 = 1849.
Extract root,
14a; + l = ±43,
14 a; = 42 or -44.
.-. a; = 3or-3f
Substitute value of a; in (1).
.-. y = 4or-3Jf
TEACHERS EDITION.
303
1 1
18. ±_i = 3
(1) 19. 1-1 =
= 2}
(1)
a; y
x y
1-1 = 21
x2 y1
(2) L-L =
■81
(2)
Divide (2) by (1),
Divide (2) by (1),
1 + 1-7
(3) 1 + 1-
s7
(3)
x y
x y
2
Add (3) and (1), ? = 10.
X
Add (1) and (3),- =
x
. 12
" 2
•••* = *.
.*. 05 =
-h
Subtract (1) from (3),
Subtract (1) from (3),
?-4.
2 =
= 1.
y
y
••• y = J.
.-. y =
20.
= 2.
x*-2xy-y* = l
(1)
x + y = 2
(2)
Square (2),
x* + 2xy + y2 = 4
(3)
Add (3) and (1),
2a? = 5,
a2 = 2J.
.-. x = ±V2}.
Substitute value of x in (2),
y = 2W2J.
Exercise XCII.
a? + a;y-f2y2 = 74
(1)
2a?2 + 2ay + y2 = 73
(2)
Add,
3a^ + 3ajy+3ya = 147
Divide by 3,
a;2 + xy + y* = 49
(3)
Subtract (3) from (1),
ya - 25.
,\y = ±5.
Substitute value of y in (3),
a*±5a: + 25 = 49,
a?±5x = 24,
4a-2 ± 20a: + 25 = 121.
Extract the root,
2a; ± 5 « ± 11,
2a; = ±6or ±16
.*. a: = ±3 or ±8.
304 ALGEBRA.
a* 4- xy + 4y2 = 6 (1>
3^+8^ = 14 (2)
Substitute vx for y in both equations.
From (1), x* + vx2 + 4vV - 6.
... 3* = 5 (3)
From (2), 3 x2 + 8 tftc2 = 14.
...a2- 14 (4)
Equate values of a2,
H 6 14
l + 17+4v« 3 + 8v»
18 + 48 v2 - 14 + 14v + 56 V2,
8a2 + 14* = 4 (5)
64t? + () + 49 = 81,
8t; + 7 = ±9,
8t>.= 2or-16.
.•. t;=« J or —2.
14 14
Substitute values of v in (4), x* - — -- or — —
Then a* = 4orf
.\ x-±2t ±Vf
From (2), y = ±}, ±2V|.
3.
a»-^ + y2»2i (l)
y2-2sy = -15 (2)
Substitute vx for y in both equations.
From (1), a2 - to1 + v2*2 = 21.
...*-_?L— (3)
1— v -{-v2
From (2), vV - 2 vz2 = - 15.
.,^-pi§- (4)
v2 — 2v
Equate values of a2, ; = — — — (5)
21 v* - 42 v = - 15 + 15t; - 15v2,
36«2-57v = -15,
5184^ -() + (57)2 = 1089,
72V _ 57 = ± 33.
.-. v = f or J.
4), z2 = 16 '
Substitute values of v in (4), x2 = 16 or 27.
.•. a; = ±4 or ±3 VS.
,\ t/ = ±5 or ±V3.
teachers' edition. 305
a2-4y2-9 = 0.
xy + 2y2 - 3 = 0. *
a? - 4y2 = 9 (1)
Transpose, xy + 2y2 = 3 (2)
Substitute va? for y in both equations.
From(l), x*-lv*x* = 9.
-"-nSsi (3)
From (2), 3*t; + 2^^ = 3.
9 3
Equate values of S2, —
1-4^ v + 2v*
30^ + 9^ = 3,
10* + 3t>-l,
400v2 + ( ) + 9 - 49,
20 1; + 3 _ ± 7,
20v = 4or-10.
.\t> = *or-J.
Substitue values of v in (3), a?2 = ^ or oo.
.-. z = ±5Vf
.•.y-±vf.
5.
0^-0^ = 35 (1)
*y+y2 = 18 (2)
Substitute vx for y in both equations.
From(l), aja-va:2 = 35.
-*-T=i (3)
From (2), va^ + ^a^=18.
.•.s2=-^- (4)
Equate values of s2, -iL = -^-t
1 — v v + v*
35v2 + 53v = 18,
4900 v2 + () + (53)2 = 5329,
70v + 53 = ± 73,
70v = 20or-126.
.-. v = |or-f
Substitute values of v in (3), a? = 49 or 4£.
.\ & = ± 7 or ±5V|.
.\ y=.±2orT9V|.
306 ALGEBRA.
6.
x2 -f xy + 2y2 =- 44
2x*-xy + y2 = 16
(1)
(2)
Substitute vx
for y in both equations.
From (1),
a? + vx2 + 2v2x2 = 44.
44
(3)
From (2),
2s*-irc2+irtc* = 16.
■ *'- 16 •
2-v-\-v2
Equate value
nofo? 44 16 4
'1+V + 2V2 2-v + r2
88 - 44v + 44^ = 16 + 16v + 32w»,
12^-60v = -72,
4t?-() + 25 = l,
2t>-5 = ±l.
.-. v = 3or 2.
Substitute values of v in (3), x* = 2 or 4.
From (3). .-. a: = ±V2 or ± 2.
.•. y = ± 3 V2 or ± 4.
7.
3» + ay=15 (1)
xy-y2 = 2 (2)
Substitute vx for y in both equations.
From (1), x* + vx2 = 15.
1 +v
From (2), vx* - v*x* = 2.
'.a1 ^-r (4)
v—v*
Equate values of x*, - =
1+v v—v2
I5v-15v* = 2 + 2v,
15v»-13t; = -2,
900t>*-() + 169 = 49.
Extract the root, 30 1; - 1 3 = ± 7, .
30v=20or6.
.\ v = f or J.
Substitute values of v in (3), x2 = 9 or Af .
.\ a = ±3 or ±5\/j.
.\ y =*±2 or ±VJ.
TEACHERS* EDITION.
8.
x2 — xy + y2 = 7
(1)
3x2 + 13a?y + 8?/2 = 162
(2)
Substitute vx
for y in both equations.
From (1),
a*2 — vx + iPx2 = 7.
• r2 7
(3)
l-v+v*
From (2),
3x* + 13v + 8v2x* = 162.
■*»- 162
-
3 + 13v +
8v2
-.nf* ^ „. 162
U\
^ 1-v + v2 3 + 13v + 8i;2
.-. 106V2- 253v = - 141,
44944^ -()+(253)2 = 4225.
Extract the root, 212v - 253 = ± 65.
Substitute, values of v in (3), x2 = 4 or ^ft».
.-. ar=±2or ±2Jf
.-. y = ±3or ±2^.
9.
2a2 + 3sy+y2 = 70 (1)
6a^ + ary-y2 = 50 (2)
Substitute vx for y in both equations.
From (1),
2&2+3trc, + v,a,=
= 70.
70
2 + 3v
+ v*
From (2),
6^ + 1?^ — v*x" =
= 50.
50
6 + v-
v2
70
50
~~ ' 2 + 3tf+va 6 + v-
-V2'
420 + 70i;-70t>* =
= 100 + 1500 + 50^,
12v* + 8v =
= 32,
36^ + () + (2)* =
6v + 2 =
aoo,
.±10.
.*. v =
• ljor-
-2.
Substitute vali
ae oft? in (3), a* =
» 9 or oo.
.\ « =
= ±3.
.-.y-
= ±4.
(3)
(*)
S08 ALGEBRA.
Substitute vx
From (1),
10.
x* — xy — y* = 5
for y in both equations.
a1 — wx* — v*x* — 5.
- *» 5
a)
(2)
(3)
From (2),
1-v-v8
23* + 3itt* + t*r* = 28.
. ^ 28
2 + 3v + v*
M
^ l-v-vi 2 + 3v + v*
10 + 15v + 5v« - 28 - 28v - 28V8,
33v* + 43t> = 18,
4356v» + () + (43)* = 4225I
66t> + 43 = ± 65.
.'.•-!« -if-
Substitute values of v in (3), x* = 9 or - 121.
.-. x = ±3or±llV^l
.-. y = ±l or =f18\^I.
Exercise XCIII.
1.
4sy = 96-a:V W
* + y = 6 (2)
Let » = (« + v),
and y = (u-v).
From (2), n + v+u-v-6,
2« = 6,
u = 3.
From (1), 4(u* - v8) = 96 - w* + 2wV-^,
4m» - 4v» = 96 - u* + 2mV-v*.
Substitute value of u, v* - 22v* - - 21,
4r2-()+(22)* = 400,
2v»-22 = ±20,
2v* - 22 ± 20,
^ = 21 or_l.
•\ v = ±V21 or ±1.
... <c = m + v = 3±V21, 4, 2,
and y - w - v = 3 t V21, 2, 4.
teachers' edition. 309
a»+y»«18-*-y (1)
^ -6 (2)
Put u + v for ar, and w — v for y.
(1) becomes (u + v)2 + {u — v)2 = 18 — 2w,
u% + v* + u = 9 (3)
(2) becomes (u + v) (u — v) = 6,
or u* - v2 = 6 (4)
Add (3) and (4), 2w2 + u = 15.
Complete tbe square, 16w2 + () + l = 121,
4w+l = ±ll.
.-. w=2Jor-3.
Substitute value of u in (4), — v* = 6 — *£- or 6 — 9.
.\ i> = ± J or ±V3.
.-. a: = u + i> = 3, 2, or -3±V3,
and y = u — v= 2, 3, or — 3=fV3.
3.
2(s2 + y2) = 5xy
4(x-y) = xy
Put u + v for a, and u — r for y,
2(2u2+2v2) = 5(m2-«2)
4(2v) = u2-t;2
(i)
(2)
(3)
(4)
Transpose and combine," 9v2 — u2 = 0
+ w2-t? = 8tf
(5)
(6)
Add (5) and (6), 8i>* = 8v
8v2-8v = 0.
.•. v = 0 or 1.
Substitute value of v in (6), wv = 8 1; + v2,
u2 = 0 or 9,
u = 0 or ± 3.
.\ a> = w + v = 0, 4, — 2,
and y — u — v = 0, 2, — 4.
310 ALGEBRA.
4.
4(*4-y) = 3*y (1)
x + y + x>+y* = 26 (2)
Put u -+- v for a, and « — v for y.
(1) becomes 8 u = 3 u* — 3 v2.
.-. 8u-3u2 + 3v2 = 0 (3)
(2) becomes 2u + 2^+2u2 = 26 (4)
Multiply (4) by 3, 6w + 6u2 + 6v2 = 78
Multiply (3) by 2, 16u-6u2 + 6^ = 0
Subtract, 12u2-10u = 78
Complete the square, 144u2 - ( ) 4- 25 = 961,
12w-5 = ±31,
12u = 36or-26.
.-. M = 3or-2J.
Substitute value of u in (3), 3 1? = 3.
.*. v = ± 1.
Substitute - 2J for u in (3), 3t? = i}fi.
.-. v = ±jV377.
.-. a - u + v = 4, 2, or K-13 ±V377),
and y = m - v - 2, 4, or i(-l3 * V577).
5.
4a2 + ay + 4y2 = 58 (1)
5a2 + 5y2 = 65 (2)
Multiply (1) by 5, 20a2 + 5w+ 20y2 = 290
Multiply (2) by 4, 20a2 + 20y2 = 260
Subtract, 5 ay =30
.\ xy = 6 (5)
Divide (2) by 5, a2 + y2 - 13 (6)
Substitute w + v for a, and w — v for y in (5) and (6),
u2 - v2 = 6 fl)
2u2 + 2v2 = 13 P9
Multiply (7) by 2, 2 u2 - 2 v2 - 1 2 (9)
Add, 4u2 - 25
.'. M = ± }.
Subtract (9) from (8), 4v2 = 1,
v2 = }.
.\ v = ± J.
.-. a==w + v = ±|±J = ±3or±2,
and y = u — v = ±\t £ = ±2 or ±3.
TEACHERS EDITION.
311
6.
ay(* + y)-30 (1)
a* + y3 = 35 (2)
Substitute u + v for .r, and u — v for y.
(1) becomes (u+v) (u—v) {(u+v) + (u— v)} =» 30,
or 2«s-2m?«30 (3)
(2) becomes (?* + vf + (w — vf = 35,
ot 2w8+6uv2=35 (4)
Multiply (3) by 3, 6w8-6uv2 = 90 #
\4)is 2ws + 6w^= 35
A.dd, 8w8
250
Substitute value of u in (3),
w 8 2
-125
2tt = 5,
u = l
10v2
30,
250-40v2 = 240,
40v2 - 10.
.-. v = ±£.
.•. a? = w + v = 3 or 2,
and y = u — v =» 2 or 3.
Exercise XCIV.
1. a:-y = 7
«* + ay + y* = 13
Square (1),
x*-2xy + y2 = 49
Subtract (2) from (3),
-3ay = 36.
Divide by - 3, xy = - 12
Add (4) and (2),
a;2 + 2a;y + y2 = 1.
Extract root, x + y = ± 1
Add (5) and (1), 2 a = 8 or 6.
.\ <r = 4 or 3.
Substitute value of x in (1),
y=*-3or-4.
(1)
(2)
(3)
W
(5)
2. a?2 + xy =
jry - y2 =
Substitute i>a; for
From (1), z2 =
From (2), a2 =
• 35
1 + v
35v2-29v =
4900v2-( )+(29)2 -
70v =
Substitute value
z2 =
a? =
y = va; =
i1)
(2)
(3)
35
6
5l
1 + v
6
v — v2
6
i> — t^
-6,
1,
±1,
30 or 28,
forf
of v in (3),
^or25.
±7Vj or ±5.
±3V|or±2.
312
ALGEBRA..
X-
Transpoae in (1),
-12 = 0
2y = 5
«y =12,
^ 12
(1)
(2)
Substitute value of y in (2),
x = 5.
x
Simplify, ars-24 = 5a;.
Transpose, # x* — 5 x — 24.
Complete the square,
4a*- () + 25 = 121,
2s -5- ±11,
, 2a; =16 or -6.
.-. a: = 8 or -3.
Substitute value of x in (2),
y=lJor-4.
4. ay-7 = 0 (1)
a?+y2 = 50 (2)
Transpose in (1), xy = 7 (3)
Multiply (3) by 2,
2xy = 14 (4)
Add (4) and (2),
a* + 2 ay + y* = 64.
.\a;+y = ±8 (5)
Subtract (4) from (2),
a*-23y + y* = 36.
.-. a--y = ±6 (6)
Add (5) and (6), 2a; = ±14or±2.
.\ a? = ±7or±l.
Subtract (6) from (5),
2y = ±2or±14.
.•. y = ±lor±7.
5.
2a;-5y = 9
x*-xy + y*=7
From(l), v-^T*
Substitute value of x in (2),
Simplify, 19y* + 72y = - 53.
Complete the square,
1444y» + ( ) + (72)» = 1156,
38y + 72 = ±34,
38y = -38or-106.
.\y = -lor-2}J.
Substitute value of y in (1), 2 x = 9 + (— 5) or 9 + (-
.-. x = 2or -Zfg.
0)
(2)
(3)
imr
Transpose (2),
Square (1),
Multiply (3) by 4,
Add,
Extract root,
Add (1) and (4),
Subtract (1) from (4),
6.
x — y = 9
ajy + 8 = 0
sy = -8
a?-2sy + y2 = 81
ixy =-32
a? + 2ay + y» = 49
x+y=± 7
2a?=16or 2.
.\ x = 8 or 1.
2y = - 2 or - 16.
.\ y = — 1 or — 8.
(3)
W
TEACHERS EDITION.
313
7. 5a;-7y = 0 (1)
5a;3_13^ = 4_7y2(2)
4
9. a* + 4sy = 3 (1)
4sy+y* = 2i (2)
Substitute vx for y,
2^ + 4^ = 3 (3)
From(l), x=?£
5
SimpLfy (2),
20x2-13ajy + 28y2 = 16
(3)
4ta» + tW = 2t (4)
From (3) and (4),
l + 4v
Substitute value of x in (4),
«g£ ^' + 28y' = 16,
1225y* = 400,
35y = ± 20.
a* - 9 (6)
16v+4v* v ;
. 3 9
1 + 4v 16v + 4v2
48i; +12va = 9 + 36t;,
12v2 + 12t> = 9,
-•.y-±f
4t?+4v = 3,
Substitute value of y in (1),
4i;a + () + l=4,
2v + 1 = ± 2.
.*. v = £ or — 1£.
.-. *-±f
Substitute values of v in (5),
x = ± 1 or±V— f,
and y = ±£or=Fj\/-$.
8. *-y-l (1)
s» + y' = 8J (2)
Square (1),
x*-2xy + y* = 1 (3)
(2) is s2 + y2 = 8}
Subt., -2zy = - 7J (4)
Subtract (4) from (2),
a^ + 2a^ + y2 = 16.
Extract root, x + y = ± 4 (5)
Add (5) and (1), 2 a; = 5 or -3.
,\ ic = 2Jor— 1$.
Subtract (5) from (1),
-2y = -3 or 5.
.-.y = lJor-2}.
10. s2-a;y + y2 = 48 (1)
a-y-8=0 (2)
(l)isa2— xy+y* = 48
Sq.(2), x*-2xy+y* = 64
Subt., xy = - 16
Multiply by 3,
3sy = -48 (3)
Add (3) and (1),
x* + 2xy + ya = 0.
Extract root,
x + y = 0 (4)
Add (4) and (2),
2x = 8.
.*. a? = 4.
Subtract (2) from (4),
2y = -8.
.-. y = -4.
AlA7l!iI)n&.
11.
x2 + Sxy + y2 = 1
(1)
3x* + xy + 3y2=13
(2)
Subtract (1) from (2),
2sa-2xy + 2y2 = 12.
Divide by 2,
x7- xy+y2 = 6
(3)
(l)i8
x* + 3xy + y*=l
(4)
Subtract,
-4rcy =5
Add 4 x (1) to (4),
4s2 + 8zy + 4y» = 9
(5)
extract the root,
2a:+2y = ±3.
Divide by 2,
x+y=±$
(6)
Add J of (4) to (3),
^-2rcy + y* = ^,
a;-y = ± JV29
(7)
Add (6) and (7),
2a: = ±}±J\/29.
.-. a; = J(±3±\/29). "
Subtract (7) from (6),
2y = ±}T }V29.
.-. y = J(±3=FV29.)
12.
x*-2xy + 3y2 = lf
(1)
a£ + ay-y* = i
(2)
Substitute vx for y.
From (1),
a*-2va*-f-3vV=yk
From (2),
x2 + vx2 — v232 = J.
Whence
9-18v+27i*
(3)
and
*■_ 1
(4)
9H 9v- 9V2
11 r 1
9-18V+27V1 9+9V-9V2
99 + 99„ _ 99 tf . 9 - 18 v + 27 1*.
-126v2 + 117tf = -90.
Divide by — 9,
14v2-13i; = 10.
Complete the square,
784v»-() + 169 = 729.
Extract the root,
28v-13 = ±27,
28^=40 or -14.
... Vs=^L0r -J.
Substitute value of v in (4), x2 = ^T or $.
.-. » = ±J\/TVor±f
and y = ±-yV^or tJ.
TEACHERS EDITION.
315
13.
x + y = a (1)
4ay = a2-4&2(2)
Square (1),
s* + 2:ry + y2 = a2 (3)
(l)ia 4zy = a2-462
14.
*-y = l (1)
2*. (2)
y + 1.
- + *
y a;
Subt., re2 — 2 ay + y2 =
462 (4)
Extract root, a? — y = ± 2 b (5)
Add (5) and (1), 2x = a±2b.
Subtract (5) from (1),
2y=*a*2b.
In (1), *
Substitute in (2),
y + 1 y 13
y y + 1 6*
6y2 + 12y + 6 + 6y2 = 13y2+13y,
y2 + y = 6.
Complete the square,
4y2 + ( ) + 1 = 25,
2y + 1 = ±5.
.-. y = 2or-3.
,\ x = 3 or -2.
Subtract (2) from (1),
15.
^ + 9^ = 340 (1)
7zy-y2 = 171 (2)
x2 + 2zy + y2 = 169,
s + y = ±13 (3)
.-. a; = 13-y or-(13+y).
Substitute in (1) the first value of a,
(13 - y)2 + 9 (13 -y)y = 340,
169 - 26y + y2 + 117y - 9y2 = 340,
8y2-91y = 171.
Complete the square,
256y2-() + (91)2 = 2809,
16y - 91 - ± 53.
.-.y = 9or2f (4)
Substitute in (1) the second value of x,
(13 + y)2 - 9 (13 +y)y = 340,
169 + 26y + y2 - 117y - 9y2 = 340,
8y2 + 91y = -171.
Whence, y = - 9 or - 2|.
Substitute values of y in (3), x = ± 4 or ± lOf .
316
ALGEBRA.
16. s + y-6
*» + 2? = 72
Divide (2) by (1),
a^-sy+y'- 12
Sq.(l), a*+2sy+y* - 36
Subtract, -3sy - - 24
.-.ay = 8
Subtract (5) from (3),
x*-2ay + y* = 4.
Extract root, x — y = ± 2
Add (6) and (1), 2x = 8 or 4.
.\ x = 4 or 2.
Subtract (6) from (1),
2y = 4 or 8.
.\y «2 or 4.
18.
8
(5)
(6)
17. 3xy + 2z+y =
3x-2y-
In(2), ■-
Substitute value of s
6y2 + 7y -
Complete the square,
144y2 + ()+49 =
12y + 7 =
12y-
.-. y =
Substitute value of y
3 =
485
0
2*
3
in (1),
485,
1455.
(1)
(2)
(3)
34969,
±187.
180 or -194.
15 or - 16J.
in (3),
lOor-lOJ.
a?-y=-l
s»-y» = 19
Divide (2) by (1),
x*+ xy+y2 = 19
Sq. (1), x*-2xy+yi= 1
Subt., 3xy = 18
.\xy = 6
Add (4) and (3),
a:2 + 2ay + y* = 25.
Extract root, x + y = ± 5
Add (5) and (1), 2x = 6 or - 4.
.-. * = 3 or -2.
Subtract (1) from (5),
2y = 4 or — 6.
.-. y = 2or — 3.
19,
(2)
(3)
W
(5)
ar^ + y8'
x*-xy+y* =
Divide (1) by (2),
x + y=*
Square (3),
x* + 2xy -f y* =
Subtract (4) from (2),
-3a?y =
Divide by 3, —xy =
Add (5) and (2),
3*-2sy + y,=
Extract root, x—y =
Add (3) and (6), 2 x -
.\ x =
Subtract (6) from (3),
2y =
20.
Square (1),
Subtract (2),
Subtract (4) from (2),
Extract root,
From (1) and (6),
x + y =*a
x*+y' = 6*
x* + 2xy + y2 = a1
x% + y* = b*
2xy = a* - 6*
x2-2x.y + y2 = 262--q2
x — y = ±^/262 — a2
2x = a±*/2d2-a2.
.-.z = i(q±«/2ff-a?).
2y = ag:vr2&2 — a2.
.•.y = 4(a^v^262 — a2).
2728 (1)
124 (2)
22 (3)
484 (4)
-360.
-120 (5)
±2 (6)
24 or 20.
12 or 10.
'20 or 24.
. 10 or 12.
(1)
(2)
(3)
(5)
(6)
teachers' edition. 317
21.
a?-y2 = 0 (1)
3a?-4zy + 5y2 = 9 (2)
From (1), a? - y2.
Hence, in (2), 3a? ± 4a2 + 5a? = 9,
' 12a;2 or 4a? = 9.
.\ » = ± jV3 or ±§,
and y = ± } V3 or ± }.
X + V j x-y 10
(1)
x-y a; + y 3
x2 + y2 = 45
(2)
From (1), 3 (a?+2a#+y2)+3 (a?-2a?y +y2) = 10 a? - 10y2,
3a? + 6a^ + 3y2
+ 3a? - Gxy + 3y2 = 10a;2 - 10y2,
-4a? + 16y2 = 0,
-a? + 4y2 = 0
(3)
Add (2) and (3),
5y2 - 45.
.\ y = ±3.
n (2), j? + 9 = 45.
Substitute values of y i
.\ x = ±6.
23.
1 ! K
a; y
1 ( 1 ^17
a; + l y + l = 12
Clear of fractions and unite, x + y = 5a?y
(2)
5a; + 5y = 7 — I7xy
Divide (2) by (1),
5 = 7-17ay
5ay
25a^ = 7- 17 xy
(3)
W
42xy = 7,
sy = J
(5)
From (1),
a + y = f
(6)
Square (6),
Multiply (5) by 4,
a? + 2xy + y2 = f$
Subtract,
a? - 2xy + y2 = ^
a;-y = ±i
(7)
Add (7) and (6),
2a; =1 or J.
.-. a; = J or J.
2y - J or 1.
Subtract (7) from (6).
-•-J/-Jor|.
318 ALGEBRA.
8
24.
ar»-sy + y2 = 7 m
.T*+a^» + y* = 133 (2)
Divide (2) by (1), x* + xy + y2 = 19
Subtract (1 ) from (3), 2 zy = 12.
.\zy = 6 (4)
Add (4) to (3), and Bubtract (4) from (1),
x* + 2xy + y2 = 25,
a*-2asy +y*«l.
Whence as + y = ± 5,
a? — y = ± 1.
.•. jt = ±3 or ± 2,
and y = ± 2 or ± 3.
25.
* + y = 4 (1)
z* + y* = 82 (2)
Put u + v for a, and u — v for y .
(1) becomes 2u = 4.
.-. u = 2.
(2) becomes u* + 6uV + tr* = 41 (3)
Substitute 2 for u in (3), 16 + 24 v8 + «• = 41,
v* + 24^ = 25.
Complete the square, *>* + () + 144 = 169,
* + 12-±13,
* = 1 or - 25.
.\ v = ± 1 or ±V-25.
.-. a; = 3, 1, or 2±V^25,
and y = l, 3, or 2 tV- 25.
Divide (1) by (2),
Square (2),
Subtract,
Add (3) and (4),
Extract the root,
Subtract (2) from (5),
Add (2) and (5),
x* — y8 — a*
(2)
x-y^a
x* + xy + y2 = a2
(3)
x2 — 2ay + y2 = a2
3ay =0
<cy = 0
W
x2 + 2sy + y2 = a2.
x + y = ±a
(5)
2y = 0or-2a.
.«. y = 0 or — a.
2a; = 2a or 0.
,%x = aor 0.
TEACHERS EDITION.
319
27.
x2 — xy =
xy — yl =
Subtract (2) from (1), x2 - 2 xy + y% =
Extract root, % — y =
(l)is . x(x-y) =
Substitute value of x — y in (1), ± x[a — o) =
,\ x =
y(x-y) =
±y(a-b) =
.-. y =
28.
a5
(2) is
Substitute value of (a; — y) in (2),
* + &
2ab (2
a2-2ab + b2.
±(a-b).
a2 + 62.
. a2 + &>.
a — 6
2a&.
2a6.
. 2ab
x*-y*-
xy--
y-
(a2-b2)2
In (2),
Substitute value of y in (1), x2 -
x*-a* + 2a2b*"-b* =
x* — 4 afor2 =
Complete the square, «* — () + 4a2£>2 =
Extract root, a2 — 2a& =
z2 =
.\ x =
Substitute value of x in (1), [a + bf — y2 =
y2 =
.-. y-
■Aab
a2-b2
a2-b2
(2)
= 4a&,
4a&r2,
a*_2a2&2 + 64.
a* + 2a2&2 + 6*.
±(a2 + 62),
±(a2 + 2a& + 62).
- ± (a + 6).
4 aft,
a2-2a& + 62.
.±(a-6).
29. ajy =
s2+y2:
Multiply (1) by 2, 2a!y =
Add (3) and (2),
x2 + 2xy + y2 =
Extract root, a; + y =
Multiply (1) by 4, 4zy =
Subtract (6) from (4),
x2 — 2xy + y2 =
Extract root, x — y-
Add(5)and(7), 2x-
.\ x--
Subtract(7)from(5),
2y =
= 0
= 16
= 0
= 16
= ±4
= 0
(1)
(2)
(3)
(5)
(6)
30.
•y-
= 16
s±4 (7)
■ ± 8 or 0.
* ± 4 or 0.
= 0 or ± 8.
0 or ± 4.
a:2 + xy + y2 =
e4 -f £2y2 + y* =
Divide (2) by (1),
x2 - xy + y2 =
(1) is a2 + xy + y* =
Subt., -2xy =
.\ —xy =
Add (3) and (4),
x2 — 2xy + y2 =
Extract root, x—y =
Subtract (4) from (1),
x2 + 2zy + y2 =
Extract root, x + y =
Add (5) and (6), 2s =
Subt.(5)fr.(6),*2y =
.-. y =
1.
±1
37
481
13 (3)
37
-24
-12 (4)
(5)
49.
±7 (6)
± 8 or ± 6.
± 4 or ± 3.
± 6 or ± 8.
± 3 or ± 4
820 ALGEBRA.
31.
x* = ax + by (1)
y* = ay + bx (2)
If x = 0, y must equal 0.
If a: — y, and does not equal 0, then x=*a + b, and y = a + b.
If a; does not equal y , subtract (2) from (1), and divide by x - y.
x + y = a — b (3)
Add (1) and (2), x* + f = a(x + y) + b(x + y).
Substitute a- 6 for a; + y, = a(a - b) + 6(a - b).
Thatis, a8 + ya = «2 - fc2 (4)
Square (3), x* + 2sy + y» = a'-2a6 + 6* (5)
Subtract (4) from (5), 2sy - -2aZ> + 2&» (6)
Subtract (6) from (4), x*- 2xy + y» = a* +2a6-3&«.
Extract root, as - y = ± y/a% + 2.ab-$W (7)
Add (7) and (3), 2 jc = a - b ± Va* + 2 aft HSff.
.-. x - }(a~ & ±VaJ + 2a&-3P).
Subtract (7) from (3), 2y - a - 6 t Vaa + 2a6-3y.
.'. y = \{a-b*y/a* + 2ab-ZV).
32.
a._y_2 = 0 (1)
15(rf-y")-16xy (2)
Transpose (1), a>-y = 2 (3)
Divide (2) by (3), 15 (x + y) = 8 ay,
15a; + 15y-8;ry = 0 (4)
From(l), z = y+2.
Substitute value of x in (4),
15y + 30 + 15y - 8y* - 16y - 0,
8y*-14y = 30.
Complete the square, 64y2- ( ) + (7)* = 289,
8y-7 = ±17,
8y = 24or-10.
.-. y = 3or-lJ.
Substitute value of y in (1). .'. x = 5 or f.
teachers' edition.
321
33.
x + y jc-y_89
x — y x + y 40
(1)
6a; = 20y+9
(2)
Simplify (1),
9a*-169y2 = 0,
9z* = 169y2
(3)
Extract the root,
3s = ±13y,
3a?Tl3y = 0.
Multiply by 2,
ex*26y = 0
W
Transpose in (2),
6x-20y = 9
(5)
Subtract (4) from
(5),
6y = 9,
or -46y = 9.
.-. y = lJor-A.
Substitute values of y in (2), x = 6} or - 2^.
34.
a b
a)
a b .
- + - = 4
x y
(2)
Simplify (1),
bx + ay — ab
(3)
Simplify (2),
bx + ay = 4ay
.\ 4a?y = a&,
andy = -^-
y 4a;
(4)
Substitute value of y in
(3).
4a;
Simplify,
4a£ + a* = 4 ax.
Transpose,
4a? — 4aa? = — a2.
Complete the square,
4a»-() + a» = 0.
Extract the root,
2a; -a = 0.
a
.•. a; = --
2
Substitute value of x in
(3),
-i
322 ALGEBRA.
35.
s2 + y2 = 7 + xy (1)
a» + y»-6xy-l (2)
Transpose ay in (1), a2 — ay + y* = 7 (3)
Divide (2) by (3), <c + y - 6ay"""1-
Simplify, 7a? + 7y = 6sy - 1 (4)
Put u + v for re, and u — v for y, in (4),
7(u + t>) + 7(u-v) = 6(u*-T*)-l,
6m»-6^-14w = 1 (5)
Put u + 1> for a\ and w — v for y, in (3),
(u + vf -(&-<*) + (u-vf-7,
u* + 3v» - 7 (6)
Multiply (6) by 2, 2u» + 6t* = 14 (7)
Add (5) and (7), 8 m* - 14 u = 15.
Complete tbe square, 256 u» - ( ) 4- (14)* = 676.
Extract the root, 16 w - 14 = ± 26,
16w = 40or-12,
u = { or — }.
Substitute J for u in (6), ^ + 3 v* = 7,
3*-f,
+ -i-
Extract the root, v = ± }.
Substitute - f for u in (6), -ft + 3 v2 = 7,
Extract the root, v = ± J V1}*.
Since x = m + 1>, substitute J for ti and ± J for i>,
»-§ + (*».
x = 3 or 2.
Substitute value of — } for w, and ijV1}1 for v,
ar = l(-3±Vi^).
Since y = u — v, substitute f for u, and ± J for v}
y-i-(±».
y=2or 3.
Substitute — f for w, and ± jV^J* for v,
y — i-(±y/m,
y-*(-«TVSj»).
TEACHERS1 EDITION. 323
36.
a*-y5 = 3093
(1)
x-y = S
(2)
Let x = u + vt and y = u — v.
From (2), u + v — u + v=3,
2v = 3.
From (1), u5+5w4i;+10ttV+10tt2ys+5ttv4+t£
-(m5-5u*i;+10wV-10mV+5mv*-^) = 3093.
Transpose and combine, 10u*i>+20uV+2v* = 3093.
Substitute value oft,, 5^ + 5^3? + ?*5 _ 3098.
2 8 16
Simplify, 240u4 + 1080u2 = 49245.
Divide by 15, 16u* + 72u2 = 3283.
Complete the square, 16 w4 + ( ) + 81 = 3364.
Extract the root, 4 u2 + 9 = ± 58,
u2 - ^ or - 3£ .
.-. u = ±}or ± JV-67,
x = u + v - 5, -2, or £(3±V^67),
y - u - v = 2, -5, or J(-3± V^67).
37.
l(.-l)-*(* + l)(y-l)--ll (1)
i(y + 2) = J(* + 2) (2)
From (1), 9 a; - 9 - 10 xy - lOy + 10a; + 10 - - 165,
or 19a; - lOay - lOy - - 166 (3)
From (2), 4y + 8 = 3a> + 6.
3a;-2
4
Substitute value of y in (3),
19,-10 J§f^-io(5til)--ie6b
76a; - 30a? + 20a; - 30 s + 20 = - 664,
-303? + 66 a; = -684,
5a? -11a; =144.
Complete the square, 100a2 - ( ) + (ll)2 = 2401,
10a; -11 = ±49,
10a; = 60 or -38.
.-. g»6or-3f
Substitute values of x in (4), y = 4 or — 3^.
(4)
324 ALGEBRA.
38.
10a*+15*y = 3aft-2a»
(1)
10y* + 15xy = 3ab-2b*
(2)
Let
ux = y.
(1) becomes
10x*+ 15a*i = 3aft-2a*f
^ 3aft-2a*
10 + 15u
(3)
(2) becomes
lOxV + 15x*u = 3 aft - 26s,
10u»+15u
W
Eau&te values o
f^ 3aft-2a*_ 3aft-2ft»
n 10+15U 10u»+15u
Simplify,
30aftu* - 20a»u* - 30o«u + 30ft*u = 30 aft - 20 ft*.
Divide by 10,
3aftu* - 2a*u* - 3a*u + Situ = 3 aft - 26*,
or uVSaft^a'J-Sttfa'-ft^Saft^ft*.
Complete the square,
4m,(3o6 - 2a*f - ( ) + 9(a* - ft')1 = 9a*-24asft+34a*ft*-24afts+9&«
Extract the root,
2u(3oft - 2a*) - 3(a* - 5*) = ±(3a* -4aft + 3ft8),
2w(3aft - 2a8) - 6a* - 4aft or 4aft - 66*.
3a -2ft ft
3ft-2a a
Substitute value of3a~~2& for u in (3),
3ft -2a
Extract the root,
25
Substitute - - for w in (3), a1 = - -•
a 5
s = ±a
^1
Q. 3a-2ft v A /3ft-2a\
Since u* = y, y = ___ x ± ^_ __j.
3ft-2a
3a-2ft
.\ y = ±-
ory = --x(±aV=~|).
.•. y = =FftV-J.
teachers' edition. 325
Exercise XCV.
1. If the length and breadth of a rectangle were each in-
creased by 1, the area would be 48 ; if they were each diminished
by 1, the area would be 24. Find the length and breadth.
Let x = length of rectangle, r
and y = width of rectangle.
Then (a + l)(y + 1) = 48
(i)
and (a;-l)(y-l) = 24
(2)
Simplify (1), sy + a; + y + l=48
(3)
Simplify (2), xy-x-y + 1 = 24
Add, 2xy +2 = 72
xy = 35
(4)
Substitute value of xy in (3),
35 + x + y + 1 = 48,
a + y=12
(5)
Square (5), a2 + 2xy + y2 = 144
Subtract 4 X (4), 4xy = 140
x2 — 2xy + y2 = 4
Extract the root, x — y = ± 2
(6)
From (5) and (6), x = 7 or 5,
y = 5 or 7.
2. The sum of the squares of the two digits of a
number is
25, and the product of the digits is 12. Find the number.
Let x = digit in tens' place,
and y = digit in units' place.
x* + y8 = 25
a)
ay = 12
(2)
Multiply (2) by 2, 2xy = 24
(3)
Add (3) and (1), a:2 + 2sy + y* = 49.
Extract the root, x + y = ± 7
W
Subtract (3) from (1), a;2 - 2a;y + y2 = 1.
Extract the root, x — y = ± 1
(5)
From (4) and (5), 2 a; = ± 8 or ± 6.
.\ a? = ±4 or ±3,
y = ± 3 or ± 4.
Hence, the required number is 43 or 34.
326 ALGEBRA.
8
3. The sum, the product, and the difference of the squares of
two numbers are all equal. Find the numbers.
Let x + y = one number,
and x — y = the other number.
Then 2x = the sum of the numbers,
a* — y* — the product of the numbers,
and 4xy»= the difference of the squares.
2x = a? - y2
z*-y* = 4ay
Transpose in (1), xt — 2x — y2 = 0
Transpose in (2), a* — 4xy — y2 = 0
Subtract, 2x — 4xy =0
l-2y = 0,
2y = l.
Substitute value of y in (1), 2x = x% — \1
x*-2x = \.
Complete the square, x* — 2 a; + 1 = $.
Extract the root, x — 1 = ± J V5.
.-. «=l±iV5,
a; + y - f ± £ V5 or J (3 ±V5),
*-y-J±jV5or }(1±VB).
4. The difference of two numbers is f of the greater, and the
sum of their squares is 356. What are the numbers?
Let x = greater number,
y «- lesser number,
and x — y = difference of the numbers.
Then *-y = T W
and sc'+y'-SSe (2)
Simplify (1), 8*-8y = 3*.
.•.«-2* (3)
5
Substitute value of x in (2), ^1 + y* = 356.
Simplify, 64y2 + 25y2 = 8900,
89y2 = 8900,
y2 = 100.
Extract the root, y = ± 10.
Substitute value of y in (3), bx = ± 80.
.-. a; = ±16.
I
i
teachers' edition. 327
5. The numerator and denominator of one fraction are each
greater by 1 than those of another, and the sum of the two frac-
tions is 1^ ; if the numerators were interchanged the sum of the
fractions would be 1J. Find the fractions.
Let - = one fraction,
y
r 4- 1
and = the other fraction.
y + 1
Then - + ^ = T? «
y y + 1 12
and £±i + _^. = § (2)
y y + i 2
Simplify (1), I2xy+\2x+\2xy+\2y = 17y2 + 17y.
Simplify (2), 2xy+2y+2x+2+2xy = 3y2 + 3y.
Transpose and combine,
_17y2 _ 5y + 24ary + 12a; = 0 (3)
- 3y2 - y + 4a;y + 2a; = - 2 (4)
Multiply (4) by 6,
- 18y2 - 6y + 24sy + 12a - - 12 (5)
Subtract (5) from (3), y2 + y - 12 (6)
4y2 + ()+l = 49,
2y + 1 = ± 7,
2y = 6or-8.
.*. y = 3 or —4.
Substitute 3 for y in (1), ~ + £±i - i~
Simplify, , 4a; + 3a; + 3 = 17,
7a; = 14.
.-. a; = 2.
Hence, the fractions are § and J.
6. A man starts from the foot of a mountain to walk to its
summit. His rate of walking during the second half of the dis-
tance is £ mile per hour less than his rate during the first half,
and he reaches the summit in 5} hours. He descends in 3|
hours, by walking 1 mile more per hour than during the first
half of the ascent. Find the distance to the top and the rates of
walking.
328 ALGEBRA.
Let 2x = distance,
and y = rate at first.
Then - = number of hours he was walking 1st half,
y
and — — = number of hours he was walking 2d half.
y-i
Hence, £ + _£_ = 5}. (1)
y y-i
Also, -^r=3} (2)
y + i
Clear (1) of fractions, 4 xy — 2 x + 4 xy = 22 y* — 11 y,
22y*-8ay+2a?-lly = 0 (3)
Clear (2) of fractions, 8 x = 15y + 15.
. r - lgy + 15 m
Substitute value of a; in (3), " " x g W
22y*_8y(^^y2(l^)-lly = 0,
176y» - 120ys - 120y + 30y + 30 - 88y = 0,
56y*-178y = -30.
Complete the square, 3136y8 - ( ) + (89)» = 6241.
Extract the root, 56y - 89 - ± 79.
.-.y-s.
Substitute value of y in (2), ^ = — .
4 4
2a: = 15.
Hence, the distance is 15 miles ; and the rates of walking, 3,
2}, and 4 miles.
7. The sum of two numbers which are formed by the same
two digits in reverse order is $f of their difference ; and the dif-
ference of the squares of the numbers is 3960. Determine the
numbers.
Let x = digit in ten's place,
and y = digit in unit's place.
Then 10 a; + y = first number,
10 y -f x = second number,
llaj-flly^ sum of the numbers,
9x — 9y = difference of the numbers.
(10a: + y)2 - (x -f 10y)3 = difference of the squares.
.*. lla: + lly = ft(9a;-9y) (1)
teachers' edition. 329
and (10a + yf - {x + 10y)2 = 3960 (2)
Simplify (1), s+y = -5a;~5y,
7y-3s = 0 (3)
Substitute value of x in (2),
# «! = 3960,
y* = 9.
.-. y = ±3.
From (3), 3.u = 7y.
.■. x = ±7.
Hence, the numbers are 73 and 37.
8. The hypotenuse of a right triangle is 20, and the area of
the triangle is 96. Determine the sides.
Let x = longer side,
and y = shorter side.
Since sum of squares on sides equals square on hypotenuse,
x2+ya = 400 (1)
Since area of triangle equals one-half product of sides,
^ - 96 (2)
xy = 192.
Multiply (2) by 2, 2xy - 384 (3)
Add (1) and (3), x2 + 2xy + y* = 784.
Extract the root, x + y — ± 28 (4)
Subtract (3) from (1), x2 - 2xy + y2 = 16.
Extract the root, x — y = ± 4 (5)
From (5) and (4), 2x - ± 32 or ± 24.
.-. a; = ±16 or ±12.
2y = ±24or ±32.
.-. y = ±12or ±16.
Hence, the sides are 16 and 12.
330 ALGEBRA.
9. Two boys run in opposite directions round a rectangular
field the area of which is an acre ; they start from one corner
and meet 13 yards from the opposite corner ; and the rate of one
is $ of the rate of the other. Determine the dimensions of the
field.
Let x =■ length of first side,
and y — length of second side.
x + y + 13 = number of yards one boy runs,
x + y — 13 «= number of yards the other boy runs.
s + y-13 = $(a + y + 13).
.\ 6x + 6y - 78 = 5x 4- 5y + 65, •
and x + y = 143 (1)
xy = area of field of one acre.
(Since 4840 Bq. yds. = 1 acre),
xy - 4840 (2)
Square (1), x* + 2xy + y* = 20449
(2) x 4 is 4ay 19360
a*-2xy+y*= 1089
x - y = ± 33 (3)
From (1) and (3), 2a? « 176 or 110.
.-. a; = 88 or 55.
2y=110or 176.
.-. y = 55or 88.
Hence, the dimensions are 88 yds. by 55 yds.
10. A, in running a race with B, to a post and back, met him
10 yards from the post. To make it a dead heat, B must have
increased his rate from this point 41 f yards per minute ; and if,
without changing his pace, he had turned back on meeting A, he
would have come 4 seconds after him. How far was it to the
post?
Let x — number of yards to the post.
Then 2x = number of yards to the post and back.
Let y = number of yards A runs per minute.
Then — = number of minutes A is running the race.
y
B runs (x — 10) yards while A is running (a; + 10) yards.
Hence, B runs ^-^ — of v yards ** %2Lz — U yards per minute.
x + 10 * J aj + 10' *
teachers' edition. 331
A has (a; — 10) yards to run when B meets him ; and, as he runs
y yards per minute, it will take him ?-Z — minutes to finish the
race.
B has (x + 10) yards to run; and, if he increases his pace 41f
yds. per min., he will be running at the rate of I **& ~ & + 41f J
yards per minute ; and, as he has (x + 10) yards to run, it will
take him (x + 10) -s- ( xV~l V + 41* J minutes to finish the race.
\ x + 10 J
But this change of rate will make it a dead heat ; therefore,
(*+M^+41*)=*-f (1)
Since 4 seconds = ^ minute, B, without changing his rate,
will be -j^ of a minute longer than A in running the (jc— 10)
yards which A has to run when he meets B ; therefore,
(g-io) + fo-10yV*-10-JL (2)
V ' \ x + 10 ) y 15 K)
a- vr /o\ a;+10 2-10 1 /oX
Simplify (2), - — (3)
y y lb
.-.y = 300.
Simplify (1),
( iqn /7y(»-10)+290ap + 2900\ ac-10
{X+ >'{ 7(» + 10) ) y '
7 (a; + 10)' a- 10
7y(a;-10) + 290a; + 2900 = y
Substitute 300 for y, ^* + ™Z-*=™
y 2390a; -18100 300
210a? + 4200a; + 21000 - 239 x«- 4200 x 4- 18100,
29 a?- 8400 a; = 2900,
^_ 8400a =1(X)
29
*-( ) + (*«*)■- (M**)1.
.-. x = 290 or -£$.
Hence, the distance to the post was 290 yards.
332 ALGEBRA.
11. The fore wheel of a carnage turns in a mile 132 times
more than the hind wheel ; but if the circumferences were each
increased by 2 feet it would turn only 88 times more. Find the
circumference of each.
Let x = circumference in feet of the fore wheel,
and y — circumference in feet of the hind wheel.
Then 5*2-5*2_im (1)
x y
52§P_5280 = 88 (2)
x + 2 y + 2 w
Simplify (1), 5280y - 5280a; = 132xy.
Divide by 132, 40y - 40a: = xy (3)
Simplify (2), 5280y+10560-5280a;-10560 - 88xy+176a;+176y+352.
Divide by 88, 60y - 60a: = xy + 2a: + 2y +4.
Transpose and combine, 58 y — 62 a: = xy + 4 (4)
(3) is 40y-40x = zy
Subtract, 18y-22x= 4
2 + 11*
y = -
9
Substitute value of y in (3),
Simplify, 80 + 440a; - 360a; = 2a: + 11 a*
11a? -78 a? =80.
Multiply by 11, 121 x* - 858 x - 880.
Complete the square, 121 x* - ( ) + (39)* = 2401.
Extract the root, 1 1 x - 39 = ± 49,
11a; = 88 or -10.
.-. a; = 8or -|f
Substitute 8 for x in (3). . \ y = 10.
12. A person has $ 6500, which he divides into two parts and
loans at different rates of interest, so that the two parts produce
equal returns. If trie first part had been loaned at the second
rate of interest, it would have produced $ 180 ; and if the second
part had been loaned at the first rate of interest, it would have
produced $ 245. Find the rates of interest.
teachers' edition. 333
Let x = number of dollars in one part of the capital,
6500 — z = number of dollars in the other part,
and y = return from each part.
y
Then - = rate of interest on first part.
Also. „,.j[ — = rate of interest on second part,
6500 — x r
x ( 3| J =« return of first part when loaned at second rate.
•"(ssfc ,) = 180 (1>
y
(6500 — x)- = return of second part when loaned at first rate.
.\ (6500-.x)| = 245 (2)
Simplify both equations and add,
xy = 1170000 -180a? (3)
-ay = + 245a;-6500y (4)
0 = 1170000+ 65a;-6500y
Transpose and divide by 65,
100y-a = 18000 (5)
1170000-180o:
From (3) y =
Substitute value of y in (5),
100 (UTOOOO-IM^., 18(X)0
Simplify, x1 + 36000*, •= 117000000,
x* + ( ) + (18000)2 - 441000000.
Extract the root, x + 18000 = ± 21000.
.-. x = 3000,
and 6500 - x = 3500.
From (5), y - 210.
•••1-0.07.
and65bfc = 0.06.
Hence, the rates of interest are 7% and 6%.
334
ALGEBRA.
Exercise XCVI.
1. 2*+lly = 49.
Transpose, 2 x = 49 — 1 1 y .
.-. a:-24-5y + i^.
Let *-^ = m,
2
l-y = 2m.
.-. y = l — 2m.
Substitute value of y in original
equation,
2* + 11 -22m = 49.
.-. x =19 + 11m.
If m = 0, x - 19, y - 1.
If m = -l, 3 = 8, y = 3.
2. 7s + 3y = 40.
Transpose, 3y = 40 — 7 a;.
:13-2jb +
Let
Lz£.
3
1-x*
1-x
-3 m.
,\ x= 1 — 3m.
Substitute value of * in original
equation,
7-21m + 3y = 40,
3y = 21m + 33.
.*. y = 7m + 11.
If m = 0, y = ll,a;=l.
If m = — 1, y = 4, x = 4.
3. 5s + 7y = 53.
Transpose, 5x — 53 — 7y.
...,«io-y+l^a!
*_10 + y = 3^.
5
Multiply by 3,
3x-30 + 3y = 2^^
= l-y+izX
Let
i^a
<m,
4 — y = 5m.
.\ y = 4 — 5m.
From given equation,
x = 5 + 7m.
If m = 0, a; = 5, y = 4.
4. a; + lOy = 29.
Transpose, x = 29 — lOy.
If y = l,a; = 19.
If y = 2, a: = 9.
If y = 3,a; = -l.
. \ y can only = 1 or 2,
x can only = 19 or 9.
TEACHERS EDITION.
335
5. 3*+8y = 61.
3* = 61-8y.
.-. * = 20-2y+*^.
.3
*_20 + 2y = 1-^i&
* 3
Multiply by 2,
2*-40 + 4y = ^i#
y 3
._, + Sjl.
Let ^ = m,
3
2-y = 3m.
.-. y = 2-3m.
Substitute in original equation,
3s + 16 -24m = 61,
3a = 45 + 24 m.
.\ a; =15 + 8m?
If ?n = 0, *=15, y = 2.
If m = — 1, *=7, y = 5.
7: 16* + 7y = 110.
7y = 110-16*.
.•.y«15-2* + ?=??.
Transpose,
y + 2*-15 =
Multiply by 4,
5-2*
4y + 8*-60 =
20-8*
Let
6-
=2-*+^
= m,
6 — * = 7m.
.\ * = 6 — 7m.
Substitute in original equation,
96~112m + 7y = 110,
7y = 14 + 112m.
.\y = 2 + 16?n.
If m = 0, x = 6, y = 2.
8* + 5y = 97.
5y = 97-8*.
.-. y = 19— * +
2-2*
2-3*
y-19 + * =
Multiply by 2,
2y-38 + 2*:
4-6*
= -* +
4-*
Let
4-*
.*. * = 4 — 5m.
Substitute in original equation,
32-40m + 5y = 97,
5y = 65 + 40m.
.\y = 13 + 8m.
If m = 0, * = 4, y = 13.
If m = - 1, * = 9, y = 5.
8. 7* + 10y = 206.
7* = 206-10y.
= 29-
y+
3-3y
Let
*_29+y = ^l_^).
l^ = m.
7
.-. y = 1- 7m.
Substitute in original equation,
7* + 10 -70m = 206,
7* =196 + 70m.
.-. * = 28 + 10m.
If m ^0, * = 28, y = 1.
If ?n = -l, *=18, y = 8.
If m = - 2, * = 8, y - 15.
336
ALGEBRA.
9. 12x-7y-l.
Transpose, 7y = 12x-l
Multiply by 3,
5s -1
11. 23y-13a;=3.
Transpose, 13 a; = 23 y - 3.
10y-3
.*. X — V = *-
3y-3a; = 2a; + ^— -•
Let 2L=J* = m.
.-. x -7m + 3.
Multiply by 4,
13
Let
4a: — 4y =«3y + *
V-12 _
-12
13
13
.-.y- 13m + 12.
Substitute this value of x in origi- Substitute this value of y in orig-
nal equation, inal equation,
84m + 36 - 7y - 1, 23(13m + 12) -13*- 3,
7y - 35 + 84m. 13a. = 299m + 273.
.-. y « 5 + 12m. ^ x _ 23m + 21.
If
m = 0, x = 3, y = 5. If *»_<), *-21, y-12.
10. 5a>-17y = 23.
5x = 23 + 17y.
,-. s = 4 + 3y+'
.3±&
3 + 2S
5
a._4_3y =
Multiply by 3,
3*-12-9y = l+y+^-
Let ±P*=rn.
o
Then y = 5m-4.
Substitute this value of y in origi-
nal equation,
5a;-17(5m-4) = 23,
5a;-85?» + 68 = 23,
5a; = 85m -45.
.*. x«= 17m — 9.
I! «i-lf *-8, y=l.
12. 23s-9y = 929.
9y = 23a;-929.
.-.y = 2x-103+^.
y-2* + 103 = ^p.
Multiply by 2,
2y-4a; + 206 = a;+^—
Let = m.
9
Then x-4 = 9m.
,\ a — 9m + 4.
Substitute this value of y in orig
inal equation,
207m+92-9y - 929,
9y = 207m -837.
.-. y = 23m-93.
Jf m = 5, x-49, y-22.
teachers' edition. 337
13.
23 a; -332/ = 43.
23a: = 33y +43.
y 23
Let £-±_2 = m.
23
Then y = 23m-2.
Substitute this value of y in original equation,
23*-33(23m-2) = 43,
23 a; -759m + 66 -43,
23jc = 759m-23.
.-. a;=33m-l.
If m--lf a = 32, y = 21.
14.
555a:-22y = 73.
22y = 555s-73.
...y = 25s-3+^l2.
9 22
Transpose. y - 25 a: + 3 = 5a?~^
Multiply by 9, 9y-225<c + 27 = 2a; + 2 + ^=i?.
Let *=l?-m.
22
Then * -19 = 22m.
. .\a;=19 + 22m.
Substitute value of x in original equation,
555(19 + 22m) - 22y = 73,
10545 + 12210m - 22y = 73,
22y = 10472 + 12210m.
.\ y=» 476 + 555m.
If m-0, x = 19, y = 476.
338 ALGEBRA.
15. How many fractions are there with denominators 12 and
18 whose sum is {§?
Let *. + .£_*
12 18 36
Simplify, 3a: + 2y = 25,
2y = 25-3s.
Let — — — = m.
2
Then l-ar = 2m.
.-. x=l-2m.
Substitute value of x in original equation,
3-6m + 2y = 25.
.\y = ll + 3m.
If to = 0, x = 1, y =- 11.
If m = -l, x = 3, y = 8.
If to ="- 2, a; = 5, y = 5.
If to = -3, a?=7, y = 2.
Hence, the pairs of fractions are
AH; A. A; A, A; A A
16. What is the least number which, when divided by 3 and
*. leaves remainders 2 and 3 respectively?
Let n = number,
^-2-* (i)
n-=^-y (2)
From (1) and (2), n = Sx + 2 and 5y + 3.
.-. 3z + 2 = 5y + 3,
3* = 5y + l (3)
Transpose, x — 1
y-f 2
Multiply by 2, 2a; - 2 = y +
Let ^ = m- ^
Then y = 3m-2.
From (3), 3a -15 m -9.
,\ a; = 5m-3.
If * m-1, ar = 2, y=L
But n = 3a; + 2.
.•. n-8.
teachers' editiok. 339
17. A person counting a basket of eggs, which he knows are
between 50 and 60, finds that when he counts them 3 at a time
there are 2 over ; but when he counts them 5 at a time there
are 4 over. How many are there in all?
• . " t * n — 2
Let = x.
3
and — -y.
Then n = 2 4- 3a: or 4 + 5y.
.*. 2 + 3a; = 4 + 5y,
3aj = 2 + 5y (1)
* -* + *G±A
Let i-^-2 = m.
3
Then y-3m-l.
Substitute value of y in (1), 3 a; = 2 + 5 (3 m — 1),
3a; = 15m- 3.
.*. a; = 5m — 1.
If m = 4, a; = 19, y-11.
Hence, the number of eggs is 59.
18. A person bought 40 animals, consisting of pigs, geese,
and chickens, for $40. The pigs cost #5 apiece, the geese $1,
and the chickens 25 cents each. Find the number he bought of
each.
Let x = number of pigs,
and y = number of geese.
Then 40 — x — y = number of chickens.
5* + y + 10*- 2f-40 (1)
4 4
or 20a; + 4y + 40 - x -y = 160,
or 19a; + 3y = 120,
3y = 120-19a; (2)
y = 40-6s-5.
y 3
Let - =» m.
3
.*. ic = 3m.
Substitute value of x in (2), 3y = 120 — 57 m.
.-. y = 40-19m.
If m = 1, x = 3, y = 21.
If m = 2, x = 6, y = 2. .
Hence, he bought 3 pigs, 21 geese, and 16 chickens ; or 6 pigs,
2 geese, and 32 chickens.
340 ALGEBRA.
19. Find the least multiple of 7 which, when divided by 2, 3,
4, 5, 6, leaves in each case 1 for a remainder.
Let 7x = least multiple of 7,
and y = sum of quotients.
Then
7x-l , 7s-l 7x-l , 7x-l . 7a; -1
2~+~3~ + ~lT+~5~+_6~=y-
Simplify,
210* - 30 + 140* - 20 + 105a: - 15
+ 84* - 12 + 70a: - 10 = 60y,
609x-60y = 87.
(1)
Divide by 3,
Transpose,
203s-20y = 29
-20y = -203x + 29.
.-.y-10* l+3*~9.
20
y 20
Let
20
x- 3 = 20m.
.-. x = 20m + 3.
n(l),
+ 609-20y = 29,
Then
Substitute value of x i
4060971
20y = - 4060m -580 (2)
.\y = 203m + 29 (3)
If m = 2, x = 43, y = 435.
Hence, the number is 301.
20. In how many ways may 100 be divided into two parts, one
of which shall be a multiple of 7 and the other of 9?
Let 7ar = one part,
and 9y = the other part.
.-. 7x + 9y=100.
7x = 100-9y.
■-.«-14-y + 2(1-y).
Let "~y = m.
7
Then 1— y = 7m.
.-. y = l -7m.
Substitute value of y in the original equation,
7a: + 9(1 -7m) = 100,
7a: = 100 -9(1 -7m),
7a;=91+63m.
.-. x =13 + 9m.
If m = 0, s=13, y = l.
If m« — 1, a: = 4, y = 8.
Hence, the parts are 91 and 9, or 28 and 72.
teachers' edition. 341
21. Solve 18 a;
-5y
~ 70 so
that y may be a multiple of x, and
both positive.
18a
- by - 70.
Let
y = mx.
Substitute value
of y in 1
this equation,
18s-
-5ma; = 70.
ar(18-
-5m) = 70.
18-5m
and
^70m_.
y 18 -5m
Now, if m =
■2,
rr = ^ or 8},
and
y«l^orl7J.
And, if m =-
'3,
*- V or23i.
and
y - *f* or 70.
22. Solve 8ac+12y = 23so that ac and y may be positive, and
their sum an integer.
8a; + 12y = 23 (1)
Let x + y = m.
Transpose, at = m — y (2)
Substitute value of x in (1),
8m-8y + 12y-23,
4y = 23-8m.
. 23-8m
••y 4
Substitute value of y in (1),
8a; + 69 -24m = 23,
8a; = 24m -46.
24m -46
..... — •
Let m - 2.
48-46 1
Then a; =
and y =-
8 4
23-16 7
342 ALGEBRA.
23. Divide 70 iuto three parts which shall give integral quo-
tients when divided by 6, 7, 8, respectively, and the sum of the
quotients shall be 10.
Let x = first part,
y = second part,
and 70 — x — y = third part.
6 7 8 W
Simplify,
24a: + 24y + 1470 - 21 x - 21y = 1680,
7* + 3y = 210 (2)
3y = 210-7a;.
.\ y = 70-2z---
y 3
Let § = w- *
3
.*. x = 3m.
Substitute value of m in (2),
21m + 3y = 210,
3y = 210-21m.
.\y = 70-7m.
If m = 2, 4, 6, 8,
(the lowest values that will produce multiples of the numbers),
a= 6, 12, 18, 24,
y - 56, 42, 28, 14,
70-a;-y = 8, 16, 24, 32.
24. Divide 200 into three parts which shall give integral quo-
tients when divided by 6, 7, 11, respectively, and the sum of the
quotients shall be 20.
Let
a; =
<= first part,
and
y =
= second part.
Then
200-a:-y =
= third part,
E + 3U
200-ar-.y_
= 20.
5 7
11
Simplify,
77 a: + 55y
+ 7000
-35x — 35y =
= 7700,
42a; + 20y =
= 700,
21a; + 10y =
= 350
.-. y =
= 35_2a:-A.
10
(1)
Let — = m,
10
a: = 10m.
Substitute value of a; in (1), y = 35 — 21 m.
If m = 1, x = 10, y = 14.
200-a:-y=176.
teachers' edition. 343
25. A number consisting of three digits, of which the middle
one is 4, has the digits in the units' and hundreds' places inter-
changed by adding 792. i1ind the number.
Let x = digit in hundreds' place,
and y = digit in units' place.
.•. 100 x -f 40 + y — the number.
lOOy + 40 + x = 792 + 100a; + 40 + y.
Transpose and combine,
99y-99a:=792.
Divide by 99, y - x = 8 (1)
y = x + 8.
Let x + 8 = m,
x = m-S (2)
and y = m.
From (2), m must be equal to 9, in order to make x positive.
Then x = 1,
Hence, the number is 149.
26. Some men earning each $2.50 a day, and some women
earning each $1.76 a day, receive all together for their daily
wages $44.75. Determine the number of men and the number
of women.
Let x = number of men,
and y = number of women.
Then 5£ + l2 = I7§,
2 4 4
10a+7y-179,
y = 25-x + l=l£
4 _ Q.B
Transpose, y — 25 + x = - — — •
Multiply by 5, 5y- 125 + 5z = 2- 2a? + 6~x
Let 6 —
7
x
- = m
7
x = 6 — 7m.
Substitute 6- 7 m for x in the original equation,
60-70m + 7y = 179,
7y = 119+70m.
.-. y= 17 + 10m.
If w = 0, x = 6, y = 17.
If m = - 1, x = 13, y = 7.
344 ALGEBRA.
27. A wishes to pay B a debt of £1 12 s., but has only half-
crowns in his pocket, while B has only four-penny pieces. How
may they settle the matter most simply?
Let x = number of half-crowns,
and y = number of four-penny pieces.
Then half-crowns = 30 x pence,
and four-penny pieces = 4y pence.
£1 + \2b. = 384 pence.
But £1 + 12«. = 30a: - 4y.
.-.30x-4y = 384 (1)
4y = 30a:-384.
30 s -384
••*- — i —
ory = 7a;-»96-|-|.
Let - = m.
2
Then x = 2m.
Substitute value of a? in (1),
60m-4y = 384.
.\ y = 15m-96.
If m - 7, x = 14, y = 9.
Hence, A can give B 14 half-crowns, and receive from B
9 four-penny pieces.
29. A farmer buys oxen, sheep, and hens. The whole num-
ber bought is 100, and the whole price £100. If the oxen cost
£5, the sheep £1, and the hens 1 s. each, how many of each did
he buy?
Let x — number of oxen,
and y = number of sheep.
Then 100 — x — y = number of hens.
5* + y + 100~20~'V = 100 (1)
100a: + 20y + 100 - x - y = 2000,
99a; +19y = 1900 (2)
Transpose, 19 y = 1900 -99 a;.
Divide by 19, y = 100 - 5 x - — •
9 19
teachers' edition. 345
Transpose, 100 — 5 x — y = — •
Multiply by 5,
600-25 x-5y = x+~
* 19
Let — = m.
19
Then a;=19m.
Substitute value of x in (2),
1881m + 19y = 1900.
Transpose, 19 y = 1900 - 1881 m,
y = 100 -99m.
If m = l, 3=19, y=l,
and 100-x-y = 80.
Hence, he buys 19 oxen, 1 sheep, and 80 hens.
30. A number of lengths 3 feet, 5 feet, and 8 feet are cut ;
how may 48 of them be taken so as to measure 175 feet all
together?
Let x = number 8 feet long,
y — number 5 feet long,
and 48 — x — y = number 3 feet long.
%x + by
+ 3(48 -a- y) = 175 (1)
Simplify,
5a + 2y = 31.
Transpose,
2y = 31-5a; (2)
.•.y-16-2» + !=5.
Let
l-s_TO
2
Then
1 — x = 2m.
.•. x— 1 — 2m.
Substitute value of x in (2), 2y = 31 — 5 + 10m.
y = 13 + 5m (3)
If
m = 0, -1, -2,
x =
» 1, 3, 5 — number of 8-ft. lengths,
y-
= 13, 8, 3 = number of 5-ft. lengths,
48-a-
-y =
- 34, 37, 40 = number of 3-ft. lengths.
346 ALGEBRA.
31. A field containing an integral number of acres less than
10 is divided into 8 lots of one size, and 7 of 4 times that size ;
and has also a road passing through it containing 1300 square
yards. Find the size of the lots in square yards.
Let x = number of acres.
.•. 4840 x = number of square yards.
y = number of square yards in 1 lot,
8y = number of square yards in 8 lots,
28 y = number of square yards in 7 lots of
second kind.
Sy + 28y + 1300 = 4840a; (1)
9y + 325 = 1210*.
9y = 1210a; - 325 (2)
y = 134a?-36 + 4z~1.
y-134x + 36 = i^— i-
* 9
Multiply by 7,
7y - 938 a; + 252 = 3x + ^?.
9 9
Let ^? = m.
9
Then x- 7 = 9m.
.*. a: = 9m + 7.
Substitute value of x in (2),
9y = 10890m + 8470 - 325,
9y - 10890m + 8145.
.-. y = 1210m + 905.
If m = 0,
x = 7 = number of acres.
y = 905 = number of sq. yds. in 1st lot.
4y = 3620 = number of sq. yds. in 2d lot
32. Two wheels are to be made, the circumference of one of
which is to be a multiple of the other. What circumferences
may be taken so that when the first has gone round three times
and the other five, the difference in the length of rope coiled on
them may be 17 feet?
teachers' edition. 347
Let x = circumference of the first wheel,
and y = circumference of the second wheel.
Zx — by = 17, difference of length of rope coiled on them
when first wheel goes round three times, and
second five times.
Let x = my,
3a:-5y = 17,
3 my — by = 17,
y(3m-5) = 17.
y 3m-5
If m = 2, y = 17, s = 34.
33. In how many ways can a person pay a sum of £ 15 in
half-crowns, shillings, and sixpences, so that the number of
shillings and sixpences together shall be equal to the number of
half-crowns?
Let x = number of shillings,
and y — number of sixpences.
Then x + y = number of half-crowns,
y sixpences = J y shillings.
(x + y) half-crowns = J(a? + y) shillings.
Then a; + £y + J (x + y) = whole number of shillings.
But 300 = whole number of shillings.
.*. z + iy + i(a; + y)=300.
2x + y -f 5a + 5y = 600,
7a? + 6y = 600.
a; = 85 + *•
7
Transpose, and multiply by 6,
6z~510 = 4-5y + ^=^.
7
Let —^ «= m.
7
.\ y = 2 — 7m.
Substitute value of y,
7x+12-42m = 600.
.•. a;=6m + 84.
Ifm = 0, -2,-2,-3,-4, -5, -6,-7, -8,-9, -10,-11,
- 12, — 13, then x and y would each have 14 positive values.
Hence, there are 14 ways.
348 ALGEBRA.
Exercise XCVII.
1.
a* + Zb*iB>2b(a + b).
If
a2 + 3o2is>2ao + 2&2,
if (transposing),
a* + fi,is>2a&.
But
a* + b*is>2ab.
.\ a2 + 3o2is>2i(<M b).
§249.
a»o + aft* is > 2 a2&*.
If (dividing both sides by ab),
a2 + o2is>2ao.
But a* + o*is>2ao. §249.
.-. a86 + a6»is>2aW
3.
(a* + 6*)(a* + 64)is>(a8 + &8)*.
If (simplifying),
a« + a4^ + a«j4+ J« i8 >a« + 2a86s + 6«
if (transposing), a46* + o*64 is > 2a868,
if (dividing by a262), a2 + 6* is > 2a6.
But a2 + &2is>2a&. §249.
.-. (a2 + 6*)(a* + b*) is >(a8 + 6s)8.
4.
o*6 + a2c 4- a&2 + o*c + ac2 + ic2 is > 6aoc.
a(6* + c2) + 6(a2 + c2) + c(a* + i2) is > Qabc.
Since (62 + c2) is > 2 oc, §249.
.-. a^ + c2) is >2abc.
Since (a* + c2) is > 2ac,
.-. 6(a* + c2)is>2a6c.
Since (a2 + 62) is >2ab,
.\ c(as + 62)is>2a6c.
Therefore (by adding),
a(b* + c2) + 6(<z2 + c2) + c(o2 + &2) is > 6abc.
teachers' edition. 349
5. The sum of any fraction and its reciprocal is > 2.
Let - = the fraction.
6
Then - = the reciprocal,
and ? + ^is>2,
6 a
if (multiplying by a6),a2 + 62 is > 2ab.
But a2 + 62is>2a6. J 249.
.. « + *i8>2.
b a
If 32=a2+62, and y2=c*+cP, xy is not less than oc+6o*, or ad+bc.
Now, if xy equals or is > ac + bd,
then x*y* equals or is > (ac + 5c?)2,
and (by substituting the values of a3 and y2),
(a2 + ft2) (c2 + d2) equals or is > (ac + 6cJ)2,
or (simplifying),
o2^ + a2^2 + c2^2 + o2^2 equals or is > aV + 2a6cd + 62d2,
and a2cP + 62c2 equals or is > 2a6co*.
But a*& + 62c2 equals or is > 2abcd. J 249.
.\ xy equals or is > ac + 6a\
7.
o6 + ac + 6c is < (a+6-c)2+(a+c-6)2+(6+c— a)*,
if (by expanding and combining),
oft + ac + 5c is <3a2+362+3c2-2a6-2ac-26c,
if 3a6 + 3ac + 36c is <3 a2 + 362 + 3c2,
if a6 + ac + be is < a2 + 62 + c2.
But 2a6 is< a2 + 62
2ac is< a2 + c2
26c is < ¥ 4- c2
2a6 + 2ac + 26c is < 2a2 + 26s + 2c2
a6 + ac + 6c is < a2 + 62 + c2.
.•. a6 + ac + 6c is < (a+6— c)2+(a+c-6)2+(6+c-a)2.
350 ALGEBRA.
8. Which is the greater,
(a8 + 6»)(c* + d8) or (ac + bdyi
Simplify, aV + 6*c* + a*d* + ^d8 is > aV + 2a6cd + 68d8.
If W + (W» is > 2 a&cd
But 68c8 + a«d» is > 2 aicrf, J 249.
.-. (a8 + &)(<? + d8) is > (ac + 6d)».
9. Which is the greater,
m8 + m or m8 + 1 ?
m8 + m is > or < wi8 + 1,
b m(m + 1) is > or <(m8 — m + l)(m + 1),
b m > or < m8 — to + 1,
b 2m is > or < m8 + 1.
But m8 + 1 is > 2m,
• \ to8 + 1 is > m8 + to.
10. Which is the greater,
a* — b* or 4a8(a — 6), when a is > 6?
4 (^(a — 6) is > or < a4 — b\
as (dividing by a — 6), 4 a8 is > or < a8 + a86 4- a&8 + ft8,
as (subtracting a8 from both sides),
$a*\B>or<a2b + ab* + V,
as (transposing a*6), 3 a8 — a86 is > or < a&2 + 6s,
as a8(3a - b) is > or < 68(a + 6),
if the factor a8 be taken out from the -left side, and the factor 6* from
the right side, since a is > bt the left side will have been divided by
a greater number than the right ; so that, if the left is greater than
the right, after both factors have been taken out, it must have been
greater before.
If, therefore, 3a — 6 is > a + 6,
if (by adding 6 — a to both sides),
2a is > 26.
But 2a is > 2b.
.-. 4a8(a-6)is>a*-&*.
teachers' edition. 351
11. Which is the greater,
• or Va+Vb?
U + \ is>or<vS+V5,
a8(squaring), T -\-V2ab + — is >or <a +V2ab + 6,
o a
as (transposing), ^- + — is > or < a + 6,
b a
as (multiplying by ab\ a8 + 6s is > or < a26 + a&2,
as (a + 6)(a2-a6 + 62) is>or<a& (a + 6),
as (a2 + 62)is>or<2a&.
But (a2 + 62) is > 2 a6, § 249.
.^+^£»>Va+V5.
12. Which is the greater,
«±*or-?«*-?
2 a + 6
"Jt*is>or<2a\
2 a + 6
as
a2 + 2ab + 62 is >or <4a6,
as *
a2 + 62 is>or<2a6.
But
a2 + 62is>2a6,
• ° + ^ in ^ 2o^
2 a + 6*
13. Which is the greater,
6* a2 6 a
a 6 • ^ ^ 1 1
62 a2 6 a
J249.
b a8 + 68 is > or < a2b + a62,
b (a + 6)(a2 - a6 + 62) is > or < (a + b)ab,
9 a2 + 62is>or<2a6.
But a2 + 62 is > 2a6, J 249
6J a2 6 a
352
ALGEBRA.
Exercise XCVHI.
1. >/? = x*
(x/x)* =x«.
^ -A
^iy? = x*y*2.
Vx*y2z4 =* ary*2*.
3. a*
aH*
4x*y~*=4\/xy~6.
4. a
a*
1
a1'
3x"1y-»
6x_8y
xy8
X8
x*y5
2a~1x
X*
V
6x7s
S^y"1
1 a6»
= 3xy2~8.
z
= x-*y_42.
a
6c
= ab~lc~l.
= alV<?.
x"«
= ^§yl
x"»
= a:-2f/-*.
6. a*xa* =o*.
6*X6* -ft*.
d*x <** = <*'.
7. m*x m~* = m*.
n* X w" * = fA
a°xal = a*.
cfixa-i -i-
a*
8. a* xVa =a*xa* -a.
c-ixVc 4xc» -1.
y* x^y = y*xy* -jf1-
x* xVxri = a*xx"*=»'.
teachers' editiok. 358
9. ab*cxa-lbc* «a*&M. 14. Qr*)-i -1>*.
aiblc-*xalb-*cld = aJd. (?*)* = ?*•
(*"*V)'* -*Vf-
(alxa*)-tt-(d»)-M-.a-«
10. 3*yMxaf*3f*z'* = x~M2~*.
xiyMxaT«yiz~*=ajV*- 15. (4a"V* = 4"*a=~
8
11. a»x«-»X«-*x.-»-«-* IW* -64"*C"V
(fN^W
1
X"TXaHi
~fo 16. /w^V'-iS^
U^8/ 81-*V*
^81* a8 ft*
12. a* 4- a* =a*. " ^i
c« ^c* = c*\ 27 a8 ft*
n&.±- a* = n~*. 8
\\6b-'J i6-»6*
16»
13. (a«)U(a«)» = (a«)-* = a-» 9,<*61
= 1. 64
= a 27 a6 6*'
l**t =c"*=7» (3la-»)-l=3-^ = J
(m"ty -m"1-™ f^Y1 _ 256"* = 625*
1 \625/ 625"* 256*
(n*)-» =w-1=-- ^125
354 ALGEBRA.
Exercise XCIX.
. 1. 4.
x* + xPyP + y*r 8a* + 4a1 6* + 5a* 6* + 9&*
x»-g*y* + y» 2a*- 6*
x+ + x»yp+x**y* 16a + 8a*&* + 10a*6* + 18a*5*
- **y*- x**y*- x*y» _8a*6*-4aW-5aW-9i
a^y^+ar'y^+y4*
a4* -fs^y2* +y*
5.
2. l+oi-' + a^-2
2* _j_ y**~
l+abl+a*b*
+ x~»-»y~«-* + fl»J> + a86-s + a*6-4
-V* 1 + a'&» +a46-«
gum — xH W* -f- x^11*-* y"1
*i_2**+i ' oif:+'+o";s
1 . a*6-* + 2aJ6", + l
x-2x*+ x* _2a,?r1-4-2a-,5I
- s* + 2x*-l -l-2a-8y-a-464
z _ 3xt + 3«* - 1 a46"4 -4-4aJ6,-a*54
4x-s + 3x"s + 2a;-1 + l
x-»_ x^ + l
4ar6 + 3x-* + 2ar8+ x~*
-4aT4-3x-8-2aT2- x'1
+4x-8 + 3aT2+2x-1 + l
4a:"6- x-4 + 3x-8 + 2x"2+ x~l + 1
TEACHERS* EDITION. ' 355
8.
a;4n _ yAn | #»* _ yn
xAn__x9nyn x9n + x2nyH + ^yin + y3
xSnyti _
xSn yn _
-y4n
- jc^y^*
x2ny2n .
x2ny2n _
- a^*y**
9.
|a?* + y*+z*
a; + x'y*
+«••»
a£ — a;*y* — z*2*
+ yl
-yM
+ **
— aj*y*
— x*y*
-«m
-x*y*
-3a:*yM + y + z
— afly*z'
— x*z* ■
f x*y*
-2s*yM+y + z
— a:*y*z*
a:*y*
jc*y*
+ je* 2* — x*yM + y + s
+ yM + y
— x*y*z* + a" 2* —
— jc*y»2» —
yl2*
yM
+ 2
-yM
+ yM
+ yM
+ 2
+ 2
10.
3 + y | a:* - x*yl + x^y* — sM + y*
aj — a£y* + ajf yf _ ^1 yi + &*y* x* + y*
g4yi _ x% yl -f #1 y$ _ #4y* + y
a;tyi _ jpf yl + aj$yf — aj*y* + y
856 ALGEBRA.
11. x*y-* + 2 + x-iy* \xy~l +xly
rr*y-4 + l xyl+x~ly
1+*"V
12. a-* + a-*&-* + &-* |a-»-g-i6-i + &-«
a"* + a*b* - a"8*"1 a"8 + a1 6"1 + 6"*
<r26-2- a1 &-» + &-*
a"
13. ^aft'1)? =16a«6*.
(a*-&*)2 =a-2a*5*+6.
(a + a"1)* - a2 + 2 + a"1.
(2a*6*-a-*6ty = 4aZ>* - 46 + a"1^
14. aH =4*x2 =4.
Soft"1 =5x4x} = 10.
2(a6)* =2^8 = 4.
a-^c^JxJXl -J.
12<r'&-s = 12xAxi = A.
15. (a*-2>ty = a*-3a&* + 3ai&*-6.
(2a;-1 + x)* - (2a:-1)* + 4(2aT1)»(aO + 6(2aTlJ»(*)P
+ 4(2aT1)(a:)s + tf*
- 16aT* + 32aT2 + 24 + 8x* + x*.
(aJr1-^1)6 =a66-«-6(a56-6x5y-1)+15(a*6-4xJ2y-1)
-20(a86-sx6sy8) + 15(a26-2xJV4)
-6(ai-1x&5y-5) + J6y"6
«a«5-6_6a56-4y-i+15a46-.y-._20ay8
+ 15a2 6V4 - 6a6*y6 + &V«
teachers' edition.
357
16.
9aT4 -18g'8y*-{- 15aT2y - 63;"^* + y2|3aTa- 3ar'y* + y
9a;-*
Qx^-dx^y^
6aT*
-lSx-*yk + 15 x'%y
-18ar8y* + 9aT2y
■Gar^vi
y* + 2/
6rc~2y — 6a?_1y* + y2
CaT2y — GaT^ + y2
3(2a;)» = 12x2
(6*+l)(l)=__+6^+l
17.
^g + l-Sg-1
8a?8 + 12a* - 30a? - 35 + 45a--"1 + 27aT2 - 27 ar*
8ar>
~ 12a* -30 a; -35
12a*+6aH-l
12a?+12x+ 3
12a* + 6a; + 1
-18+9a;-1+9a;-4
12a^+12a;-15^9a;-1+9a;-a
- 36a; - 36 + 45a;"1 + 27 a;"8 - 27 a;"8
- 36a; - 36 + 45a;1 + 27 a;'2 - 27a;-8
18. v'l2=v'3x2x2 = 3*x2».
v/72=v^3«x^ = 3*x2*
</m=</3x¥ = 3*x2*.
^64«^2* =2*
= 2»x3 =
19. [(a**)ix(a»)-»]»«-2
= [(a«) x (a?-10*)^
24. ~a*».
^)
20. (x*** X a;-11)**-2
l
= (*18.-12)8.-2
21. 3 (a* + &*)» - 4(a* + 6*) (a* - &*) + (a* - 2 &ty
= 3a+6ai&i + 3&-4a + 46 + a-4aM + 4&
= 2a*&* + 116.
{(am)m~m},"+1
= am_1.
23. ^Y+f— lT'f
358 ALGEBRA.
24. [«a--)-}i»]t + [{(*»H-»]-t
-1.
25.
a-SKt-D _y*(|,-i)
ajjKff-l) -fy«(l»-l)
By factoring the numerator,
(arKf-i) + y«(j»-D)(a;J*«-i)_?/«(j>-i))
"* ar^'-1) + y«(p-D
= x*«-i) - y^F"1).
Exercise C.
1. 3V5 -\/5x(3J» =V55. 2. Sy'v^ =^8liy.
3V2l=V21x(3)» =Vl89. 2x\fejj =^32^.
5V32=V32x(5)2 -V800. o8^5? =v^.
a*6 V£- -Vfcc x (a*&)» =Vtf&c. 3<*\fcbc =\^27Sc7.
*^y*=^*yOc)» =v^iy. SabcVcficr* =V25aWc.
3. *>/£ - VAX^_ - VJ^
lfiVgf - V256 X Hi =V244f
'os + 2ay + y8 * * xa + 2sy + y8 '
4. VaPifz = xy*Vz.
y/&a*b =VWx2a& = 2aV2a£
v^54a^y =v^27ay x 2 ax* - 3ay v^?.
V24 =V4x6 - 2V6.
\/l25a*5» = \/25a*d*x5a* = 5a2aV5d.
5. vTOOO^" = v'10xl0xl0xa = 10v'a".
v^l60x*y7 -v^20ayx8ay =2o^2^20^.
\^108 m9n10 =v^27m97i9x4n -SflAi'Mn]
^1372 au&w =v^343a16615 x 46 = IcP&Vlb.
6. \V- 3as6 + 3aW - ab* = Va(a*-Za?b + Zab*--V) = (a-6)W>
V50a2-100a& + 50&2 - V2 X 25(a2 - 2a6 + 62) = 5 (a -6) VS.
TEACHERS* EDITION. 359
7. 2^80 aW<* = 2v'l6a*c4 X 5aiV = iacy/ba^?.
7\/396^ = 7V36xlla; - 42VIT¥.
9\/sTtftf~z - 9^27 y»X 3&*« = 27y #TSg.
5 V726 = 5V121 X 6 - 55 V6.
|V^0f=tVlF^i^2xy29»tV58.
2y/\ -^xl=^4.
,. ^ .^ .i^.
WcPbx /3 cPbcxy a ._
V5 5
2
O701
-vs-vii-*^
11. (ax)x(M»)*-(aW)*-a&.rv'5.
(2a»&*) x (8V)* = 2(a«6Mx8)* = 2a*64j?v^.
360
ALGE&ttA.
12. Show that V20, V45, V$ are similar surds.
V20=V4~x~5 = 2\/5.
Vi5=>/5x^ = 3V5.
Therefore, they are similar surds because they have the same sard
factor.
13. Show that 2\faWt \/$¥, JVft
2Vtfb* = 2aW.
#8P = 2bW.
are similar surds.
•v?
Since they all have the same surd factor, they are similar surds.
14. If y/2 - 1.414213, find the values of
V50; 4^288; -7=; -; '
* V2' V450
V60
= 5\/2
= 5x1.414213
- 7.071065.
1
V2
f V288 = JV144X2
= 30V2
= 30 x 1.414213
- 42.426390.
= JV2
_^ 1.414213
2
= 0.707107.
V450
= VT^X2
= TVV2
-^X 1.414213
-0.1414213.
TEACHERS EDITION.
361
Exercise CI.
1. Which is the greater,
3>/7or2\/l5?
3V7 =V9~X7 =\/63.
2Vl5=V4xl5=V60.
Since 63 is > 60,
.-. 3V7>2VIB.
2. Arrange in order of magni-
tude
9V3, 6V7, 5>/l0.
9V5 =>/243.
6V7 =V252.
5VI5-V250.
Since V252 > V250 > V243,
.-. 6\/7>5Vl0>9V3.
3. Arrange in order of magni-
tude
4^4, 3^5, 5^3.
4^4-^256.
3^5=^135.
5^3 = v^376.
Since #135 < #256 < #375,
.\ 3#5<4#4<5#3.
4. 3V2X4V6 =12Vl2
= 24V3.
-VS.
5. 5VjxfVT62.
5Vf = *Vl4;
$Vl62 = ^V2.
$ Vl4 x V- V2 = W V28
= J#V7.
t#4x2#2 = #8 = 2.
6. 2V5-*-3Vl5 = jVj
= $V3.
tVa + AV5-|V85^
- 1* Vl5.
7. J\/3x$V5-^>/2
= ^Vi6H-fV2
-t^Vso.
8 2VJ0 7VJ8
4vT5
3V27 5VT4 15V21
-9xixV)Vtfxttx«
9. 2#4x5#32-s-#l08.
2#4x5#32 = 10#l28.
10#128 -s-v^T08 = 10 #^f
= 10#|f
-10^x4
= 6J#4.
962
ALGEBRA.
Exercise CII
1. 2v^- 2(3)* = 2(3)* =^576.
3V2 - 3(2)* - 3(2)* =v^58|_
§ y/l - § (2)* - $ (2)* - v^l953j.
.-. the order of magnitude is 3V2, |v^i, 2^3.
2. VJ -(f)* = (!)* -J'ffi.
3. 2^^-^l76-v^l76*=v/30976.
3^7 -^189 =^189* = ^35721.
4V2 -V32 «v^3? =#32768.
.-. the order of magnitude is 3v^7, 4>/2, 2^22.
4. 3 Vl9 - Vm - 171* =- 171* - vT71» =#5000211.
5#2 -#^ = 250* = 250*=#^=#625W.
3#3 =#81 -81* -81* =#81» =v6561.
.-. the order of magnitude is 3\/l9, 5 #2, 3 #3.
5. 2Vax x #3a*6 x V26x; -&a*xy* xy/a*xy.
2\/as =2aM.
#3^& = 3*a*6*.
V26i = 2*6***.
^26x3*x28xa*x6*Xa: = 2ax #72^.
#S^» = (aV)* - (as*y»)* -vWey*
y/a*xy = (atxyfi = (a'sy)^ — y/ahfy*.
v/5s^Xv^y=v/a»xa*xyu= a^Vy5.
TEACHERS* EDITION. 363
6. 3(4a&2)»-*-(2a8&)*.
3 (4 a&2)* - 3 (4 a&2)* = 3 y/WaW.
(2a36)* = (2a86)* =v^8^P - a^W.
a a*
(2a»ft>)*x(aW)*-i-(eiW)*
(2W)x(a*&) = 2*a**&*.
= (2a6)*x(3a62)^(5a6»)» # ' W VW U^V
= ^(2^8x^^+v/5^55 ' _ aty 6M aM
= ^Wxv/3W-j-v/5^ x* v1 y*
= v'28x32xa5x&7 +V$a&
-*F*
a6»
32 x PaW a&M
5«
= ^225000 a*5*.
4Vl2^-2V3 = ^^V-
= 2V5=2x2 = 4.
a#
9. (7V2-5\/6-3V8+4\/20)x3\/2.
7V2-5V^-3V8 + 4V20
3v^2
42T30\/3-36 + 24Vl0
= 6-30V3 + 24VlO.
= (f)7x(f)6 = a8&^2.
dHir
^(l^S2? X V^(2a26)« = a'1 ft* cl* (H.
= ^(8a3ft8)*
-(2oft)"
364 ALGEBRA.
Exercise CUT.
1. V27 - V9x3
= 3\/3
2V48 -2V16X3
= 8V3
3V108-3V36X3
= 18V3
Sum
= 29V3
3V1000- 3VlOOxlO= 30V10
4V50 - 4V25x2 - 20V2
12V288 =12Vl44x2 = 144V2
Sum -164V2 + 30V10
2. ^128=^64x2 = 4v^2
4. 2V3+3VIJ-V5
^686=^343x2 = 7v^2
-2V3 + 3V|-Vy
y/16 -v^ - 2v^2
:=2V3 + $V3-|V3
Sum - 13^2
= fV3.
2VJ+Veo->/i5-V|
7^54 -7^27x2 = 21^2
= }Vl5 + 2Vi5-Vl5
3^16 =3v^8x2 - 6^2
-JV15
= f|Vl5.
^432- ^216x2= 6v^2
Sum -33^2
5- ^-gv*
Vg-g^
la*cd* _ad .—
3. 12V72 =12(2»x32)4 = 72>/2
3 Vl28 - 3(2*x42x2)*=24V2
.^IK ^/aV ^/H?
Difference = 48V2
7^/81 =7(3*)* =21^3
3V| + 2v^S-4VA
3 #1029 = 3(3 x 78)* - 21 #3
-SVIo + jVIo-jVio
Difference *- 0
-|VTa
teachers' edition.
365
6. V±M+V2&ab*-(a-5b)\/ab
= 2aVab + 5 6\/o6 — (a - 5b) Va&
= (a + 106)Va6.
7. cv^V-a\/^F? + 6#aW
= abcVab2<? — abcVab'W + abcVabW
= abcy/ab'W.
8. 2#40 = 4#5.
3#l08 ~3#33~x~22* = 9#4.
#500 - #5rx~2* = 5#4.
-#320 = -#28~x5 =»-4#5.
- 2#l37~2 - - 2#2*x78 - - 14 #4.
And 4#5 + 9#4 + 5#4 -4#5 -14#4 = 0.
9. (2#3a*&)» = [2(3a*6)*]8
= 28(3a46)* = 8a*V3l
(3#3)2 = [3(3)*]»
-3*(3)* = 9#3.
-(f)' -(t)'-»^-
(V27)*=(27i)i = 27* = #27.
11. (#81)* = (#3*)* =#3.
(#512)* = (#*)* = 2* -#8.
#16 -(2*)* =#2.
#27 =(33)A =#3.
12. #4 = (2*)*=#2.
#36 = (62)*=#6.
#32 -(25)*=#2.
#243=,(35)*=V3.
#125 = (58)» =#5.
#49 -(7»)* =#7.
13. #8? = (23a*)* = #2^".
#9a2&* = (32d*6*)* = #3^5*.
#l6aIi=#2V7i = 2*a*
= a#2a.
#320^- (2*a10)* - 2a2.
366 ALGEBRA.
14. (y/S)* =(v/2»)* = 2*-2^2.
(^27)* = ( #38)« = 3* - 3v^3.
(^6l)» = (^26)s = 2* = 4v/2.
(\/iy =-(\/^)J-2*=2v/2.
15. (av^)"8 =(a*)-» -a"*.
(a-'v^78)-* -(v'St*)-*-^"*) -*-a*«.
16. (Va+Vb? = (at + blf
- a* + 5a26* + 10a*6 + lOai* + 5a*&* + 61
- a2Va + 5a2 V5 + lOai Va + lOabVb + 5^ + b*Vb.
(-Vtd+Vity-lmi + df
- (m*)8 + 3(mtyx* + 3m*(z*)2 + (aty
= m2 + Smxy/mta? + 3a?Vm* -f a^Vx.
(Va^-2V&)6 = (a*-2&ty
= (ai)5-5(ai)*(26*)+ 10(ai)»(26i)2-10(ai)J (26*)8
+ 5(a*)(2ftty-(2&4)5
- a2 VS - 10a2\/5 + 40 obVa- 80 a&V& + 80 62Va" - 324* V8.
17. (2a*-lVaf
= (2a2)8 - 6(2a2)5(J Va) + 15 (2 a1)4 (J Va)2 - 20(2a2)8(jVa")8
+ 15(2a^(JvG)4-6(2a»)(}vG)H(JV,a)1
- 64a12 - 96al0Va + 60a» - 20a7 Va + 3fa8 - f a*Va + ~
64
= (2^ - 4(2,*)' (*) + 6(2,*)» (*)'- 4(2**) (£)*+ (f )'
16
TEACHERS* EDITION. 367
(~^)'=(2*y'-y»)'
- (2*V)« - 6(2a!y 7>(y») + 15(2a!»y-1)«(yl), - 20(2*y >)»(y»)»
+ 15(2^-')J(y1)4- 6(2*y ')(y»)» + (3/1)4
= teas'y*- ma^V^y1 + 240 a»y 'v^T' - leOafy1
+ 60**^ -12ay v'y+y*.
18. fv^6 S_V
V 2Vb)
(«i_viy
-(S)(T)'-(¥y
.W_"4W(g)!W(0_4(^(0+(*)*
= a«6* - 2a*6* + §**-£!&? + ?'*
2 16
19.
d^-viy
=(!^y-(!^y(Vf)+3(!^)(^y-(v?y
- a*b*cid-* - 3a*6-« + $ab~lc'*d* - cH\
368 ALGEBRA.
(n'-a"5)*
-(a»)*-4(a5)»(a"*) + 6(a5)'(a'*),-4(a^)(a"*)» + (a"7
= a**-4a* + 6 — 4a-»+a-*1.
'(t)K-H!W;)'
16a4 32a* 8 a* 8 a* a«6«
ft8 36* 36* 27 81
a26*
81
20.
a.__ VcV
be Sab)
\(6c)V \(6e)V\3a6/ \(&c)V\3aJ UaJ
(6c)« Uc 3a67 \bW\9aWj
a* 3ac* 3a*c c*
' (6c)* ~ 3a6*c 9 a*&M 27 a*V
a* 1 c* c*
+ -
&M 62c* 3a*6* 27a86»
(^H^^w^-^'
86* 46* 2
TEACHERS EDITION.
869
(aVa_Vb\*
U* 2a)
6* 26** ' 4a*6* 8aS
a* 3a2 + 3
21.
a** + 6a*»y»» + lla^y2*1 + 6xmySH -hy4*^*2"1 + 3a*»yn + ySn
2xSw+3a-»y»
2aA» + Qxmyn + y2n
2a^"*y2n + 6a?*y3B + y4*
2a£"ys,t + 6a*lySM + y4"
22.
|l + 2x"* -3or* + 4arl
1 + 4a;-* _ 2x'i - Ax'1 + 25a:-* - 24aT* + 16a;-2
1
2 + 2aT*
4a;-* 4- 4.t""
2 + 4ar*-3ar*
_8a;-*- 4a;1 + 25 aT*
-6ar*-12arl + 9 a?-*
2 + 4a;"*- 6a;-* + 4a;"1
8a;"1 + 16a;"*- 24a;"* + 16a;-2
8a;"1 + 16a;"* -24aT* + 16 aT2
370 ALGEBRA.
Exercise CIV.
2.
(i) -^— 0)
V7+V5 V6-v^
3(V7-V5) g(V6+V^)
" (V7 + V5)(V7 - V5) (V^ + Vc)(V6 -Vf)
3V7-3V5 -«(^+^).
" 2 6~c
-}(V7-V5). (2) a + *
a-Vb
7 a + 6 a+Vft
(2) - 7?X y=
2V5-V6 a-V6 a+V6
7 2V5+V6 _(a+&)(a+V6)
:X
2V5-V6 2V5+V6 a ~°
_7(2>/5+>/6) (3) 2x-VwxV^ + 2y
14 Vxy — 2y Vxy + 2y
= i (2 V5 + V&). = 2{x-y)Vxy + 3xy
xy-Af
(3) ±=* ' 3.
1+V2
(4-\/2)(l-\/2) (1)
2(V5)
(1+V2)(1-V2) * <^<^
v v ' 2\/3 3.463
4-5V2 + 2
-1
= 5>/2-6.
(2)
3 3
- 1.154
1
(4)
VE-V2
-2V6 _ 1(V5+V2)
6
5 + 2\/§ (V5-V2)(V5+V2)
5-2V6 5 + 2V6 V5+V2
3
30 + 12>/6
3.649..
1 3
= 30 + 12V6. - 1.216..
teachers' edition. 371
(3) ^ ' (4) I+2VI0
V7+V3 7-2VI0
7V5(V7-V3) (7-f2VJ0)(7 + 2Vl0)
(V7 + V3)(v^7-V3) (7-2VT0)(7 + 2Vl0)
-/ /** /Tin _49 + 28VT0+40
= 7(V35-Vl5) j^^
7(5.91 'o -3.8729 ) = S^VW
--* 4 9
89+28x3.162..
= 3.576 9
- 19.726
Exercise CV.
1. 4+V=3 = 4 + 3*V=T.
4_V^3 = 4-3*\/^T.
(4 + 3*V^T)(4 - 3* V^T) = 16 - 3(- 1) - 19.
V3 - 2V=^2 - V3 - 2 x 2*\£T
V3 + 2V^2 - VS + 2 X 2* y/-T.
( V3 - 2 x 2*V^1)(V3 + 2 x 2*V^I) = 3 - 8(- 1) = 11.
2. V54x V^^ = 3>/6xV=l = 3v^T2 = 6\/^3.
aV^b x sv^~y = (a&*\/^l) X (aj^V^T) = a&*xy*(- 1)
= — axVby.
3. V^S+vcrs^aiv^rT + ftiv^T.
V=^ - V^6 - a* V^l - b*V^i.
(a*V=7 + &*V=T)(a*V=n[ - 6*\/^) = a(-l)- &(-l)«6-a.
02ft2 VZ1 x a26* V^l - a*6*(- 1) = - a4M VS.
4. ( V^IO) ( V=^2) - (10* V=T)(2* V=3) = 20* (- 1) - - 2 V5.
(2V3-6V^5)(4V3-V^~5)
= (2 X 3* - 6 X 5* V-l) (4 X 3* - 5* V^T)
- 24 - (24 X 3* x 5* V^l) - [2 X 3* x 5* V^l + 30(- 1)]
. _ 6 - 26(3* X 6* V-T) - - 6 - 26 V=HJ5 - - 2(3 + 13 V^15).
370
1.
(1)
Vl +V5
3(V7-VE)
(a/7+V">)(VT- V5)
3>/7-3V5
2
-}(>/7-V5).
Exercise CIV
(1)
(2)
2V5-V6
7
2\^5-V6 2V5+V6
_1(2V5+y/6)
14
■l(2V5+v/G).
2.
(2)
g(V6+Ve)
(V5+Vc)(V5-Vf)
_g(\/6+\/c)
a + b
a-y/b
a + b a + Vfe
a— Vb a+Vh
_(a+b)(a+Vb^
a* -6
(3) 2x-ViyyV^
Vxy-2y v^
_ 2(x — y)Vxij
(3, *^
1 fV2
4-nV2 + 2
-i
<*)
5
* 5^2"V^ * 5 + 2v4
.30+ IfVfi
-30 + J2>/tf
(1) -L=^(v
V3 (V3)
_ 2n^ =
3
- L154L...
(2)
1
Vs~V5
, !_
'. ' V ■ - *
-\¥>:
i
1
I
rrnra-, ^
-20.
13-2V30.
13 + 2 V30.
V169-120.
7.
-13.
-10,
y = 3.
y=\/l0-V3.
10.
c_v^=Vll-6>/2.
/i+Vy-Vii + eV2.
iltiplying,
x-y=Vm-12.
.:x-y = l.
x + y — 11.
.-. a: = 9,
and y = 2.
.Vx-Vy = 3-V2.
374
ALGEBRA.
11.
Let Vx - Vy -Vu"-4V6.
Then VS +Vy =Vl4 + 4V<3.
By multiplying,
x-y-Vl96-96.
.\x-y-10.
But x + y — 14.
.\x=12,
and y =» 2.
.\ Vx -Vy = 2V3 -V5.
14.
Let Vx-V^=V57-12Vl5.
Then Vi + Vy =V57 + 12V15.
By multiplying,
x-y=V3249-2160.
... x-y = 33.
But x + y = 57.
.-. x = 45,
and y = 12.
.\V£-Vy-3V5~2V3.
12.
15.
Let Vx"-Vy=V38-12VT6. Let VJ-Vy =V|-VlO.
Then Vi"+Vy =V38 + 12Vl0. Then VS + Vy = Vj + VIO.
By multiplying, By multiplying,
x-y=Vl444-1440. x-y=* V^-10.
.\s-y = 2. .-. x-y=f.
But x + y = 38. But * + y = j.
.\x = 20, .-. x = J,
and y = 18. and y — 1.
.\Vx-Vy = 2V5-3V2. .\Vx-Vy = JVI6-1.
13.
16.
Let Vx-Vy=Vl03-12VlT. Let VS + Vy =V2a+2VSM*.
Then Vx + Vy =Vl03+12VIT. Then VS - Vy = V2a-2Va^P.
By multiplying, By multiplying,
x - y = V10609-1584. x - y = V4a2-4a*+4P.
.\x-y = 95. .•• x — y = 26.
But x + y = 103. But x + y = 2a.
.\x = 99, .\ x = a + 6,
and y = 4. and y = a — 6.
.•.Vx-Vy-3VII — 2. .•.Vx+V^=Va+6+Va^6.
teachers' edition. 375
17.
Let Vx -Vy = Va2 - 2bVa2~^T\
Then Vx + Vy = Va2 + 26 Va2 - b*.
By multiplying,
a; — y =Va4 - 4a262 + 46*.
.\a?-y = a2-262.
But a; + y = a2.
.-.a; = a2-62,
and y — 62.
.;Vx—Vy = Va2 — b* — b.
18.
Let v^ - V£ = V87 - 12 V42.
Then x-y =V7569-6048.
.\#-y = 39.
But a; + y = 87.
.-.a? = 63,
and y = 24.
.-.v^-Vy=>/63-V24
= 3V7-2\/6.
19.
Let Vx -Vy = V(a + b)2 - 4(a - 6) Vab.
Then Vx+Vy =V(a + 6)2 + 4(a - 6) Vab.
.'.x-y = Va* - 12a86 + 38a262 - 12a6s + b*
= a2-6a& + 62.
But x + y = a* + 2ab + b*.
.:x = (a-b)\
and y = 4 aft.
,\Vx — Vy = a — b — 2Vab.
37G
ALGEBRA.
Exercise CVTL
1. Vx~=5 = 2.
Squaring, x — 5 = 4,
* = 9.
2V3x + 4-ar + 4.
Squaring,
12x + 16 = a^ + 8a? + 16,
a*-4a; = 0,
x(*-4) = 0.
,\x = 4or0.
3. 'S-VxT^i = 2x,
3-2a?=Va^-T.
Squaring,
9 -12* + 4a* = a*-l,
3x»-12x=-10,
9a* -36a: = -30,
9**-() + 36 = 6,
3x-6 = ±V6,
3a? = 6±V6.
.-.a;-2±iV6.
4. V3x-2 = 2(a;-4).
Squaring,
3a?-2 = 4x«-32x + 64
4a* -35 a; = -66,
64xa-() + (35)» = 169,
8a;-, 35 = ±13.
.-.a; = 6 or 2}.
5. 4a;-12Vx = 16.
Divide by 4,
x-3Vx = 4,
ZVi = x-i.
Squaring, 9 a; = a* — 8 a; + 16,
a*- 17s = -16,
4aJ-68a; + 289 = 225,
2* -17 = ±15.
.•. a; = 16 or 1.
6.
Squaring,
VaT+l+V2a;-l = 6.
x + 4 + 2 V2x2 + 7z-4 + 2 re - 1 - 36,
Squaring,
2\/2a* + 7a;-4 = 33 - 3ar.
8 a* + 28a; - 16 = 1089 - 198a; + 9a*
a* -226 a; = -1105,
a*-() + (113)' = (113)* -1105,
x - 113 - ± 108.
.-.a; = 221 or 5.
TEACHERS EDITION.
377
7.
Vl3x-1 -V2Z-1 = 5.
Squaring,
13a;- l-2V26a?- 15a; + l + 2a; -1 = 25
15a; - 27 - 2V26a? - 15 a; + 1.
Squaring, 225a? - 810a; + 729 = 104a;2 - 60a; + 4,
121 a? -750 a; --725,
58564a? - ( ) + (750)2 = 211600,
242 a; -750 -±460,
242a;=1210or290.
.\a; = 5orl12^:.
8.
Squaring,
Squaring,
V4 + a? + Vx = 3.
A + x + 2V4a? + a? + a; — 9,
2V4a? + a? = 5-2a;.
16a: + 4a? = 25- 20 a; + 4a?,
36 x =25.
9.
V25 + g +V25-a; = 8.
Squaring, 25 +a? + 2 V625 - a? + 25 - x = 64,
2V625-a? = 14,
2500 -4a? = 196,
4a? = 2304,
2a; = ±48.
.\ a; = ±24.
10.
a? = 21-fVs2^T9.
a?-2t=Vi?^9,
a?- 42 a;2 + 441 = a? -9,
x*- 43a? = -450,
4a? -172a? = -1800,
4a?-() + (43)2 = 49,
2a?-43 = ±7,
a? = 25 or 18.
•\ a;«*±5 or ± 3V2.
378 ALGEBRA.
11.
2x-#S*»~+26 + 2 = 0.
^8** + 26 = 2* + 2,
Sx* + 26 = 8** + 24** +'24* +8,
24*a + 24*=18,
4*» + 4* = 3,
2x + 1 = ± 2.
.\* = J or — f.
12.
V* + l +V* + 16 = V* + 25.
* + l + 2V*a + 17* + 16 + * + 16 = * + 25,
2V** + 17* + 16 = 8-*,
4** + 68a? + 64 - 64 - 16* + **,
3** + 84* = 0,
** + 28* = 0,
*(* + 28) = 0.
.\ a; = 0 or -28.
13.
V2xTl - VxTl = i V* - 3.
2*+l-2V2*» + 9* + 4+* + 4 = *~~3
9
27* + 45 - 18V2** + 9*T4 = * - 3,
18\/2*» + 9*+4 = 26* + 48,
9V2*« + 9* + 4-13* + 24,
162*2 + 729* + 324 - 169*2 + 624* + 576,
-7** + 105* = 252,
3»_15* = -36,
4**-()+225 = 81,
2* -15 = ±9.
.\* = 12or 3.
14.
V* + 3 +V* + 8 =» 5V*.
* + 3 + 2\/*2 + ll* + 24 + * + 8 = 25*,
2\/** + ll* + 24 = 23*-ll,
4** + 44* + 96 = 529** - 506* + 1215
525*2-550* = -25,
21a* - 22a- -1,
1764 x2 -() + 484 = 400,
42* -22 = ±20.
.\ *=1 or fa.
TEACHERS EDITION.
379
15.
y/3+x+Vx*
6
Vs + ~i
Clear of fractions, 3 + x +V3a + ;c2 = 6,
y/3x + x* = 3 - a:,
3a; + a;2 = 9-6a; + a:2,
9a; =9.
.-. «-l.
16.
V?^l
a2 - 1 + eVjF^l = 16,
6V^=l = 17-a;a>
36a* -,36 = 289-343* + **,
a?* -70 a;2 = -325,
tf*-() + (35)2 = 900,
x* - 35 = ± 30,
a2 = 65 or 5.
.*. a = ±V65 or ±>/5.
17.
1
1
Vx + 1 Va? — 1 Va;2 — 1
Va; — 1 +Vx + 1 = 1,
/x — 1 = 1 — Va; + 1,
/x+ 1 + a? + 1,
Va; + 2a-Va; — 2a
a;
:2a"
Vx — 2a + VaT+2a
2aVr + 2a — 2aVx — 2a=*xVx — 2a + ffVaT+lJa,
(2a - a;)VaT2a = (2a + a)Va;-2a,
(4a2 - 4aa; + a*)(x + 2a) = (4a2 + 4aa; + a;2)(ar - 2a),
x9 - 2az2 - 4a2a; + 8a8 = x* + 2a£ - 4a2x - 8as,
4aa;2 = 16a3.
.*. x = ±2a.-
380 ALGEBRA.
19.
Zx+Vlx — x**
Sx-Vlx-x*
3s +V4s-s2 = 6s - 2V4s-s*,
3>/4a-s,~ 3s,
V4s — a2 =• x,
4s — a^ — s2,
2s2-4s = 0,
2s(s-2) = 0.
.-. s = 0or 2.
20.
V7s2 + 4 + 2V3s-l_7
V7x2 + 4-2V3x-l '
V7s2 + 4 + 2Vj3T^T= 7V7s2 + 4 - 14V3s-l,
16V3s-l = 6V7s2+4,
8V3a;-l = 3\/7^ + 4f
192x-64 = 63s2 + 36,
63^ -192s- -100,
4(63s)2-() + (192)2 = 11664,
126 s -192 = ±108.
.-. s = 2^ or}.
21.
V(s — a)2 + 2ab + 6* = x — a + b.
x1 - 2as + a2 + 2ab + 6* = s2 + a2 + 6*-2as— 2ab + 2bx,
26s==4a6,
2s = 4a.
.-. x = 2a.
22.
V(s + a)2 + 2ab + 62 = 6 - a - s.
a* + 2as + a2 + 2ab + 62 - i2 + a2 + s2-2a6 + 2ax-26x;
26s = -4a6.
.-. s 2a.
teachers' edition. 381
23.
VF3+a
»-36 =
-v?
=VSx.
V33+36+V3;
3a; + 36
+ 2V9a*-1296 + 3;
r-36-
= 8a;,
-2*
-a?,
2V9JC2-
-1296*
>/9a*-
- 1296 -
9a*-
-1296 =
a* =
.\ a? =
24.
- 162.
= ±9V2
4ar*-3(a;i + l)(a
r» - 2) -
-a;* (10
-3a;*).
4a;*-3aj+3j
C* + 6 =
3ari-
26.
= 10a;*-
= 6,
-36.
-4.
-3a;,
(a;1 -2) (a;
*-4) =
*x*(x*-
-1)*-12.
a;*-2a:,-r4:
c* + 8 =
5a;* =
*•-
= a;*-2
= 20,
-4.
a;§ + a-! _ ;
12,
Raise to third power,
3* =
»64.
.\ a? =
■±8.
26.
a*-4a# = 96.
27.
a; + x~l ■-
= 2.9.
a*-
-4a;* + 4 = 100,
x*-2 = ±10,
a;
= 2.9,
a;* = 12 or - *
a* + l =
= 2.9 a;,
5,
10a*- 29 a; =
= -10,
a;* = vT2or-
-2- 400a* -|
[) + (29)* =
= 441,
.\ a; = (#144) or 4
20 a;-- 29 =
-±21.
= 2^18 or
4.
.•. a? =
= Jor2}.
382 ALGEBRA.
x* + 2a*ar* = 3a.
Multiply by re*, a; + 2 a* = 3 ax*
x-3ax* = -2a*,
4x-() + 9a* = a»,
y/±x — 3a = ±a,
V4x = 3a±a,
4x = 16a* or 4a2.
.\ x = 4a* or a*
81^ + -^ = 52*.
Vx
81a?* + 81 = 52s*.
Transpose, 52x* - 81 x1 = 81,
(104)»x*-() + (81)» = 23409,
104 x1- 81 = ±153,
104 a* = 234 or -72,
st-ftor-A.
Exercise CVIII.
1.
a-*_3x-6Vxa-3x-3 + 2 = 0.
Add —3 to both sides,
(JB«_3x-3)-6(x8-3x-3)i = -5,
(a4_3a._3)_(\ + 9 = 4,
(x»-3x-3)*-3 = ±2,
(x»-3x-3)* = 5orl,
a-*--3x-3 = 25or 1,
x*-3x = 28or4,
4x*-() + 9 = 121or 25,
2x -3 = ±11 or ±5,
2x = 14, -8, 8, -2.
.-. x = 7,-4, 4,-1.
TEACHERS EDITION.
883
x x2 36
Subtract 2 from
Since
From
H)
i, and from
36
65
36
a(.-i) + 8.
16
2a;-
13
s-*or-
3 3
6a;2-6 = -5a;or-13a?,
6a? + 5a; = 6,
144 a? + 0 + 25 = 169,
12a? + 5 = ±13.
.-. a? = }or-li.
6a?-6 = - 13a?,
v 6a;2 + 13a; = 6,
144 a? + () + 169 = 31 3,
12a; + 13 = ±V313.
.•.*-A(-13±VSl3).
(2a;2- 3a;)2- 2(2a?-3a;) - 15.
(2a?-3a;)2-0 + l = 16t
(2a?-3a;)-l = ±4,
2a,-2 -3a; = 5 or -3,
16a;2-() + 9 = 49or-15,
4a; — 3 = ± 7 or ±V— 15,
4a; =10, -4, 3 ±yClg.
.-.x = 2J, -1, J(3±V-15).
384 ALGEBRA.
(a*-6)»+4a(as-&)=^-.
4
4(ax - ft)P + 16a(ax - &) = 9as,
4(ax - by + ( ) + 16a* - 25a»,
2(ar — fc) + 4a = ±5a,
2(ax — &) = a or — 9a,
2ax =-a + 2& or 2&-9a.
. - a + 2b M2b-9a
••• x = — or — •
2a 2a
5.
3(2c«-*)-(2zt-*)t-2.
36(2^-»)-() + l-25,
6(2**-a;)*-l = ±5t
6(2x*-x)* = 6or-4,
(2a?-*)*- 1 or -|,
2x*-a; = l orf,
16x»-8x-8or ^,
16««-() + l = 9orV,
4«-l-±3or±jV41.
.-.*-1,-*.«1±*VE)l
6.
15x-3s* + 4(a*-5a; + 5)* = 16.
Change signs and add 15 to both sides,
(3z,-15x + 15)-4(*f-6* + 5)*--l,
3(a?-5x + 5)-4(*,-5x + 5)* = -l,
36(s»-5a; + 5)-() + 16 = 4,
6(a*-5a; + 5)* = 6or2,
(x*-5* + 5)*-lor J,
a£ — 5x + 5 — 1 or £,
a? — 5a; =* — 4 or — 4$,
4x«-()+25 = 9or^,
2a; — 5 = ±3 or ± J.
.\x = 4, 1, 3f, 1}.
teachers' edition. 385
•
7.
X2 + X~2 + X + 0T1 =
-4.
(*+?)+(*+;)-
-4,
{x4j+ix+l)'
= 6,
4('+i),+4(*+-)"
= 24,
4^+iy+()+1=
= 25.
Extract the root, 2( x + - ] =
= 4 or -6,
1
= 2 or -3,
• X
a» + l-
= 2x or — 3a?.
For first value, x2 + 1 =
= 20?,
a?-2a; =
•-1,
a*-2» + l = 0,
a?-l =
= 0.
.\ a? =
■ 1.
For second value, x2 + 1 =
= -3a>,
aa + 3a? =
■-1,
4a» + () + 9-
= 5,
2a? + 3 =
= ±V5.
.*. <c =
= J(-3±V5).
8.
x2 +Vx* — 7 =
-19.
Subtract 7 from each side,
(a* _ 7) + (a* -7)* -12,
4(s2-7) + () + l =
2(a*-7)* + l =
-49,
= ±7,
2(«"-7)* =
= 6 or -8,
(a?_7)* =
= 3 or -4,
i»-7«
= 9 or 16,
a*.
-16 or 23.
.-. a> =
-±4or±\/§3.
J
386 ALGEBRA.
9.
6(a? + *) + (*■ + *)*« 7.
144(s»+aO + Q + l = 169,
12(^ + ^ + 1 = ^13,
(0^ + ^ = 1 or -{,
(x8 + x) = lorft,
4a* + 4a; + l = 5or_W,
2x + l = ±V5or ±$V58.
... ^(-li^orJC-l**^)
10.
(s + l)* + (*-l)l = 5-
Squaring, * + l + 2(*»-l)* + *-l = 25, '
2(x*-l)4 = 25^2xf
4a? - 4 = 625 - 100* + 4a2,
100x = 629.
.•.a? = 6^fiy.
11.
a;-l-2+2flri.
a._l = 2 + 4»
2 x
.-2-3,
x*
a;i_2 = 3a;i,
<pi_3a-* = 2.
Squaring, a»-6*»+9*-4l
a*-6x2 + 9s-4 = 0,
(s-l)(a*-5s + 4) = 0.
...(a._l)or(x2-5a; + 4) = 0.
If (*-i)-o,
*-l.
If a2- 5a? + 4 = 0,
B*-5a:~-4,
4a;2 -20 a? = -16,
4aj2-20x + 25 = 9,
2x-5 = ±3,
2s = 8 or 2.
,\ x = 4 or 1.
TEACHERS* EDITION. * 387
12.
V3a; + 5 W3a;-5 + 4,
(3a + 5) = (3 a; -5) + 8V3aT^5 + 16,
6 = 8V3rc-5,
36= 192 a; -320,
-192a = -356.
13.
(** + l)-»(^ + 1)--2a*
Transpose, a* + 2a? + 1 - x(x* + 1) = 0,
(a* + l)2-a;(a;2 + l) = 0.
Multiply by 4, and complete the square,
Extract the root, 2 (a* + 1) - x = ± x,
2x* + 2 = 2x or 0.
For first value, 2a? - 2x - - 2.
Multiply by 2, and complete the square,
432_() + l = _3.
Extract the root, 2a;-l = ±V-3.^
••• &= *(!*</- 3).
For second value, 2 a2 = — 2,
a*--l.
,\ a; = ±V^l.
14.
2a?2 - 2V2z2-5.r = 5 (.r + 3).
2a* _ 5x - 2V2a;2-5a; = 15,
(2a? - 5a) - 2(2a? - 5s)* + 1-16,
(2a?-5a;)*-l.= ±4,
(2a;2 -5a;)* = 5 or -3,
2a?- bx = 25 or 9,
16a?-()+25 = 225or97,
4s-5 = ±15or ±V97, _
4 a; = 20, -10, (5 + V97), (5-V97).
.-. x - 5, -2J, J(5+ V97), J (5- V97).
388 ALGEBRA.
15.
x + 2 - 4x>/x + 2 = 12a*
Complete the square,
(x + 2) - 4xVx + 2 -f 4x» = 16x*,
Vx + 2-2x = ± 4x,
y/x + 2 = 6x or -2x,
x + 2 = 36X2 or 4x»
36s«-x = 2,
5184x»-144x=s288,
5184x2-() + l = 289,
72x-l = ±17,
72x = 18or-16.
.\ x = J or — f.
Also, x + 2 = 4x*,
4x2-x = 2,
64x*-16x = 32,
64x*-() + l = 33,
8x-l = ±V33. __
.-.x = J(l±V33).
16.
V2x + a + V2x — a = 6.
2x + a + 2V4X2 - a2~+ 2x - a = #,
2V4x2-a2 = *2-4x,
lBx2 - 4a2 = 6* - 8&*x + 16x»,
862x = 4a2 + 6*.
4a2 + &*
86«
17.
V9x* + 21x + l - V9^T6x+T = 3x.
9x2-f21x-fl-2V81x* + 243xs + 144xa + 27x + l + 9x2 + 6x+l=9gg
2V81x* + 243x» + 144X2 + 27x + 1 - 9x* + 27x + 2,
324 x* + 972x» + 576x* + l08x +'4 = 81 x* + 729 x* + 4 + 486 Xs
243x* + 486x» - 189 x2 = 0, + 36a;2 + 108*'
27x2(9x2 + 18x-7) = 0.
.\x = 0.
Or, 9x2+18x=7,
9x2 + () + 9 = 16,
3# + 3 = ±4,
3x = l or -7.
.-. » = Jor-2|.
TKACHERS' KDITI(JN. 389
18.
x* — 4 a;' + x' * + 4aT* = — J.
(•S)-e4)~*-
Since re1 - 2 + \ - (a* - i Y.
(■Sy-'t-S)--*
(^i,)-2.,,
2a>*-2 = 3a*or 5a;*,
2^-3^ = 2,
16*1-0 + 9-25,
4** -3- ±5,
4a;* = 8 or -2,
at = 2 or - },
x* = 8 or - |.
.-. a? = ±2V2or ±J\/^2.
Also, 2**-2 = 5a'»
2a;* -5a;' = 2,
16**-() + 25-.41,
4a-'-5 = ±V41,
a;» = J(5±Vil).
.•.*-[! (5 ±VE)]».
390 ALGEBRA.
19.
(2* + 3y)«-2(2x + 3y) = 8 (1)
x*-y* = 21 (2)
Add 1 to both sides of (1),
(2x + 3y)» - 2(2* + 3y) + 1 = 9.
Extract the root. (2* + 3y) - 1 = ± 3,
2x + 3y = 4or-2 (3)
2 2
^ 16-24y+9y» or 4+12y+9y»
4 4
Substitute value of z* in (2),
16-24y + 9y» .a21
4 y
5y*-24y = 68,
100y» -0 + 576 = 1936,
lOy - 24 = ± 44,
10y = 68or-20.
... y = 6$or-2.
Substitute second value of ae1 in (2),
4 + 12y + 9y» a_gl
4 y
5y2 + i2y - 80,
100^ + 0 + 144 = 1744,
10y + 12 = ±4VT09,
10y = -12±4Vl09.
.•.y = -U±$VT09
= j(-3±>/l09).
Substitute values of y in (3), a? = - 8J, 5, } (4 if 3 VT09).
teachers' edition. 391
20.
x + y + Vx + y = a,
a — y +y/x — y = 6.
4(a; + y) + () + l = 4a + l,
4(g-y) + () + l-4ft + l,
2(a; + y)* + 1 = ±V4a + l,
2(a?-y)* + l=±V46 + l,
2(a? + y)* = - 1 ±V4o + lt
2Qc-y)* = -l±V4& + l,
4(a? + y) - 4a + 1 * 2V4a +1+1,
4(jt -y) - 46 + 1 * 2V46TT + 1,
8aj = 4a + 46+4*2\/4a+1^2V46+l.
.-. s- J(a + 6+l)^J(V4^+I*V46+l).
8y-4o-46*2V5o + i*2Vl6 + l.
.-. y -J(«-ft)*H V4a~+i * V4F+1).
21.
a>*-afy2 + y4=-13 (1)
a»-ay + y2 = 3 (2>
Square (2), x*+3xy+y*-2x*y-2xy* = 9 (3)
Subtract (3) from (1), 2arJy+2&ys-4afy2 = 4.
2
Divide by 2 <ry, a2 — 2 ay +y2 = — (4)
xy
Subtract (4) from (2), xy = 3
.\ aty2-3ajy = -2,
4s2y2-() + 9 = l,
2a:y-3 = ±lf
ay = 2 or 1 (5)
Subtract (5) from (2), x* - 2 xy + y2 - 1 or 2,
a— y = ±lor±V2 (6)
Multiply (5) by 3, and add to (2),
x* + 2xy +y2 = 9or6,
x + y = ±3 or ±\/§ (7)
Add (6) and (7), 2a? = ± 4, ± 2, or ± \/2 ± y/6.
.-. a; = ±2, ±1, or J(±\/2±V6).
Subtract (6) from (7), 2y = ± 2, ± 4, or t V2 ± V&
.-. y = ± 1, ±2, or J(tV2±V6).'
392 ALGEBRA.
22.
x»+y* + x + y = 48
(1)
• 2zy = 24
(2)
Add (1) and (2), a*
'+ 2xy +ya + a? + y = 72,
(s + y)* + (s + y)=72.
Complete the square,
4(x+y)* + () + l = 289.
Extract the root,
2(x + y) + l = ±17,
x + y = 8 or — 9
(3)
From (2),
12
a? = —
y
Substitute value of x in (3), — + y = 8 or - 9,
12+y* = 8y or-9y,
y»-8y = -12,
3^-0 + 16=- 4,
y-4 = ±2.
.:y = 6 or 2.
Also,
12 + y* = -9y,
y* + 9y = -12.
Complete the square,
4y* + () + 81=33,
2y + 9«±V33,
2y 9±V33.
.-.y = J(-9±V33).
Substitute values of y
n(3), *=2, 6, }(~9tV33).
23.
.r2 + xy + y* = a*
a)
a; +>/aJy +y = 6
(2)
Divide (1) by 2,
x-Vxy +y = ^
• 0
P)
Subtract (3) from (2),
2V^-<
Divide by 2,
Squaring,
w
TEACHERS EDITION.
393
Add (1) and (4), x* + 2xy + y2
Extract the root, x + y — ±
From (4),
Add (1) and (6), x* - 2 xy + y'
Extract the root,
(5) is
Add,
Subtract (7) from (5),
a* + 2a262 + b*
(5)
(6)
46*
a* + 62
2b
-3sv = -3(*4-2a262+"4)
y 46*
10q262-3a4-364
' 462
aj-y = ±lyi0a262-3a4-36* (7)
a* + 62
2a; = ±^-^±i-Vl0aa62-3a*--36*.
26 26
2y = ± £±^ t i-Vl0a262-3a4-364.
* 26 26
... x = -^ [±(af + 62)± Vl0a262-3a4-364].
46
•"• y - 71 [*(«* + i2)=FVl0o262-3a4-364].
46
24.
(*-tf-3(*-y)-10 (1)
a?y-3a!y = 54 (2)
Complete the squareof (1), 4(a?-y)2 - ( ) + 9 = 49,
2(x-y) + 3 = ±7.
.vx-y = 5or-2 (3)
Complete the square of (2), 4afy2 - ( ) + 9 = 225,
2 ay - 3 = ± 15.
.-. zy=9or-6 (4)
From (3), y = x— 5 or x + 2.
Hence from (4), ■ a2 — 5a? = 9 or —6.
Therefore x2 — 5x + ^=-^- or h
x-$ = ±i V61, or±J,_
s = 3, 2, orJ(5± V61).
Putting x + 2 for y in (4), we get
3 = -l± VlO, or-l± V-5.
Whence y = -2, — 3, i(-5± V61),
1 ± VlO, 1 ± V^5.
394 ALGEBRA.
.26.
yfx —Vy = x^(x^ + y*)
(l)
(*+y)j = 2(x-y)»
(2)
From (I),
Vx—Vy = x + x*y*
(3)
Square (3),
x — 2Vxy + y = a? + 2xVxy
4-^y.
x — x1 — xy+y = 2xVxy +2Vx~y
(4)
From (2),
x* -f 2xy + y% = 2x* — 4xy +
2y»,
x*+y* = 6xy
(5)
Subtract 2 xy from both sides of (5),
a*-2zy + y, = 4ay.
Extract the root,
a;-y = ±2\/zy.
Substitute x — y i
or 2Vxy in (4),
x— a£ — xy + y = x* — xy + x-
-y.
or ar1 — y = 0.
.-. y = x*.
Substitute x* for
y in (5), a? + 3^ = 63*
(«)
a? = Oorl + a* =
■ 6*.
If
a; = 0, y = 0.
From
l + a? = 6x,
x»_6x = -l,
x»-6a; + 9 = 8,
a-3 = ±>/8,
x = (3±2V2).
Since
y = (3 ± 2 V2)*.
26.
ginvf-
(1)
sy-(a; + y) = 54
(2)
Square (1),
x + y 3*
Simplify,
9x* + ** + 2ay + y* = 6 x» + 6xy,
4a? — 4xy + y* = 0.
2 x - y = 0
Extract the root,
(3)
\ y = 2<c.
Substitute value of y in (2), 2x* — 3a; = 54,
16^-() + 9 = 441,
4s-3 = ±21,
4a; = 24 or -18.
.-. x = 6 or -4}.
.\ y = 12or-9.
TEACHERS* EDITION. 395
27.
.
x + y +Vxy = 28
(1)
a* + y2 + xy = 336
(2)
Divide (2) by (1),
x—Vxy + y = 12
(3)
Subtract (3) from (1),
2Vxy = 16,
a^ = 64
(4)
Add (4) and (2),
a? + 2a;y +y2 = 400.
Extract the root,
x + y = ± 20
(5)
Multiply (4) by 3, and subtract from (2),
x* — 2xy + y2 = 144.
Extract the root, x — y = ± 12 (6)
Add (5) and (6), 2 a? = ± 32 or ± 8.
.\ x = ±16 or ±4.
Subtract (6) from (5), 2y = ±8 or ± 32.
.\ y = ±4 or ±16.
- - Zax = V4a* + 9oa;2 + ?^t
a 4
4a;2 - 12a2a; =4aa?V4a? + 9a -f 27a3,
12a2a; + 27a8 - 4aa?V4a;-+9a = 4x2,
3aa(4a? + 9a) -4aa;V4a; + 9a = 4 a?,
36a2(4a?+9a)-48aa;V4a? + 9a + 16a;2 - 64a;2.
Extract the root, 6a(4a; + 9a)* — 4a; = ± 8a;.
Divide by 2, 6a(4a; + 9a)* = 12a; or - 4a;.
Square, 3a(4a; + 9a)* =* 6a; or -2ar
Transpose, • 36a2x + 81a8 = 36 a? or 4 a2
36a?-36a2a;=81a8,
and 4a?-36a2a; = 81a8.
36a;2- 36a2a; + 9a4 - 9a4 + 81a8,
6a; - 3a2 = ± 3aVa2 + 9a,
6a; = 3a2±3aVa2+9a.
.-. a; = |(a±N/a2+9a).
From 4a;2-36a2a; = 81a3,
4a;2 -( ) + 81a4 - 81a2(a2 + a),
2a;-9a2 = ±9a(a2 + a)*.
.% a,* = -^(a±Va2 + a).
4
396 ALGEBRA.
(x + l+a^Xs-l+ar1)5"5*-
? V'
4x* + 4a* + 4 = 21*»,
4**-17x» = -4,
64a?4- () 4- 289 = 225.
Extract the root, 8 Xs - 17 - ± 15,
8a* = 32 or 2,
x* = 4or}.
, ,\ a? = ±2 or ± J.
30.
2(x*- l)-i-2(x*-4)-1 = 3(x* -2)"1.
2 2 3
xl_l a;4_4 a;*-2
2a: - 12s* + 16 - 2 x + 6 a* - 4 = 3 a; - 15 x* + 12,
-3a; + 9a:* = 0,
3aj*(a* + 3} = 0,
a?i =, _ 3 or 0.
,\ a> = 9or0.
Exercise CIX.
1. log 6 = log (3x2) 3. log 21 = log (7X3)
- log3 + log 2. - log 7 + log 3.
log 3 = 0.4771 log 7 = 0.8451
log 2 = 0.3010 log 3 = 0.4771
.-.log 6 = 0.7781 .-.log 21 = 1.3222
2. log 15 = log (3x5) 4. log 14 = log (7x2)
= log3+log5. =log7+log2.
log 3 = 0.4771 log 7 = 0.8451
log 5 = 0.6990 log 2 = 0.3010
.-. log 15 = 1.1761 .% log 14 = 1.1461
teachers' edition. 397
5. log 35 = log (7x5) 11. log42 = log(7x2x3)
— log 7 + log 5. = log 7 + log 2 + log 3.
log 7 = 0.8451 log 7 = 0.8451
log 5 = 06990 " log 3 = 0.4771
.•.log35-L6441 lo* 2 = O3010
.-.log 42 =1.6232
, log 9 = log(3x3) » ^42°:S^f0-35X7)
-log 3 + log 3. 8+log?+log7.
log 3 = 0.4771 log 2' = 0.6020
log 3 = 0.4771 log 3 =0.4771
.-.log 9 = 0.9542 }°g I -£££?
6 log 7 = 0.8451
.-.log 420 = 2.6232
7- ^ 8:iSf+X4X2?Iogl 13.1ogl2 = log(2x2x3)
lot 2-03010 =log2 + log2 + log3.
b| 2 = a3olo H 1 = 0.3010
lol 2 = 03010 Jog 2 = 0.3010
.Mog 8 = 0.9030 ...log 12 = 1^1
14. Iog60 = log(2x 3X2X5)
8. log49 = log(7x7) = log2»+log3 + log5.
= log 7 + log 7. log 2 = 0.3010
log 7 = 08451 log 2 = 0.3010
log 7 = 0.8451 }°g 3 = 04771
e log 5 = 0.6990
. log 49 = 1.6902
, log 60 = 1.7781
15. log75 = log(5x5x3)
9. log 25 = log (5x5) =log5 + log5+log3.
= log 5 + log 5. ! 5 = 0.6990
log 5 = 0.6990 log 5 = 0.6990
log 5 = 0.6990 log 3 = 0.4771
.\ log 25 = 1.3980 ... log 75 = 1.8751
16. log7.5 = log(3X 5X5X0.1)
10.1og30 = log(2x3x5) = ^+io#+logO 1
= log2 + log3+log5. j 3 -0.4m
log 2 = 0.3010 log 5 = 0.6990
log 3 = 0.4771 log 5 = 0.6990
log 5 = 0.6990 log .1^9.0000-10
.-. log 30 = 1.4771 ... log 7.5 = 0.8751
ay
O " ALGEBRA.
17.
logO. 021 =log(7 X 3 X 0.001)
=log7+log3
+log0.001
log 7=0.8451
log 3=0.4771
log0.001=7.0000-10
22.
23.
24.
)
25.
26.
logl2.5=log(5*X0.1)
=log6+log5+log5
+logO,l.
•log 5=0.6990
log 6=0.6990
log 5=0.6990
. Iog0.021=8.3222-10
log0.36=log(7x 5X0.01)
=log7+log5
+logt).01
log 7=0.8451
log 6=0.6990
log0.01=8.0000-10
log 0.1— 9.0000— 10
18.
.log 12. 5= 1.0970
log 1.25=log(5*X 0.01)
=log6-Hog5-flog5
-HogO.Ol.
log 5=0.6990
log 5=0.6990
log 6=0.6990
log0.01=8.0000-10
. Iog0.35=9.5441-10
log0.0035=log(7 X 6 X 0.0001
=log7+log6
-HogO.0001.
log ' 7=0.8451
log 5=0.6990
logO. 0001 =6.0000-10
19.
.log 1.25=0.0970
Iog37.5=log(58x3x0.1)
=log5+log6+log6
+logS+log0.1.
log 6=0.6990
log 5=0.6990
log 5=0.6990
log 3=0.4771
log 0.1=9.0000-10
.
.log0.0035=7.5441-10
log0.004=log(2 X 2 X 0.001)
=log2+log2
+log0.001.
log 2=0.3010
log 2=0.3010
log0.001 = 7.0000-10
20.
.log37.5=1.5741
log2.1=log (3X7X0.1)
=log3+log7+log0.1.
log 3=0.4771
•log 7=0.8451
logO. 1=9.0000-10
.log2. 1=0.3222
21.
.log0.004=7.6020-10
Iog0.05=log(5x0.01)
=log5+log0.01.
log 5=0.6990
logO.Ol =8.0000-10
logl6=log(2*)
=log2+lpg2
+log2+log2.
log 2=0.3010
log 2=0.3010
log 2=0.3010
log 2=0.3010
.•
. Iog0.05=8.6990-10
.log 16=1. 2040
teachers' edition.
399
27. logO. 056=log(2X2X2X 7x0.001)
=log2+log2+log2+log7+log0.001.
log 2=0.3010
log 2=0.3010
log 2=0.3010
log 7=0.8451
log0.001=7.0000-10
.-.log0.066=8. 7481-10
28. Iog0.63=log(3x 3x7x0.01) 30.
=log3+log3
+Iog7+log0.01.
log 3=0.4771
log 3=0.4771
log 7=0.8461
logO.Ol =8.0000-10
Iogl05=log(6x3x7)
=log5xlog3xlog7.
log 6=0.6990
log 3=0.4771
log 7=0.8461
.log 105=2.0212
.-.logO. 63=9. 7993-10
31.
29. logl.75=log(5X5x7x0.01)
=log6+log5
+Iog7+log0.01.
log 5=0.6990
log 5=0.6990
log 7=0.8451
log0.01=8.0000-10
Iog0.0105=log(3x7x6
X 0.0001)
=log3+log7+log5
-flogO.0001.
log 3=0.4771
log 7=0.8451
log 5=0.6990
logO. 0001 =6.0000-10
.logl.75=0.2431
. Iog0.0105=8.0212-10
32. Iogl.06=log(7x3x5x0.01)
=log7+log3+log6+log0.01.
log 7=0.8451
log 3=0.4771
log 6=0.6990
log0.01=8.0000-10
.-.logl.05=0.0212
Exercise CX.
1. log28=3xlog2
=3X0.3010
=0.9030.
log74=4xiog7
=4X0.8451
=3.3804.
2. log52=2Xlog6
=2X0.6990
=1.3980.
4. log38=8xlog3
=8X0.4771
=3.8168.
400 ALGEBRA.
5. log7* = 3xlog7 16. W 7* = *oflog7
= 3x0.8451 6 = I of 0.8451
= 2.5353. =5.2415.
6. log 5s = 5 xlog5. ,
= 5x0.6990 17. log 5* = $ of log 5
= 3.4950. =fof0.699C
= 1.1650.
7. log2* = lofloe2
= i of 0.3010 18. log 3* = -X of log 3
= 0.1003. B = A of 0.477
l\ OCK\A
8. log 5* = J of log 5
[771
= 0.3904.
03195 19-l08 ^ = ioflog7
°-3495- = J of 0.8451
= 2.9579.
9. log 3* = i of log 3
1^96 »k8 3* = 4oflo?3
= 0.059b. = I of 0.4771
= 0.6361.
10. log 7* = J of log 7
- f of 0.8451 -
= 0.1690. 21. log 5*= $ of log 5
= |of0/™
11. log 5* = t of log 5
= *of0.P~
(990
2.4465.
,■ ux v.6990 u
6.1398. 22. log 2 V= J^. of log 2
= V of 0.3010
,0 , *A 1 m * =0.4730.
12. log 7A= -A- of log 7
= -A of 0.8451
-0.0768. 23. log 5t = foflog5
= I of 0.6990
13. log 2* = } of log 2 =0.5243.
= | of 0.3010
~ °2258- 24. log 7*= V of log 7
= V- of 0.8451
14. log 5* = i of log 5 =1.3280.
-J of 0.6990
" °,466a 25. log 21* = i of log(7x 3).
log 7 =0.8451
15. log 3? = 4 of log 3 log 3 = 0.4771
= 4 of 0.4771 log 21 =1.3222
= 0.2045. i of 1.3222 = 1.1569.
teachers' edition. 401
Exercise CXI.
1. log $ = log 2 + colog 5. 8. log § = log 5 + colog 2.
log 2 = 0.3010 log 5 - 0.6990
colog 5 = 9.3010 - 10 colog 2 = 9.6990 - 10
. .. log i = 9.6020 - 10 . \ log { = 0.3980
2. log $ = log 2 + colog 7. 9. log } = log 7 + colog 3.
log 2 = 0.3010 log 7 = 0.8451
colog 7 - 9.1549 - 10 coiog 3 - 9.5229 - 10
.-. log * = 9.4559 - 10 ... i0g j = 0.3680
3. log f = log 3 + colog 5. ,*,.,* i o
log 3 -0.4771 10- log* -log 7 + colog 2.
colog 5 = 9.3010 -10 log 7 = 0.8451
5 colog 2 = 9.6990 -10
.-.log $ = 9.7781 -10
6. log $ = log 7 + colog 5.
log 7 = 0.8451
colog 5 = 9.3010 -10
log} = 0.5441
4. log $ = log 3 -f colog 7.
log 3 = 0.4771 n. bg f = log 3 + colog 2.
colog 7 - 9.1549 - 10 log 3 = 0 4771
. .. log } = 9.6320 - 10 colog 2 = 9.6990-10
.-.log J = 0.1761
5. log f = log 5 + colog 7.
log 5 = 0.6990
colog7 = 9.1549-10 ^ log^=log7 + cok*a*
,.log* = 9.8539-10 JogWMSl
colog 0.5= 0.3010
log 7 = 0.8451 . • . log j^: = 1. 1461
.-.logJ = 0.1461 006
1$. Iog-jp = log0.05+colog3.
7. log i - log 5 + colog 3. Iog0.06 = 8.6990-10
log 5 = 0.6990 colog 3 _ 9 5229-IO
colog 3 -9.5229-10 n nft
, . ZZZ2 .-.log^ = 8.2219- 10
.\ log f -0.2219 3
402
ALGEBRA.
14. log
0.005
Iog0.005+colog2. 20. log
logO. 005 = 7.6990 -10
colog 2 = 9.6990 — 10
.••log^ = 7.3980-10
0.007
: logO. 007 -fcologO.02.
0.02
log0.007 = 7.8451 -10
colog 0.02 = 1.6990
••1°^7=e-5441-10
15. log -J7— = log0.07 + colog 5.
• log 0.07 = 8.8451-10
colog 5 = 9.3010 — 10
...log 5^1=8.1461-10
o
21. Iog^| = log0.02+colog0.0f;7.
log 0.02 = 8.3010-10
colog0.007 = 2.1549
•••l0W7 = °-4569
16. log — =log5 + colog0.07.
log 5 = 0.6990
colog 0.07 = 1.1549
22. Iog^^ = log0.005+colog0.07.
Iog0.005 = 7.6990- 10
colog 0.07 = 1.1549
.'.log^ = 8.8539-10
17. Iogj^y = log3 + colog0.007.
log 3 = 0.4771
colog0.007 = 2.1549
3
.log
0.007
= 2.6320
23. log ^ = log0.03 + colog7.
log 0.03 = 8.4771-10
colog 7 = 9.1549-10
0.03
•.log ^=7.6320-10
18. log
0.003
logO. 003 + colog7.
Iog0.003 = 7.4771 -10
colog 7 = 9.1549-10
...log2^? = 6.6320 -10
**- log^^ = log0.0007+cologOi
logO.0007 = 6.8451-10
colog 0.2 = 0.6990
, 0.0007 Z~77T<i ™
.•.log-^y- = 7.5441-10
19- Iog^| = log0.05+colog0.003. 25. Iog^^=log0.022 + colc
log 0.05 = 8.6990-10
colog 0.003 = 2.5229
logO. 022 = 6.0O2O -10
colog 38 = 8.5687-10
0 022 "
...log^- = 5.1707-10
TEACHERS EDITION.
403
33
26. log — = log 33 + cologO.022.
log 38=1.4313
colog0.022 = 3,3980
•••logofe=4-
27. Iog-^ = log78 + colog0.022.
log 78 = 2.5353
colog0.022 = 3.3980
78
.log
0.02*"
: 5.9333
28. Iog|^8 = log0.078 + colog0.003».
log 0.078 = 6.6353-10
cologO. 0038 =7.5687
0.078
.log^ro = 4.1040
'O.OOS3
29. Iog^^ = log0.0052+colog78.
logO. 0062 = 6.3980 -10
colog 78 = 7.4647-10
...log2^ = 2.«7-10
30. log-
*0.0062
log 78=2.5363
colog0.0052 = 4.6020
= log78 + colog0.006a.
Exercise CXII.
log 60 = 1.7782.
- 6.
log 3780 = 3.5775.
log 101 = 2.0043.
3.
log 999 = 2.9996.
log 54327 = 4. 7348 + Jfo of 0.0008
= 4.7350.
Iog90801 = 4.9581 + Tfo of 0.0005
= 4.9581.
Iog9901 = 3.9966 + Jfc of 0.0005 9t
= 3 9957 log 10001 = 4.0000 + T^ of 0.0043
= 4.0000.
5. 10.
Iog5406 = 3.7324 + ^ of 0.0008 log 10010 = 4.0000 + ^ft of 0.0043
= 3.7329. =4.0004.
404 ALGEBRA.
11. 22.
log 70633=4.8488+^ of 0.0006 Antilogarithm of 3.6330.
=4.8490. Number corresponding to 0.633Q
12.
log 12028=4.0792+^ of 0.0036
=4.0802.
is 4290+T5ff of 10=4295.
. * . number required is 4295.
13.
Antilogarithm of 2.5310.
, ~ ~~.™ -~~,« .~ Number corresponding to 0.5310
log0.00987=7.9843-10. - fa 3390+A of^3396^
]^ . • . number required is 339.6.
logO.87701 =9.9430-10. 24.
^ Antilogarithm of 1.9484.
* ,™, ~~™, , **^m* Number corresponding to 0.9484
log 1.0001=0.0000+^ of 0.0043 ^ ^
=0.0000. . . ,. QOQ
. • . number required is 88.8.
16- 26.
log 877.08=2.94^+TJ5of0.0005 AnffloglHWim rf 4W17.
Number corresponding to 0.7317
17 is 6390+ 1 of 10=5391.
log 73.896= 1.8681 +^ of 0.0005 • ' • numl)er required is 53910.
=1.8686. M
jg Antilogarithm of 1.9730.
log 7.0699=0.8488+^ of 0.0006 Number corresponding to 0.9730
=0.8494. is 9390+f of 10=9398.
. • . number required is 93.98.
19.
log 0.0897=8.9628-10. **-
Antilogarithm of 9.8800-10.
20. Number corresponding to 0.8800
log 99.778=1.0987+^ of 0.0004 is 7680+ J of 10=7686.
= 1.9990. . • . number required is 0.7586.
2L 28.
Antilogarithm of 4.2488. Antilogarithm of 0.2787.
Number corresponding to 0.2488 Number corresponding to 0.2787
is 1770+| of 10=1773. is 1890+fJ of 10=1900.
. • . number required is 17730. . • . number required is 1.9.
TEACHERS EDITION.
405
29.
Antilogarithm of 9.0410—10.
Number corresponding to 0.0410
is 1090+fJ of 10=1099.
. • . number required is 0.1099.
30.
Antilogarithm of 9.8420—10.
Number corresponding to 0.8420
is 6950.
. • . number required is 0.6960.
31.
Antilogarithm of 7.0216—10.
Number corresponding to 0.0216
is 1050+^- of 10=1061.
. • . number required is 0.001051.
32.
Antilogarithm of 8.6580—10.
Number corresponding to 0.6580
is 4550.
.•. number required is 0.0455.
33.
948.76x0.043875.
log -948.76 = 2.9772
log 0.043875 =8.6423 -10
1.6195
= log 41.64.
34.
3.4097 X 0.0087634.
log 3.4097 = 0.5328
log 0.0087634 - 7.9427 - 10
8.4755 - 10
= log 0.02989.
35.
830.75x0.0003769.
log 830.75 = 2.9195
log 0.0003769 = 6.5762 - 10
9.4957-10
- log 0.3131.
36.
8.4395 x 0.98274.
log 8.4395 = 0.9263
log 0.98274 = 9.9925 - 10
0.9188
= log 8.294.
37.
7564 x(- 0.003764).
log 7564 = 3.8787
log (-0.003764) = 7.5756* - 10
1.4543*
- log -28.47.
38.
3.7648 x(- 0.083497).
log 3.7648 = 0.5757
log (- 0.083497) = 8.9217* -10
9.4974*- 10
- log -0.3144.
39.
-5.840359 x(— 0.00178).
log (-5.840359) = 0.7664*
log (-0.00178) = 7.2504* -10
8.0168 - 10
= log 0.0104
40.
- 8945.07 X 73.846.
log (-8945.07) = 3.9515*
log 73.846 = 1.8683
5.8198*
= log -660600.
406
ALGEBRA.
41.
- log 70654 + colog 54013.
70654
54013
log 7Q§54 = 4.8491
colog 54013 = 5.2675
0.1166
= log 1.308.
10
43.
M?i^ = log 8.32165
0.07891 e -fcolog 0.07891.
log 8.32165 = 0.9202
colog 0.07891 = 1.1028
2.0230
= log 105.4.
58706
= log 58706 + colog 93078.
65039
93078
log 58706 - 4.7686
colog 93078 = 5.0312 - 10
9.7998 - 10
- log 0.6307.
44.
log 65039 + colog 90761.
4.8132
90761
log 65039
colog 90761 = 5.0421 - 10
9.8553 - 10
= log 0.7167.
7.652
45.
■ log 7.652 + colog (- 0.06875).
- 0.06875
log 7.652= 0.8838
colog (- 0.06875) = 11.162?
2.0465"
- log -111.3,
10
0.07654
83.947 X 0.8395
log 0.07654 =
colog 83.947 =
colog 0.8395 =
46.
= log 0.07654 + colog 83.947 + colog 0.8395.
8.8839 - 10
8.0760 - 10
0.0759
7.0358 - 10
dog 0.001086.
47.
7564 x 0.07643
8093 x 0.09817'
log 7564 = 3.8787
log 0.07643 = 6.8832-10
colog 8093 = 6.0919 - 10
colog 0.09817 = 1.0080
9.8618 - 10
-log 0.7277.
48.
89 X 753 x 0.0097
36709 x 0.08497 '
log 89 = 1.9494
log 753 = 2.8768
log 0.0097=7.4352-10
colog 36709 = 5.4352-10
colog 0.08497 = 1.0708
9.3190-10
= log 0.2084.
teachers' edition.
407
49.
413 x 8.17 X 3182
915 x
728 x
2.315
log
413
= 2.6160
log
8.17
= 0.9122
log
colog
3182
= 3.5027
915
= 7.0386 -
10
colog
728
= 7,1379 -
10
colog
2.315
= 9.6354 -
0.8428
= log 6.963
50.
10
212 x
(- 6.12) X(- 2008)
365 X (- 531) x 2.576
log 212 - 2.3263
log (-6.12) = 0.7868*
log (- 2008) = 3.3028*
colog 365 = 7.4377 -10
colog (-531) = 7.2749* -10
colog 2.576 = 9.5891 -10
0.7176*
= log -5.21 9.
51.
log 6.05 =0.7818
3
log 6.05s = 2.3454
- log 221.5.
log 1.051 =0.0216
7
log 1.0517 = 0.1512
= log 1.416.
53.
log 1.1768 =0.0707
5
log 1.17685 = 0.3535
= log 2.257.
54.
log 1.3178 =0.1198
10
log 1.317810= 1.1980
= log 15.78.
55.
log 0.78765 =9.8963-10
log0.787656 = 9.3778 -10
= log 0.2387.
56.
log 0.691 =9.8395-
9
10
log 0.6919 = 8.5555 -10
= log 0.03593.
57.
log (ft)11 = 11 (log 73 + colog 61)
= 11(1.8633+8.2147-10)
= 0.8580
= log 7.212.
58.
log (M)7 = 7 (log 14+ colog 51)
= 7(1.1461 + 8.2924-10)
= 6.0695-10
= log 0.0001174.
59.
(io|)* -(w.
-~)* = 4(logc
= 4(1.505f+ 9.5229 -10)
log(W
1 32 + colog 3)
= 4.1120
= log 12940.
60.
(1
log (iff = 8 (log 16 + colog 9)
= 8(1.2041 + 9.0458-10)
= 1.9992
= log 99.82
408 ALGEBRA.
61. log(Hi)6 = 6 (log 951+ colog 823)
= 6(2.9782 + 7.0846-10)
= 0.3768
- log 2.381.
62. (7*)"" =(H)0J8.
log (ff)0-88 - 0.38 (log 83 + colog 1 1)
- 0.38(1.9191 + 8.9586 - 10)
- 0.3335
= log 2.155.
63. (3|B« =(W)41T-
log (Vt1)4*17 = 41 7 (log 120 + colog 31)
- 4.17(2.0792 + 8.5086 - 10)
- 2.4511
= log 282.6.
64. (W -(H)*
log (H/8*2 = 3.2(log 13 + colog 11)
= 3.2(1.1139 + 8.9586-10)
= 0.2320
= log 1.706.
65. (8})" '-(W.
log (V)2'8 * = 2-3 (log 35 + col°g 4)
= 2.3(1.5441 + 9.3979 - 10)
- 2.1666
= log 146.8.
66. (5||f875 =(Wf875-
log(W)0'375 ■= 0.375(log 216 + colog 37)
= 0.375(2.3345 + 8.4318 - 10)
= 0.2874
= log 1.938.
teachers' edition. 409
67. log 7 = 0.8451. 71. log 906.80=2.9575.
3)0.8451 4)2.9575
log 7* = 0.2817 log 906.80* = 0.7394
= log 1.913. -log 5.487.
72. log 8.1904 = 0.9133.
68. log 11 = 1.0414. 5)0.9133
5)1.0414 log 8.1904* = 0.1826
log 11* = 0.2083 = log 1.523.
= log 1.616.
73. log 0.17643 = 9.2466 - 10
5
69. log 783 = 2.8938. 46.2330 - 50
31|^ 6)56.2330-60
log 783* = 0.9646 0.17643*= 9.3722- 10
= l089-218- =log0.2356.
74. log 2.5637 =0.4088
70. log 8379 = 3.9232. 3
10)3.9232 11)1.2264
log 8379* = 0.3923 log 2.5637* = 0.1115
= log 2.468 = log 1.293.
75. log («i)* = l(log 431 + colog 788).
log 431 -
colog 788 =
431 - 2.6345
7.1035 - 10
9.7380 - 10
10. - 10
2) 19.7380 -"20
9.8690 -10 = log 0.7397.
76. log (*#**)* = * (log 71 + colog 43406).
log 71 = 1.8513
colog 43406 = 5.3624 - 10
7.2137 - 10
4
28.8548 - 40
30. - 30
7)58.8548-70
"" 8.4078 - 10 = log 0.02558.
410 ALGEBRA.
77. (9H)* = (W)*-
log (W)* = iflog 408 + col°g 43)-
log 408 = 2.6107
colog 43 = 8.3665 - 10
5)0.9772
0.1954 = log 1.568.
8. (llftf-W)1- , , „„
log (W)1 = i (log 802 + colo8 71)'
log 802 = 2.9042
colog 71 = 8.1487 - 10
1.0529
$ (1.0529) = 0.8423 = log 6.955.
?9 6/0.0075433' x 78.343 x 8172.4* X 0.00052
* 64285* X 154.27* X 0.001 x 586.79*
log0.0075433*= 5.7552 - 10
log 78.343 = 1.8940
log 8172.4* = 1.3041
log 0.00052 = 6.7160 - 10
colog 64285* = 8.3973 - 10
colog 15427* = 1.2468 -10
colog 0.001 = 3.0000
colog 586.79* = 8.6158 -10
5)36.9292-50
7.3858 -10 = log 0.002431.
g0 •; 15.8328 x 5793.6* X 0.78426
^0000327* X 768.94* X 3015.3 X 0.007*
log 15.832s = 3.1
log 5793.6* = 1.2543
log 0.78426 = 9.8445-10
colog 0.000327* = 1.1618
colog 768.942 = 4.2282 -10
colog 3015.3 - 6.5207 - 10
colog 0.007* = 1.0774
27.7357 - 30
20. - 20
5)47.7357-50
9.5471 -10 = log 0.3525.
TEACHERS' EDITION. 411
81.
5/7.
82.
7.1895 x4764.22x 0.00326s
00048953 X 457s X 5764.42"
log 7.1895 = 0.8566
log 4764.2s = 7.3558
log 0.003265= 7.5660 -20
colog 0.00048953 = 3.3102
colog 457s = 2.0203 -10
colog 5764.43 = 2.4786 - 10
23.5875-40
10. - 10
5)33.5875-50
6.7175 -10 = log 0.0005218.
& '3.1416x4771.21x2.7183*
> 30.103* x 0.4343* x 69.897*
log 3.1416 = 0.4971
log 4771.21 = 3.6786
log 2.71834 = 0.2172
colog 30.103* = 4.0856 - 10
colog 0.4343*= 0.1811
colog 69.897* = 2.6220 - 10
11.2816-20
30. - 30
5)41.2816-50
8.2563 -10 = log 0.01804.
03 t '0.032712 X 53.429 x 0.77542s
"•a!
32.769 x 0.000371*
log 0.03271* = 7.0292-10
log 53.429= 1.7278
log 0.77542s = 9.6688-10
colog 32.769= 8.4845-10
colog 0.000371* = 13.7224
7) 10.6327
1.5190 = log 33.04.
732.056' x 0.0003572* x 89793
1.2798s x 3.4574 x 0.00265185'
log 732.0562= 5.7290
log 0.0003572*= 6.2116-20
log 89793= 4.9533
colog 42.2798s = 5.1217 -10"
colog 3.4574= 9.4612-10
colog 0.00265185 = 12.8825
3)4 3593
1.4531 = log 28.39.
412 ALGEBRA.
3- »/7932x 0.00657x0.80464
\ 0.03274x0.6428
log 7932 = 3.8994
log 0.00657 = 7.8176 - 10
log 0.80464 = 9.9056 - 10
colog 0.03274 = 1.4849 - 10
colog 0.6428 = 0.1919
3)3.2994
1.0998 = log 12.58.
86.
^
7.1206 xVO.13274 X 0.057389
V0.43468 X 17.385 x V0.0O96372
log 7.1206 = 0.8525
log V0.13274 = 9.5615 - 10
log 0.057389 = 8.7588 - 10
colog V0.43468 = 0.1809
colog 17.385 = 8.7599-10
colog V0.0096372 = 1.0080
3)29,1216-30
9.7072 - 10 = log 0.5096.
_ f 3.075526* x 5771.2* X 0.0036984* X 7.74 \ *
t 72258 X 327.93* X 86.97* J
log 3.075526s = 0.9758
log 5771.2* - 1.8806
log 0.0036984* - 9.5136 - 10
log 7.74 = 0.8887
colog 72258 = 5.1412-10
colog 327.93* - 2.4526 - 10
colog 86.97* = 0.3030 - 10
1.1555-20
3
3.4665- 60
40 - 40
5)43.4665-100
8.6933- 20
= log 0.000000000004936.
teachers' edition. 413
Exercise CXIII.
1. Write down the ratio compounded of 3 : 6 and 8 : 7. Which
of these ratios is increased and which is diminished by the com-
position?
fxf-tf.
As f-fi
and f-«.
3 : 5 is increased.
8 : 7 is decreased.
2. Compound the duplicate ratio of 4 : 15 with the triplicate
of 5 : 2.
3. Show that a duplicate ratio is greater or less than its sim-
ple ratio according as it is a ratio of greater or less inequality.
a? : b2 is > or < a : b.
As
6*
is>or<~
6
Asf
a a
bXb
is>or<£.
6
As
a
b
is > or < 1.
4. Arrange
in order of magnitude the
: ratios
3
:4;
23;
:25;
10
s 11; and 15
:16.
3
23
10
15
. •. the order of magnitude is 15 : 16, 23 :
25, 10 :
11,
3:
4.
414 ALGEBRA.
5. Arrange in order of magnitude
a + b : a — b and a2 + 6* : a* — &*, if a> b.
a + 6 : a - 6 is > or < a* + 6* : a2 - 6'.
A, «±|iB>or<4±|.
a — b a* — 6*
^ a« + 2ai + yig>or<^±y.
a* — 6* a2 — 6a
As a* + 2 aft + b* is > or < a* + 6*.
But a* + 2a& + &,is>a, + &*.
. • . a + b : a - b is > a* + &* : aJ - 6*.
6. Ratio compounded of
3:5; 10:21; 14:15,
= *xtfxtt
-A
-4:15.
7. Ratio compounded of
7:9; 102:105; 15:17,
= 2:3.
8. Ratio compounded of
a* + ax + sc1 , a2 - ax 4- a*
a8 — cfo + flKc2— a8 a + a;
_ a4 + a V + g*
a*-s*
9. Ratio compounded of
s2-9a?4-20 andga-13a;+42
sc2 — 6a: a2 — 5a?
(s-5)(g-4) (g-fl)(s-fl
s(a>-6) a>(a;-5)
-(a* -11a; + 28) jar2.
teachers' Edition. 415
10. Ratio compounded of
a + b : a - b ; a2 + b2 : (a + b)2 ; (a2 - b2)2 : a* - 6*.
= a + b a2 + b2 (a2 - V)2
a-b (a + bf a*- b*
= a + b a2 + b2 (a-b)(a + b)(a-b)(a + b)
a- 6 (a + 6)(a + 6) (a2 + b2) (a + 6) (a - 6)
= 1:1.
11. Two numbers are in the ratio 2 : 3, and if 9 is added to
each, they are in the ratio 3 : 4. Find the numbers. •
Let 2x and 3 s = the numbers.
Then 2x + 9: 3a? +9: : 3: 4,
8a; + 36 :9s + 27.
.\ x = 9.
Hence, 2 a =18,
and 3 a; = 27.
12. Show that the ratio a : b is the duplicate of the ratio
a + c : b + c, if c* == ab.
a = fa + c\2
mes, when
6 \6+i
This becomes, when c2 = aft,
■Va6V.
\-Vab)
a _a2 + 2aVa6 + aft
6 62 + 26 Va6 + a&
a(o + 2Va6 4- 6)
b(b + 2Vab + a)
13. Find two numbers in the ratio 3 : 4, of which the sum is
to the sum of their squares in the ratio of 7 : 50.
Let 3 a; = first number,
and 4 a; = second number.
Then 3a; + 4a; or 7a; = sum.
dx2 + 16a;2 or 25a;2 = sum of squares.
.-. 7a; : 25a;2 : : 7 : 50, or x : x2 : : 1 : 2, or 1 : x : : 1 : 2.
.-.a; = 2.
The required numbers are 6 and 8.
416 ALGEBRA.
14. If five gold coins and four silver ones are worth as much
as three gold coins and twelve* silver ones, find the ratio of the
value of a gold coin to that of a silver one.
Let
x = value of 1 gold coin,
and
y — value of 1 silver coin.
Then
5z + 4y = 3a:-f 12y,
2a = 8y.
.-. & = 4y.
That is,
x.y.A: 1.
15. If eight gold and nine silver coins are worth as much as
six gold and nineteen silver coins, find the ratio of the value of
a silver coin to that of a gold one.
Let x = value of gold coin,
and y = value of silver coin.
Then 8x + 9y = 6x + 19y,
or 2o; = 10y.
.-. x=*5y.
That is, y : z : : 1 : 5.
16. There are two roads from A to B, one of them 14 miles
longer than the other; and two roads from B to C, one of them
8 miles longer than the other. The distance from A to B is to
the distance from B to C, by the shorter roads, as 1 to 2 ; by the
longer roads, as 2 to 3. Find the distances.
Let x = shorter road from A to B,
and x + 14 = longer road from A to B.
Then y = shorter road from B to C,
and y + 8 = longer road from B to C.
That is, x : y : : 1 : 2,
x + 14 : y + 8 : : 2 : 3.
.'. 2x = y (1)
And 3»+42 = 2y + 16,
or 3a;-2y--26.
Substitute 2x for y, x = 26,
x + 14 - 40,
y = 52,
y + 8 = 60.
teachers' edition. 417
17. What must be added to each of the terms of the ratio
m : n, that it may become equal to the ratio p: q?
Let
x--
= number to be added.
. m + x
9.
mq + qx-
x(q-p)>
.: x--
=pn +pxy
= pn — mq.
pn — mq
18. A rectangular field contains 5270 acres, and its length is
to its breadth in the ratio of 31 : 17. Find Its dimensions.
Let 31 x = number of rods in length,
and 17 a: = number of rods in width.
Then 31a; X 17a; = number of square rods in area.
But 160 X 5270 = number of square rods in area.
.-. 31a; X 17a; = 160x5270.
527 a:2 = 160x5270,
a8 - 1600,
a? = 40,
31 x =1240 rods,
17a; = 680 rods.
Exercise CXIV.
1. Ita:b::c:d,
2. 3a + b:b::Sc + d:d.
a c
b~d
Ifa:b::c:d, £-1
b d
Multiply by -,
n
Multiply by 3,^ = ^.
o d
ma me
Add 1 to each side,
nb nd
That is,
3« + 1 = 3c+]
o a
ma : nb : : mc :
nd.
3a + o_3c + d
b d
.-. 3a + 6: o: : 3c + d: d
418 ALGEBRA.
3. Ifo:5::C:d, 5- a ■ <* + b: '■ c : c + d-
. a c
then - = -•
b a
Add 2 to each side,
6 a
o4-26 c + 2d
If a : 6 : : C : d, T =
c
6
d
By inversion, - =
d
a
c
By composition,
6+ a
d+ c
a
c
By inversion, =
J a +6
c
d + c
.-. a: a + 6:
: c: C + d.
b d
,a + 2b:b::C + 2d:d.
6. a:a — b::C:c — d.
If a:b::C:d, - = — •
b a
4. Since «: 6: :«:<*, By inversion, l-t
a c
a c
b d By division, _-H- = ~ •
a c
Cubing, gj = Jy B inversion| _«_ _ «
°^ a J a-b c—d
. ■. os : 6s : : c* : ds. • % a : a — 6 : : c : c — d
7. If a: 6:: e: (2,
a= c
b d
m ma mc
Multaplyby-, --^
By composition and division,
ma + nb _mc + nd
ma — n& mc — nc?
. \ ma + w6 : rwa — n6 : : mc + nd : me — nrf.
8. If a : 6 : : c : d,
Also,
£ = £.
b~ d
a , 3=c 3
6 2 d 2'
2a-f36_ 2c + 3d
26 2d
. 2a + 3& = 2c + 3rf
6 d
a__4 = c _4
6 3 <* 3'
(1)
teachers' edition. 419
Dividing (1) by (2),
3a-46_3c-4d
36 3d
,3q-4&_3c-4d
b d
2a + 3b 2c + 3d
(2)
3a-46 3c-4d
.-. 2a+3&:3a-46::2c + 3d:3c-4d
9. If a : b : : C : d,
By squaring,
6 d
b2 d2'
• ma' = nc>L
m&2 nd2
Let i^ = n
nd2
Hence, ma2 = mb2r, and nc2 = ndV,
ma2 + nc2 = (mb2 + nd2) r,
m&2 + nd2 62
.*. ma2 + nc2 : nib2 + nd2 : : a2 : 62.
Then
10.
If a: b:
: c : d, by alternation, a : c :
:b:d.
a _
c
b
a2
b2
= d2
Also,
c c
d c
a2
ab_
= cd
62
d2'
, ™&2
mc2
nab _
ncd
"pd2*
. ma2
1 + nab + pb2
jp62
62
mc2
+ ncd +pd2
pd2
d2
.•. ma* + nab +pb2 :
mc2 + ncd +pd2 : : b
2 : d2.
420 ALGEBRA.
11. If a:b::b:c,
by composition, a + b : a: :b + c: d;
by alternation, a + b:b + c: :a:b.
12. Ifa:6::6:C,
a=6 ;
b c
Multiply" g-g |
or a2 : ft2 : : ab : 6c.
By alternation, a2 : aft : : 62 : be.
By composition, a2 + ab:db : :b* + bc:bc.
By alternation, a* + ab : 6* + 6c : : ab : be.
Cancelling b in the terms of last ratio,
a2 + ab : b2 + be : : a : c.
13. Ifa:6::6:c, 6» = ac.
Multiply by (a — c), ab2 - b2c = a2c — ac2.
Add 2a6c to both sides,
ab2 + 2a6c — b2c = a2c + 2a6c — ac3.
Transpose — b2c and —'ac2,
all2 + 2a6c + ac2 = a2c + 2a6c + b2c,
or a(62 + 26c + c2) = c(a2 + 2a6 + 62),
or a(b + c)2 = c(a + 6)2.
Divide by c(6 + c)2, "• = (^±_^!,
c (6 + c)2
or a : c : : (a 4- 6)2 : (6 -f c)2.
14. When a, 6, and c are proportionals, and a the greatest,
show that a + c > 2 b.
a : 6 : : 6 : C.
Since ? = - and a > 6,
6 c
.-. 6>c.
Also, since by division a-^— = — ^ and 6 > c,
6 c
.-. a — 6>6 — c.
By adding, 6 + c = 6 + c,
a + c>26.
teachers' edition. 421
15. If - — y. = ? — - = - — - , and x, y, z are unequal, then I + at
I m n
+ n = 0.
Let — -i « r, * = r, -= r.
cm n
Then x — y = lr%
y — z = ww\
z — a; = nr.
* — V + y — z + z — x = (l + m + n)rt
or 0 = (Z + m + n) r.
. \ Z + m + n — 0.
16. Find a when x + 5: 2a?- 3 : : 5a? + 1 : 3*- 3.
Equate the product of the means and the product of the
extremes, 10a.a _ i3a. _ 3 = 3^2. + 12a. _ 15,
7a;2 -25* = -12,
• 196a* -() 4- 625 = 289,
14s -25 = ±17,
14a; = 42 or 8.
. *. x -» 3 or ^.
17. Find x when a; + a : 2a; — b : : 3a; + b : 4a? — a.
x + a _ 3x+ b
2x — b 4a?— a
Clear of fractions, 4 x* + Sax + a2 = Gx* — bx — 6s,
2a?-a?(3a + &) = (&2-a2),
16a*- () + (3a + bf = a2 + Gab + 9&2,
4a?-(3a + 6) = ±(a+3&),
4a; = 4a + 46 or 2a — 2b.
. r a — 6
. •. jc = a + 0 or
2
18. Find x when
\/x + V& : Vx—Vb: : a.b.
bVx + bVb =* aVx — aVb,
(a-b)Vx = (a + b)Vb,
^ __ (a -f 6) V6
a — ft
. 3._(a2 + 2a& + &*)&
a2-2a& + 62
422 ALGEBRA.
19. Find x and y when x : 27 : : y : 9, and a; : 27 : : 2 : x -y.
x : 27 : : y : 9.
.-. x = 3y (1)
x:27::2:x-y.
•\ X* — xy = 54.
Substitute 3y for x, 9y2 - 3y2 - 54,
6y* = 54,
y« = 9.
.-.y = ±3.
Substitute values of y in (1), x = ± 9.
20. Find & and y when x + y + 1 :x + y+2::6:7, and when
y + 2x:y-2x::12x + 6y-3:6y — 12ac — 1.
x + y + l:x + y + 2::6:7.
By division, x + y+l:l::6:l.
.\x + y + l = 6,
or x + y = 5 (1)
y + 2x : y - 2x : : 12x + 6y-3 : 6y-12x— 1.
By composition and division,
2y:4x::12y-4:24x-2,
or y:2x::6y-2:12x-l.
••. 12xy — y = 12xy — 4x.
.'.4x-y (2)
From (1) and (2), x = 1,
and y = 4.
21. Find x when
x2-4x + 2:x2-2x-l.:x2-4x:x2-2x-2.
By alternation,
x,-4x + 2:x2-4x::x2-2x-l:xa-2x— 2.
By division, 2 : x2 — 4 x : : 1 : a?2 — 2x — 2.
.-. 2x2-4x-4 = x*-4x.
.-.Xs -4.
.-. x = ±2.
22. A railway passenger observes that a train passes him,
moving in the opposite direction, in 2 seconds ; but moving in
the same direction with him, it passes him in 30 seconds. Com-
pare the rates of the two trains.
teachers' edition. 423
Let x = rate of the faster train,
and y = rate of the slower train.
Then x + y : x - y : : 30 : 2.
By composition and division,
2a; :2y:: 32: 28.
.\s:y::8:7.
23. A and B trade with different sums. A gains $ 200 and B
loses $ 50, and now A's stock : B's : : 2 : J. But, if A had gained
#100 and B lost #85, their stocks would have been as 15: 3 J.
Find the original stock of each.
Let
id
x = original stock of A.
y = original stock of B,
•
Then
a; + 200:y-50::2: J.
Simplify,
x + 200 = 4y - 200,
a;-4y = -400
a)
Also,
a? + 100:y-85::15:3J.
Simplify,
13x + 1300 = 60y- 5100,
13 a -60y-- 6400
(2)
Multiply (1) by 15, 15 x - 60y - - 6000
(3)
Subtract (2)
from (3), 2x = 400.
.-.a; = 200.
200-4y = -400.
.-. y - 150.
24. A quantity of milk is increased by watering in the ratio
4 •. 5, and then 3 gallons are sold ; the remainder is mixed with
3 quarts of water, and is increased in the ratio 6 : 7. How many
gallons of milk were there at first?
Let x = number of quarts of milk at first,
and y = number of quarts of water put in at first.
Then x + y = number of quarts of mixture after watering.
.*. x: x + y : : 4: 5,
x 4
or = -,
x + y 5
5a? = 4a; + 4y,
424
ALGEBRA.
x + y — 12 = number of quarts in remainder before watering.
* + y — 9 = number of quarts in remainder after watering.
. * + y-
-12
6
= 7
x + y-
-9
7x + 7y-
-84 =
= 6x + 6y-
-54.
X —
4y =
- 0
(1)
x +
5y =
= 30
= 30
= 6.
(2)
Substitute value of y in (1),
x-
-24 =
= 0.
.
\ x =
= 24 quarts
or 6
gallons.
25. In a mile race between a bicycle and a tricycle their rates
were as 5 : 4. The tricycle had half a minute start, but was beaten
by 176 yards. Find the rate of each.
Let x = number of yards bicycle goes per minute,
and y = number of yards tricycle goes per minute.
x : y : : 5 : 4,
4& = 5y.
••■- 4
= number of minutes tricycle was going after
V 2 bicycle started,
= number of minutes bicycle was going.
1584
y
1
2 =
-1760,
X
84 s-
xy _
' 2 =
- 1760y,
3168 a: -ary = 3520y.
Substitute ^ for x, 5y2 = 1760y.
... y = 352,
and x - 440.
teachers' edition. 425
26. The time which an express-train takes to travel 180 miles
is to that taken by an ordinary train as 9 : 14. The ordinary
train loses as much time from stopping as it would take to travel
30 miles ; the express-train loses only half as much time as the
other by stopping, and travels 15 miles an hour faster. What
are their respective rates?
Let y = number of miles ordinary train goes per hour,
and y + 15 = number of miles express-train goes per hour.
Then = number of hours required for ordinary train.
Also, + — =• number of hours required for express-train.
y + 15 y
180 + 30 180 15 1yl 0
.\ : =-= + — : : 14 : 9.
y y + 15 y
1890 ^ 2520 210t
y "" y + 15 y *
1680^ 2520
y y + 15'
1680y + 25200 = 2520y,
840y - 25200.
.-.y = 30,
and y + 15 = 45.
27. A line is divided into two parts in the ratio 2:3, and into
two parts in the ratio 3:4; the distance between the points of
seetion is 2. Find the length of the line.
Let
x = one part,
id
y = the other part.
.-. x : y : : 2 : 3.
3x=2y,
Sx-2y = 0
(1)
Also,
aj + 2:y-2::3:4,
4x-3y = -14
(2)
Multiply (1) by 3,
9z-6y = 0
Multiply (2) by 2,
8z-6y = -28
Subtract,
x =28
Substitute value of x in
i(l), 2y = 84,
y-42.
.% x + y - 70.
426 ALGEBRA.
28. A railway consists of two sections ; the annual expendi-
ture on one is increased this year 5%, and on the other 4%, pro-
ducing on the whole an increase of 4T8ff%. Compare the amount
expended on the two sections last year, and also this year.
Let x + y=* amount expended last year.
x = amount expended on first section last
year,
and y — amount expended on second section
last year.
Then ^-^ V- = amount expended on whole this year.
1000 . v J
But x + — — = amount expended on one part this
year.
and y + — %- = amount expended on other part this
year.
Then 105s 104 y _ 1043 s + 1043 y
100 100 1000
Simplify, 7s = 3y.
ar_3
„ . 105 s 315 „ , „ .
and WIT 728 A = 7A-
29. When a, 6, c, d are proportional and unequal, show that
no number x can be found such that a + x, 6 + x, c + x, d + x shall
be proportionals.
If a : b : : c : d, ad=bc;
and if a+x:b + x::c + x: d + x,
ad+dx + ax + x* = bc + cx + bx + x*.
Transpose, and cancel x*,
ax — bx — cx + dx = bc — ad.
But ad = be.
•\ x(a — b — c + d) = 0.
.-. a? = 0.
teachers' edition. 427
Exercise CXV.
1. If A oc B, and A = 4 when B = 6, find A when B- 12.
Here A = wi5,
.-. m = f
And if f and 12 be substituted for m and 5,
4 = $Xl2.
2. If -4 oc B, and when B = J, -4= J, find 4 when £= J.
Here A = mB.
m
.4
B
.*. m =
= i
Substitute § for
m and j for
5,
4 =
= }x
I
.-.4 =
-f
3. If -4 varies jointly as £ and C, and 3, 4, 5 are simultaneous
values of A, B, C, find A when B = C = 10.
Here 4 = ro.BC.
Substitute 3 for A, 4 and 5 for B and (7,
Then A = A X 10 X 10.
.\4 = 15.
4. If A <x — , and when <4 = 10, B = 2, find the value of B when
.4 = 4. B
Here .4 = — »
m-AB.
.•. m = 20.
Substitute values of m and -4,
a 20
4 = -,
4£=20.
.-.J? = 5.
428 ALGEBRA.
5. If^oc— , and when A = 6, # = 4, and 0= 3, find the value
of A when B=5 and C= 7.
Here A = ^
0
mB = J.C,
4m = 18.
.-. m = 4}.'
Substitute value of 2?, C, and m,
.:A = Z&.
6. If the square of X varies as the cube of F, and X = 3 when
F= 4, find the equation between Xand F.
Here X* = mY*t
.-. m = &.
Substitute value of m, X* = & 7*,
64X* = 9F».
7. If the square of X varies inversely as the cube of 7, and
X= 2 when F= 3, find the equation between X and Y.
Here X2 = -§*'
m-X'F8.
.\ m = 108.
Substitute value of m, X» = ^-
ys
8. If Z varies as X directly and F inversely, and if when Z = 2,
X= 3, and F= 4, find the value of Z when X= 15 and Y= 8.
Here S-2^.
™ ^F
.•.m = f = 2J.
Substitute values of m, X, and F,
5_ 2^x15
8
teachers' edition. 429
9. If A <x B + c where c is constant, and if A = 2 when 2? = 1,
and if A= 6 when 5 = 2, find J. when B=&.
As A = mi? + c.
Substitute first values of A and 5,
2 = m + c ■ (1)
Substitute second values of J. and B,
5 = 2m + c (2)
Subtract (1) from (2), m = 3.
Whence, from (1), c = — 1.
But A = m5 + c.
Substitute for to, JB, and c their values 3, 3, and —1,
A = 8.
10. The velocity acquired by a stone falling from rest varies
as the time of falling; and the distance fallen varies as the
square of the time. If it is found that in 3 seconds a stone has
fallen 145 feet, and acquired a velocity of 96} feet per second,
find the velocity and distance at the end of 5 seconds.
Let v = velocity,
T= time,
d = distance.
Then v <x Tt
and dccT2.
Let v = mT.
Substitute 96J for v and 3 for Tt
96* = 3 m.
.-. m = A$ft
When T= 5,
t, = ^X5 = 161f
Let
d = mT\
.-. 145 = 32m.
m = if±
When T= 5,
d =144x5* =402}.
11. If a heavier weight draw up a lighter one by means of a
string passing over a fixed wheel, the space described in a given
time will vary directly as the difference between the weights, and
inversely as their sum. If 9 ounces draw 7 ounces through 8
feet in 2 seconds, how high will 12 ounces draw 9 ounces in the
same time?
430 ALGEBRA.
Let x = heavy weight,
y — light weight,
2 = space.
x — y
zee 2.
x'+y
z_(x-y)m
x + y
x-y
Substitute values, , m-(7 + 9)8-
' 9-7
.-. m = 64.
6i (12+9)«
64 ~ 12-9-
3
72 = 64.
.-.s-9f
12. The space will vary also as the square of the time. Find
the space in Example 11, if the time in the latter case is 3
seconds.
We have from last example, 9} feet for 2 seconds.
Since space varies as square of time, we have
9} : x : : 2* : 3*.
.•.4a = 9x^f
x = 9xV
-20f
20} feet. Am.
13. Equal volumes of iron and copper are found to weigh 77
and 89 ounces respectively. Find the weight of 10} feet of round
copper rod when 9 inches of iron rod of the same diameter weigb
31^ ounces.
teachers' edition. 431
Let x — required weight.
9 inches = f of a foot.
If £ of a foot weigh 31.9 ounces, \ of a foot would weigh 10.03J
ounces, and 10J feet would weigh 446.60 ounces.
And, as equal volumes of iron and copper weigh 77 and 89
ounces respectively,
77 : 89 : : 446| : a.
.*. x = 516£ ounces.
14. The square of the time of a planet's revolution varies as
the cube of its distance from the sun. The distances of the
Earth and Mercury from the sun being 91 and 35 millions of
miles, find in days the time of Mercury's revolution.
Let x = time of Mercury's revolution.
91s : 35s : : P : x\
13s : 58 : : 1 : x*.
Whence, a* = 0.056895.
.*. x = 0.238, time in years,
= 87.1, time in days.
15. A spherical iron shell 1 foot in diameter weighs J^ of
what it would weigh if solid. Find the thickness of the metal,
it being known that the volume of a sphere varies as the cube of
its diameter.
Let D = diameter of shell,
d = diameter of sphere required to fill the shell,
and 1 represent the weight of iron sphere having diameter = D.
Then 1 — -ffa will represent the weight of iron sphere having
diameter = d.
Now the weights vary as the cubes of their diameters,
.-. 2P:#::1:1-Jft.
That is, IP : d* : : 1 : Jff ;
or, by extracting the cube root of each term,
D:d::l:#,
oxd=%D.
Since the thickness of the shell = J(D — d),
the thickness of the shell = J(l — f) = ^.
Hence, the thickness of the shell is ^ of a foot, = 1 inch.
432 ALGEBRA.
16. The volume of a sphere varies as the cube of its diameter.
Compare the volume of a sphere G inches in diameter with the
sum of the volumes of three spheres whose diameters are 3, 4, 5
inches respectively.
Let x — volume of first sphere,
and y = sum of volume of other three.
Then x:y::(6)»:(3)8 + (4)8 + (5)8,
x : y : : 216 : 216.
Therefore, the ratio is a ratio of equality.
17. Two circular gold plates, each an inch thick, the diame-
ters of which are 6 inches and 8 inches respectively, are melted
and formed into a singular circular plate 1 inch thick. Find its
diameter, having given that the area of a circle varies as the
square of its diameter.
Let Oi = area of gold plate 6 inches in diameter,
a, = area of gold plate 8 inches in diameter,
Oj = area of gold plate formed from the other two plates,
and x = diameter required.
Then c^ + a, : a, : : 6* + 8* : x*.
Since the first ratio is a ratio of equality, the second is also.
Therefore, x* = 6* + 8* = 100.
.-. x = 10.
18. The volume of a pyramid varies jointly as the area of its
base and its altitude. A pyramid, the base of which is 9 feet
square, and the height of which is 10 feet, is found to contain 10
cubic yards. What must be the height of a pyramid upon a base
3 feet square, in order that it may contain 2 cubic yards?
Let
and
Then
v = volume,
b = area of base,
a = altitude.
v oc ba,
v = mba (1)
When
and
V
v = 10 cubic yards = 270 cubic feet,
b = 9 x 9 = 81 square feet,
a =10 feet.
Then
— S-H»-*-
From (1),
When
a = —
mb
m = J,
v = 2 cubic yards = 54 cubic,feet,
6 = 3x3 = 9 square feet.
Then
°=^=18-
/
TEACHERS EDITION.
467
Let
Then
Hence,
x and y — first and last terms.
2 ay
x+ y
and
= middle term.
, , 2xy _^
*+»+jrn;-11
Square (1), and subtract (2) from the result,
6xy= 72, and ay = 12
Substitute 12 for xy in (1), and clear of fractions,
(x + y)2-ll(x + y) = -24.
Complete the square and extract the root,
0)
(2)
(3)
(1)
21. A number consists of three digits in geometrical progres-
sion. The sum of the digits is 13 ; and if 792 be added to the
number, the digits in the units' and hundreds' places will be in-
terchanged. Find the number.
Let x = first digit,
rx = second digit,
and r*x = third digit.
x + rx + r8® ^ 13
100a; + lOra; + r*x + 792 = 100 r^x + lOrx + x,
-99r*a; + 99a;*=-792.
r*x-x= 8 (2)
x + rx + r*x — 13
Subtract, 2x + rx= 5
.... *
2+r
Substitute value of x in (2),
5r* 5_ = g
2+r 2+r
5r2-5 = 16 + 8r,
5r2-8r = 21,
100r2-() + 64 = 484,
10r-8 = ±22,
10r = 30.
.-. r = 3.
From (1), x + 3a; + 9a? -= 13.
.-. s-1.
Hence, the number is 139.
432 ALGEBRA.
16. The volume of a sphere varies as the cube of its diameter.
Compare the volume of a sphere C inches in diameter with the
sum of the volumes of three spheres whose diameters are 3, 4, 5
inches respectively.
Let x = volume of first sphere,
and y = sum of volume of other three.
Then o;:y::(6)8:(3)8 + (4)8 + (5)»l
x : y : : 216 : 216.
Therefore, the ratio is a ratio of equality.
17. Two circular gold plates, each an inch thick, the diame-
ters of which are 6 inches and 8 inches respectively, are melted
and formed into a singular circular plate 1 inch thick. Find
riiam*±*r. h&yinj? idran. that the area of a circle varies as
»8.
l-r 1-i
(2) a = i,
r-f <n
"l-r 1_§ *
(3) a = t,
r J.
_a J_ W
"l-r~l-(-«
(4) a-1,
r — f (9)
•'T^"i-(-»
(5) a-i,
r-J. (10)
. a *
a =
= 0.1,
r =
= 0.1.
a
'"' l-r"
- 01 -1
1-0.1 !
a =
-0.86,
r =
= 0.01.
a
•'•l-r =
0.86
1-0.01
a =
aTTV»
r =
•it-
a
-4s-*
l-r
A
A + A =
-H-
a =
= iooO'
r =
= TW-
1 — r
= 1000 _ JL
AV **•
A + A =
-H-
/.
467
Let x and y = first and last terms.
Then — ; — = middle term.
x + y
Hence, x + y + ^ = ll (1)
and X2+y2+A^- = ^ (2)
Square (1), and subtract (2) from the result,
6 xy = 72, and xy = 12 (3)
Substitute 12 for xy in (1), and clear of fractions,
(x + y)* - 11 (x + y) = - 24.
Complete the square and extract the root,
x + y=S (4)
Square (4), x2 + 2xy + y% = 64
From (3), 4xy =48
x2-2xy + y8 = 16
x-y=±4 (5)
From (4) and (5), x = 6, and y = 2.
Hence, the numbers are 6, 3, 2.
11. When a, 6, c are in harmonical progression, show that
a:c::a — b: b—c.
If a, 6, c are a harmonical series,
1 _ 1 _ 1 _ 1.
6 a c 6
Multiply by a6c, ac — bc^ab — ac,
or c (a — b) = a (6 — c),
or a: c:: a — 6: 6 — C.
Exercise CXIX.
1. How many different permu- (Rule V) the number of different
tations can be made of the let- permutations of the letters is
ters in the word ecclesiastical, i^
taken all together ? j|j|j|j|j5j| = 454,063,600.
The word contains t once, a, e,
i; I, and s each twice, and c three 2. Of all the numbers that can
times; in all 14 letters. Hence be formed with four of the digits
468
ALGEBRA.
5, 6, 7, 8, 9, how many will begin
with 66?
The last two digits of the num-
bers may be selected in any way
from 7, 8, 9. This is possible in
.— = 6 ways.
Hence 6 numbers can be formed
of the required kind.
3. If the number of permuta-
tions of n things, taken 4 together,
is equal to 12 times the permuta-
tions of n things, taken 2 together,
find n.
By the given conditions
In In
— = 12 *- ,
|n-4 [n-2
or n(n — 1) (n — 2) (n — 3)
= 12n(n-l).
Hence (n — 2) (n — 3) = 12
n2-5n + 6=12
n2-5n — 6 = 0
(n+l)(n-6) = 0
n = — 1 or 6.
Therefore n = 6.
4. With 3 consonants and 2
vowels, how many words of 3
letters can be found, beginning
and ending with a consonant,
and having a vowel for a middle
letter?
The 2 consonants can be chosen
from 3 in 6 ways, and the vowel
can be chosen from 2 in 2 ways.
19 in
Hence the number of words of
the required kind that can be
found is 6 X 2 = 12. Ans.
5. Out of 20 men, in how many
different ways can 4 be chosen
to be on guard ? In how many
of these would one particular
man be taken, and from how
many would he be left out ?
(1) Four men can be selected
120
from 20 in rrrr^ = 4845 ways.
(2) If one particular man is to
be included, the remaining 3 can
be selected from the remaining
110
^—969 ways.
(3) If a particular man is to be
excluded, the 4 can be selected
from the remaining 19 in
[19
^ = 3876 ways.
3876 + 969 = 4845.
Hence the total number of
selections is 4845; including a
particular man, 969; excluding
a particular man, 3876.
6. Of 12 books of the same size,
a shelf will hold 5. How many
different arrangements on the
shelf may be made ?
The number of arrangements is
112.
II
= 95,040.
TEACHERS' EDITION.
469
7. Of 8 men forming a boats'
crew, 'one is selected as stroke.
How many arrangements of the
rest are possible? When the 4
who row on each side are decided
on, how many arrangements are
still possible ?
(1) The 7 remaining men can
be arranged in the 7 remaining
seats in [7 = 5040 ways.
(2) The 3 men, besides the
stroke, who row on the stroke
side can be seated in [3 = 6 ways.
The 4 men on the other side can
be seated in [4 = 24 ways. Hence
the total number of arrangements
under the given conditions is
6 X 24 = 144.
8. How many signals may be
made with 6 flags of different
colors, which can be hoisted either
singly or any number at a time ?
A set of r flags can be selected
. I?
fcli:
; ways, and each set can
be hoisted in [r different orders.
Hence the number of different
signals with r flags is tttzt '
The number of signals, therefore,
with 1 flag is 6
= 6
" 2 flags is 6 X 5
= 30
44 3 44 6X6X4
= 120
44 4 44 6X6X4X3
= 360
44 5 44 6X5X4X3X2
= 720
44 6 44 6X6X4X3X2X1
= 720
In all,
1956
9. How many signals may be made with 8 flags of different colors,
which can be hoisted either singly or any number at a time one
above another ?
The number of signals
with 1 flag is 8
2 flags is 8 X 7
8X7X6 =
8x7x6x6 =
8x7x6x5x4 =
8X7X6X5X4X3 =
8X7X6X5X4X3X2 =
8x7x6x6x4x3x2x1=
In all,
8
56
336
1,680
6,720
20,160
40,320
40,320
109,600
470 ALGEBRA.
10. How many different signals one light is used must be counted
can be made with 10 flags, of only once; all other cases four
which 3 are white, 2 red, and the times. The number of signals in
rest blue, always hoisted all to- a single line
gether and one above another ? ^th l light is 4 =4
The number of different signals " 2 lights is 4 X 3 =12
is equal to the number of arrange- " 3 " 4x3x2 =24
ments of 10 things, of which 2 " 4 " 4x3x2X1 = 24
are alike, 3 are alike, and 5 are In all, 60 + 4.
alike. This number is (Rule V) Hence the total number of
tig _ signals is 4 X 60 + 4 = 244.
I? I? 15 14. From 10 soldiers and 8
11. How many signals can be sailors» now many different par-
made with seven flags, of which ties of 3 soldiers and 3 sailors
2 are red, 1 white, 3 blue, and can *» formed ?
1 yellow, always displayed all The 3 soldiers can be selected
together and one above another? 110
t> t> i xr *v v . in foi* = 120 wavs> and toe 3
By Rule V the number is [3[7
II .OA sailors in rz77= 56 ways. Hence
f2J3 = 420. [8 [6
the party of 3 soldiers and 3 sailors
12. In how many ways may can be selected in
the 8 men serving a field-gun be 120 x 66 = 6720 ways,
arranged so that the same man
may always lay the gun ? 15- How manv sign*18 <*& be
„«-_.. , made with 3 blue and 2 white
The 7 remaining men may be _ _. , , ,. , . .
, . ,- °J/X flags, which can be displayed
arranged in 7 = 5040 ways. Jr . , v *
•— either singly or any number at
13. Find the number of signals a time one above another ?
which can be made with 4 lights a single flag maybe either blue
of different colors when displayed or white. 2 ways.
any number at a time, arranged Two flags may be both blue or
above one another, side by side, both white, or 1 blue and 1 white,
or diagonally. In the lafit caj3e they may be ar-
The number of arrangements ranged in two ways,
in vertical, horizontal, or either In all, 4 ways,
diagonal line is evidently the Three flags may be all blue, or
same. But the case where only 2 blue and 1 white or 2 white
teachers' edition. 471
and 1 blue. In the last two 18. From 12 soldiers and 8 sail-
cases they may be arranged in 3 ore, how many different parties
ways. In all, 7 ways, of 3 soldiers and 2 sailors can be
Four flags may be 3 blue and found?
1 white, 4 ways ; or 2 blue and The 3 soldiers can be selected
2 White' 1212 " 6 WayS (RU16 from 12 in }=- = 220 ways ; the
V). ^^ In all, 10 ways. [3 [9 Jf
Five flags must be 3 blue and 2 sailors from 8 in r3r = 28
JL--
1216
2 white, rzrrz = 10 ways (Rule ways. The party can therefore
V). l-L in all, 10 ways, be formed in
There are, therefore, in all, 220 X 28 = 6160 ways.
2 + 4 + 7 + 10 + 10 = 33 possible
signals. 19* Find the number of com-
„. „ ^ binations of 100 things, 97 to-
16. In how many ways can a .
party of 6 take their places at a
round table? B^ Rule vm *■» *"P***
, , number is
One person may take any one . 1Q0
of the 6 seats; the other 6 can L=^ = 10° x " x 98
then be seated in [6 = 120 differ- l?Z 12 3 x 2
ent orders. Hence the answer is = 161,700.
6 X 120 = 720. 2k wuh 2Q congonants ajld 5
17. Out of 12 Democrats and vowels, how many different words
16 Republicans, how many differ- can be formed consisting of 3
ent committees can be formed, different consonants and two dif-
each consisting of 3 Democrats ferent vowels, any arrangement
and 4 Republicans ? of letters being considered a
The 3 Democrats can be selected word ?
1 12 The 3 consonants can be selected
from 12 in rrr^ = 220 ways ; the |2Q
L-L [16 from 20 in r^== = 1140 ways;
4 Republicans from 16 in liilll 15
*— ^— the 2 vowels from 5 in .7=5 = 10
= 1820 ways. The committee, l± l£
consisting of any 3 Democrats ways. The number of combina-
and any 4 Republicans, can there- tions of 3 different consonants
fore be formed in and two different vowels is there-
220 X 1820 = 400,400 ways. for© 10 X 1140 = 11,400.
472
ALGEBRA.
The five letters of each com-
bination can then be arranged in
[6 = 120 ways. Hence the total
number of words of the required
kind is
120 X 11,400 = 1,368,000.
21. Of 30 things, how many
must be taken together in order
that, having that number for
selection, there may be the great-
est possible variety of choice ?
By § 417, the number taken
must be 30
f=15-
22. There are m things of one
kind and n of another. How
many different sets can be made
containing r of the first and s of
the second ?
The r of the first kind can be
selected from m in .-
[m
any seat ; the other 9 can then be
seated in [9 = 362,880 different
orders. For each order of the
10 persons and the reverse order,
each person has the same neigh-
bors, and this is true only in this
case. Hence the number of ar-
rangements required is
?»= 181,440.
24. The number of combina-
tions of 71 things taken r together,
is 3 times the number taken r — 1
together, and half the number
taken r + 1 together. Find n
and r.
By the given conditions
In o I*
lr[n^
= 3
= *
k| . ways;
the s of the second kind from n
\n
in — : The total number
|g |n — 3
of sets of r of the first kind and
s of the second is therefore
\m\n
\r \s \m — rln — s
r-l[n-r+l
l»
r+lln-r-1
ln-r + 1 \r
' , =S-^=-i (l)
\n— r \r— 1
In-r [r+1
= 2*=^. (2)
lr
23. In how many ways may 10
persons be seated at a round table
so that in no two of the arrange-
ments may every one have the
same neighbors ?
Only the order of seating being
of account, one person may take
\n-r-l
From (1),
n — r+ 1 = 3r,
n+l = 4r.
From (2),
n-r = 2(r+l),
n = 3r + 2.
.-. 3r+2 = 4r-l,
r = 3,
n = 3r + 2
= 11.
Therefore,
w= 11, r = 3.
TEACHERS^ EDITION. 473
25. In how many ways may 12 elapse before the same 20 men
things be divided into 3 sets of 4 ? go on guard the second time.
Here the sets are indifferent, The number of guard details
and the answer is which include a particular man is
112 189
— - = 6776. L— -
26. How many words of 6 let- 28. Supposing that a man can
ters may be formed of 3 vowels Place himself in 3 distinct atti-
and 3 consonants, the vowels tudes, how many signals can be
always having the even places ? made bv 4 men P^ced side by side ?
The 3 consonants can be ar- S^^ach man can place him-
ranged in the 3 odd places in self in 3 attitudes, the total num-
18 = 6 ways, and the 3 vowels can ber of distinct attitudes of the
be arranged in the even places in &0** of 4 men M 3*. = 81' , ,
6 ways. The total number of But one is the attitude of rest.
,.« . , * ., . , Therefore 34 — 1 = 80 signals,
different words of the required &
kind is therefore 6 X 6 = 36. 29. How many different ar-
rangements can be made of 11
27. Fromacompanyof 90men, cricketerSj SUpp0Sing the same 2
20 are detached for mounting always to bowi ?
guard each day. How long will The Q remaining cricketers can
it be before the same 20 men are ^ BrnsDSBd m m = 362,88o ways,
on guard together, supposing the And gince the two who bowl may
men to be changed a* much a* change placegj the total number
possible ;^ and how many times of arrangemente is
2X362,880=725,760.
30. Five flags of different colors
can be hoisted either singly or
90 in ,4=- ways; and this is ^ number at a time one above
1*5; 170 another. How many different
the number of days that will signals can be made with them ?
The number of signals
with 1 flag is 5 =5
u 2 flags is 6 X 4 =20
" 3 " 5X4X3 =60
"4 " 5X4X3X2 =120
44 6 " 5X4X3X2X1 = 120
In all, 325
will each man have been
guard?
20 men can be selected from
474 ALGEBRA.
3L How many signals can be with the lights one above another,
made with 5 lights of different col- Every possible arrangement mast
ors, which can be displayed either therefore be counted twice ; but
singly, or any number at a time if only one light is used, this can
side by side, or one above another ? be counted only once.
The number of signals with the The number of arrangements
lights side by side is equal to that in a single line
with 1 light is 5 =5
" 2 lights is 5 X 4 =20
"3 " 5X4X3 =60
" 4 " 5X4X3X2 =120
" 6 " 6 X 4 X 3 X 2 X 1 = 120
In all, 320 + 5.
The total number of possible signals is therefore
2X320+6=645.
32. The number of permuta- have 425 times as many hands as
tions of n things, 3 at a time, is there are cards in the pack. How
6 times the number of combina- many cards are there ?
tions, four at a time. Find n. Let n = the number of cards ;
By the conditions of the prob- l» ,
, ^ then rz\ s = the number of
lem» 3 n-~3 _ , . , ,
in \n • •— ' hands one can hold.
HenC6[3^3 = 425n
w(n-l)(n-2) = 425nl3
(n-l)(n-2)=2550
w2-3n- 2548 = 0
(n-52)(n + 49) = 0
33. At a game of cards, 3 being •'• n = 62-
dealt to each person, any one can There are 52 cards in the pack.
1. (l + 2x)* EXEBCISE CXX.
= (1)6+5 (1)4(2 x) + 10 (1)8(2 x)2+ 10 (1)2(2 x)»+ 5 (1) (2 x)4+ (2s)*
= 1 + lOx + 40 x2 + 80x» + 80 x* + 32 a*
2. (x - 3)8 = x* - 8 x7(3) + 28 x°(3)2 - 56 x*(3)8 + 70 x*(3)*
- 56x8(3)6 + 28x2(3)« - 8x(3)7 + (3)8
= x* - 24x7 + 252x6 - 1512X5 + 5670x*
- 13608 x8 + 20412 x2 - 17496 x + 6561.
h
-3
>|n-
-4
•
6
[n-8
|n-4
4 =
71—3
n =
7.
teachers' edition. 475
3. (2x- 3y)*
= (2s)4 - 4(2a0»(3y) + 6(2xf(3yy - 4(2s)(3y)» + (3y)*
= 16s* - 96afy + 216^-216^ + 81 y*.
4. (2-xf
= (2)'-3(2)*(s) + 3(2)(z)>-(*)a
= 8- 12s + 6s2 -a3.
-(!)• - 5(1). 0*) * 10(1C (^'_ 10(1). (& J
^<"(3f)'-(^T
1 15y 45y» 135.V3 405y* 243y*
4 8 32 256 1024*
* HI
+M(l)--»(fy+9(f)'-l(|j
i q , a„* 28a* 14x* 14ar> 28s6 4s' ^ ^
9 9 27 243 243 729 19683
7. The fourth term of (2 x — 5 y)1*.
Substitute in formula values of n and r,
w(n~1) fr-r + 2) an_r+i ^.i
1X2 (r-1)
12X11X10,
1x2x3
= - 14080000 afy8.
(2s)»(5y)»
8. The seventh term
_ 10 X 9 x 8 X
1x2x3x4
35 aV
" 1944 '
r(D'(f)'
ZTl\.
.(.' V! Ill* «»1 '
• j»- a j if i ur:r
I V .//
' 1 ,' ,y *
TEACHERS* EDITION. 477
15. The rth term from the end of (2 a + x)*
- n(n ~ *) (n ~ r + 2)(2aHaf^i
1X2 (r-1) V T
16. The (r + 4)th term of (a + x)»
= n(n - 1) (n-r- 2)a_^ ^
1X2 (r + 3)
17. The middle term of (a + z)*»
= 2n(2n-l) (2n-(»+l) + 2)fl^Watft
1X2 (n + 1 - 1)
,2n(2n-l) (n+1)^
1X2 n
Multiply both terms by [n,
(w + 1) th term = ===- aV.
(In)*
18. Expand (2a + x)w, and find the sum of the terms if a = 1,
x=-2.
(2a)u + 12(2a)na; + 66(2a)1(V + 220(2a)»x8 + 495(2a)8s*
+ 792(2a)7ari + 924(2a)«a* + 792(2afxt + 495(2a)*x8
+ 220(2a)sa* + 66(2a)2a^° + 12(2a)x11 + «".
Substitute 1 for a and —2 for a?,
(2-2)u=0.
Exercise CXXI.
1. (1 + as)* to four terms
1 _L _1_
2~8 16~
2. (1 + sc)3 to four terms
-1 + fas-^ + ^s"-
478 ALGEBRA.
3. (a + x)* to four terms
= | 3z 3z* 5g8
~a 4ai 32a* 128a*
,/,3x 3x2 5x« \
-aV+4^~3^+128^~ )'
4. (1 — x)-4 to four terms
By substituting 1 for a and — x for x in the formula (a+z)B
, , n(w— 1) , 0 , n(n— l)(n— 2)(n— 3) _,
= a»4-na^1x + -L^-^a»-2x2 + -* |j* ** +
= 1 + 4x + 10x2 + 20x8 4-
5. (a2 — x2)* to four terms
= (a2)1 ~ t (a2)1-1 + ^-p^a2)*^
. 5a8x2 . 15ax* 5x« ,
_Mi-iKi^(a2)Hx»+
15
6. (x2 4- xy)~i to four terms
= (x'2)-3 - }(x2)-H (xy) + ""W""*-1) (x2)-!-2 (xy)2
+ -f( I *)( f 2) x2-*-b (xy)» +
= ,-s_|x-,, + i^_§^+.
7. (2x — 3y)-i to four terms
= (2x)-i - (- i)(2g)-t(8y) + (~ *M~ ^ (2x)-t (3y)2
-(~^^~y~J)(2x)^(3y)'+
= 2*^{l + fx-iy + -^ar»^ + #ftar*y» + }
teachers' edition. 479
8. Vl — 5a; to four terms
= l-x--2JB2-6ar,-
9. ——z==zl to four terms
V(4a8-3aa;)3
= (4a2-3oa;)-3
1 X «
_ (-t)(-»(~t) (4a2)-«(3 ^ji
1x2x3 v ' v '
- J_ { l + ^£ + 135a;2 . 945 x8 )
= 8as( 8 a 128 a2 1024 a8 J
10. a/ . to four terms
>(1 — 3y)5
= l+§J/ + 5^ + ?35^
2 8 48
11. (1 + jc + a;*)* to four terms
«[1 +(* + «*)]»
- 1 + 1(0 + ^) + ii^l)(*+ ^ + *»zMzl9 (* + <*>
12. (1 — x + s2)* to four terms
-[1 -(*-*•)]•
- 1 - fa; + f a" + (far2 - £ s») - (- ^ar*)
£g0 ALGEBRA*
, .„» u!fhi of (a + x)»
Hi -1) (i -r-H 2)^-^1^-1
"1X2 (r-1)
d)(-i)(-i)(-i) m^
lr-1
n-(l)(l)(D(l) (^').^
[r-J.
Multiply both terms by ¥ _1
. / !y 1X3X5 (2r-5) fl3 2 V*
14. The rtffc term of (a — ac) -»
= (-3)(-4) (-r-1) fl_r. Jaf ^
1x2x3 r-1
_ 3X4 (r-i)(r)<r+i) >
K 1} 1X2X3X4 (r-1)
15. \/66 to five decimal places
-{64(1 + *»*
= 8(1+ A)4
-8{l+(»(*)+*£^(lW,l
-8(1+t1t-t^^)
-8(1+ 0.00781 - 0.00003)
= 8.06224
16. \fijfo to five decimal places
={1(1 + A)}i(*-D
-1 +0.01111 — 0,00012
-1.01099.
teachers' edition. 481
17. v'l29 to six decimal places
= {128(1 +TiT)}»
= {2'(1+Tk)}*
.v&ia-2Ci+T«*
= 2(1 +0.001116—0.000004)
= 2.002224.
18. (1— 2x + 3a?)i to four terms
= {l-(2x-3^)}-*
= 1 - (- 1) (2c - Zx>) + fc<).(--*'(2* - 3a»)»
.(-l)(-*)(-V)(2,_3aiy
1x2x3 v '
1 +
f4£ _ 6aA /28z* _ 84a3 63_^\ /224^ _ \
V 5 5 ) { 25 25 25 J V 125 '" J
_1 4a; 2s* 196r»
5 25 125
19. Ci+^i! to coefficient of a*
(l + 3s)3
= (l + 2a:)2(l+3a;)-3
= (1 + 4z + 4s2) (1 - 9<e + 54s2 - 270x* + 1215a4).
The terms containing re4 will be
1215s* - 4s(270s8) + 4a2(54s2)
= 351 a*.
20. (1 + as)* expanded
u».+iS^,,+K»-i)(»-»v
,HJ-l)tt-2)ft-8)
+ 0
-1+J* L-s.+ 1X3 -... 1X3X5 .
2x2* 2x3x2" 2x3x4x2«
When i = l, this becomes (1 + 1)'
_1+I 1 , 1x3 1x3x5 ,
2 2x2s 2x3x2» 2x3x4x2«
482 ALGEBRA.
Exercise CXXII.
1. If I throw a single die, what Out of 7 chances, 5 are favor-
is the chance that it will turn up able and 2 unfavorable. Hence
(i.) An ace ? the chance of success is f, and of
(ii.) An ace or a two ? failure f.
(iii.) Neither an ace nor a two ? _
5. The chance of an event is
Since a die has 6 faces, f Find the odds for or against
(i.) The chance of an ace is J. the eyeat
(ii.) The chance of an ace is 1,
and the chance of a two is £. °* ot 9 chances< 2 "» aTOr-
Hence the chance that either an able for *" event and 7 nnfayor-
. «iii able. Hence the odds against
ace or a two turns up is 4+ I *u c' XAC"°° ° c wuo °*>
_ i the event are 7 to 2.
(iii.) There being 4 faces be- 6, What is the chance of a
sides the ace and the two, the year, not a leap year, having 53
chance that neither an ace nor a Sundays ?
two turns up is J = f jf a year 0f 365 days has 53
% The chance of a plan sue- Sundays, it must begin on Sun-
ceeding is \. What is the chance day. But, as the year may begin
that it fails? . on any day of the week, the
Since there is one chance in 4 chancethat it will begin on Sun-
that it succeeds, there are 3 day ta *•
chances in 4 that it fails. Hence 7. Two numbers are chosen at
the chance of failure is £ . random. Find the chance that
« ,, , , , ~ . their sum is even.
3. If the odds are 10 to 1 against
an event, what is the probability The numbers may *
of its happening? (i.) both even,
(ii.) both odd,
(iii.) the first odd, the second
even,
(iv.) the first even, the second
probability of its happening is 1 _, , .
Out of 11 chances there are 10
that the event will not happen
and 1 that it will. Hence the
in 1 1 or T*T. and an these f QUr cages are eqUaii}
4. If the odds are 5 to 2 in likely to occur. But the sum is
favor of the success of an experi- even only in the first two cases,
ment, what are the respective and the chance that one of these
chances of success or failure ? will happen is J = £.
teachers' edition. 483
8. If 4 cards are drawn from a 45. But the number of adjacent
pack, what is the chance that places is now 10, since the first
they will all be hearts ? and tenth are now adjacent.
Four cards can be selected from Hence the chance that the two
1 52 assigned persons will stand to-
the 52 in the pack in rrrjz ways, gether is JJ = J.
Four hearts can be selected from n' Show that if n persons sit
1 13 down at a round table, the odds
the 13 hearts in r^r? ways. against 2 particular persons sit-
"— ■— ting next each other are n — 3
Since any set of four cards is to 2
aslikelytobedrawnasanyother, 0ne of the two persons being
the chance that they will all be geated? there are 2 seats next to
hearts is laim and n __ 3 not next to him
iAJ _^ [jjg _ iijljg The second person being equally
|4[9 * [4[48 [9 [62 likely to take any seat, the odds
_ 13 X 12 X 1 1 x 10 that he does not sit next the first
~~" 62X51X60X49 one are n — 3 to 2.
= -— • 12. If two letters are selected
at random out of the alphabet,
9. If 10 persons stand in a line, what is the cnance that both will
what is the chance that 2 assigned be vowels ?
persons will stand together ? with the 26 letters of the alpha-
The total number of pairs of [26
1 10 bet, . . = 326 pairs can be
places among the 10 is 7575- = 45. L L_
L£l° formed and from the 5 vowels
Of these, 9 are adjacent ; viz. , the [5
first and second, second and third, |2J3 = 10 pairs* Hence the chance
etc., up to the ninth and tenth. ^ ft random pair should be
Hence the chance that the two TTrtTwrolo • 1 0 — 2
, ,,. , voweis is -tot — -gx.
assigned persons will stand to-
gether is ?93 = J. 13- Five men» A> B> c> D, E»
speak at a meeting, and it is
10. If 10 persons form a ring, known that A speaks before B.
what is the chance that 2 assigned What is the chance that A speaks
persons will stand together ? immediately before B ?
As in Ex. 9, the total number The five men can be assigned
of pairs of places among the 10 is to speak in [5 = 120 different
484 ALGEBRA.
orders. In one-half of these A prize of $100. What is his ex-
will precede B. Also there are pectation ?
4 ways in which B can immedi- From the 60 tickets in the bag,
ately follow A ; viz., A may speak [50
first and B second, or A second 2 can be drawn m f^jig = 1225
and B third, or A third and B wavs-
fourth, or A fourth and B last. Two tickets entitling to prizes
The 3 remaining speakers may be <** be drawn ^^ ^ ten m
arranged in each case in 6 differ- Ii2 __ Af>mn„a n„A +1>„ «™,rtxft
°, .-^tz = 45 ways, and the expecta-
ent orders. [2 [8
Hence the chance that A speaks tion from drawing 2 prizes is
immediately before B is T*5? x §200 = $\6^.
4X6 _ One ticket entitling to a prize
60 ' and one blank can be drawn in
14. A, B, C have equal claims 40 X 10 = 400 ways, and the
for a prize. A says to B, " You expectation from the event is
and I will draw lots, and the t\°A * $100 = fifja.
winner shall draw lots with C for Hence the total expectation is
the prize." Is this fair? $3fl0^160Q= $40.
The chance that A will win is. One of two events must
from B is £. If he wins from B, happen. If the chance of one is
the chance that he will win from $. 0f the chance of the other, find
C is also i. To get the prize he the odds on the first,
must win from B and then from Since one of the two events
C, and his chance of doing this must happen, the sum of their
is i x £ = ±. probabilities is 1 ; and as the
Similarly, B's chance of win- chance of the first is £ of the
ning is £ ; but C's chance, since chance of the second, their re-
he is sure of a trial and has to gpective chances are f and f.
win only once, is £. Hence the odds on the first are
Originally A, B, C had each a 2 to 3.
chance of £. A's proposal is ^ „„ n . „ „
• . x a j t» j 17. There are 3 events, A, B, C,
equally unfair to A and B, and luwcaicuo cu»,n, , ,
.. f . . „ one of which must happen. The
more than fair to C. , , n . , « A r
odds are 3 to 8 on A, and 2 to 5
15. A person is allowed to draw on B. Find the odds on C.
2 tickets from a bag containing The chance that A happens is
40 blank tickets, and 10 tickets fy; that B happens, f. Hence
each entitling the holder to a (Rule V), since A, B, or C must
teachers' edition.
485
happen, the chance that C hap-
pens is 1 — (f + fV) = f f ; and
the odds on C are 34 to 43.
18. In a bag are 7 white and 5
red balls. Find the chance that if
one is drawn it will be (i.) white
or (ii.) red ; or, if two are drawn,
that they will be (i.) both white,
(ii.) both red, or (iii.) one white
and the other red.
Since there are 12 balls in all,
the chance of drawing a white ball
is ^ j of drawing a red ball, T*y.
From the 12 balls, 2 can be
112
drawn in r^rr^ = 66 ways. Two
white balls can be drawn from
II
the 7 in — — = 21 ways ; two
^ L6
red balls from the 5 in r^-r = 10
[2 [3
ways ; one white ball and one red
ball in 7 X 5 = 35 ways. Hence
the chance of drawing 2 white
balls is |4 = A ; 2 red balls,
i? = -fg ; one white ball and 1
red, ff.
19. If 3 cards are drawn from
a pack, what is the chance that
they will be king, queen, and
knave of the same suit ?
Three cards can be drawn from
1 52
the 52 in the pack in -=t- = 22,100
[3|49
ways. King, queen, and knave
of the same suit can be drawn in
4 ways.
Hence the chance of drawing
king, queen, and knave of the
same suit is „$„ = 5^y.
20. A general orders 2 men by
lot out of 100 mutineers to be
shot, the real leaders of the
mutiny being ten in number.
Find the chance (i.) that 1 only,
(ii.) that two, of the leaders will
be shot.
Two men can be selected from
1100
100 in -)= = 4960 ways. One
leader and one follower can be
selected in 10 X 90 = 900 ways ;
Ii?
two leaders in rr-z = 45 ways.
If l£
Hence the chance that only one
leader will be shot is ?9g0T°0 = ^ ;
that two leaders will be shot,
?iiff — riff-
21. Show that the odds are 8
to 1 against throwing 9 in a single
throw with 2 dice.
The total number of throws
with 2 dice is 6 X 6 = 36. The
number of ways of throwing 9
is 4; viz., 3 and 6, 6 and 3, 4
and 6, 5 and 4. Hence the odds
against throwing 9 is 32 to 4, or
8tol.
22. Show that in a throw with
3 dice the chance of either a trip-
let or a doublet is f .
The total number of throws
with 3 dice is 6x6x6 = 216.
The number of ways of throwing
486 ALGEBRA.
a triplet is 6, since it may be will be alternately white and black
three l's, three 2's, etc. A i8|xjxfxjx?x|x|x}
doublet may happen with any 2 = T£7.
of the three dice, and accord-
ingly the number of ways of 24. A bag contains 2 white
throwing a doublet is equal to balls» 3 black balls, and 5 red
the number of doublets, 6, multi- Dalls- K 4 balls are drawn, find
plied by the numbers of pairs of tne chance that there will be
dice, 3, and multiplied further by among them :
the number of ways the odd die (*-) Botil the white balls,
may fall. This last Dumber is 5, ("•) Two m&V of l^e Wack
since the case of a triplet has balls,
already been considered. 0"-) Two at least of the red
Hence a triplet can be thrown balls,
in 6 ways, and a doublet, but pour Dang ^n ^ drawn from
not a triplet, in 6X3X5 = 90 mq
ways. Either a doublet or a trip- the 10 in the bag in r^rz = 210
let can be thrown in 6 + 00 = 96 ways. *-L
ways. The chance of throwing (L) M the 4 balls are to include .
a doublet or triplet is therefore the 2 white oneg? the other 2 can
*rV — %' be drawn from the remaining 8
23. In a bag are 5 white and 4 in J=_ = 28 ways. Hence the
black balls. If drawn out one L?15
by one, what is the chance that chance that the 2 white balls are
the first will be white, the second drawn among the 4 is -ff^ = fa
black, and so on, alternately ? (iL) Two black Mls can be
The chance that the first ball drawn from the 3 black balls in
will be white is f . 13
The first ball having been [2Q = 3ways- « exactly 2 black
drawn, if it is white, the chance „ ,.,,,., j
that the second one is black is baUs are to be included in the 4
4 _ 1 drawn, the other 2 can be selected
1 T^e chance that the third ball f rom the 7 white and red balls
is white is then } ; that the fourth in _L_ _ 21 wayg Hence ^
is white, £ = J ; that the fifth is I? IJ>
black, | ; that the sixth is white, chance of drawing exactly 2 black
| = i ; and so on. _ ... 3 X 21
balls is = -Ac
Hence the chance that the balls Utt ° ° 210 **"
TEACHERS7 EDITION.
487
(iii.) If no red ball is drawn,
the 4 balls can be selected from
the 5 white and black balls in
I*
— — = 5 ways. Hence the chance
[4[1
that no red ball is drawn is jfo
= ^. If exactly one red ball is
to be drawn, it can be selected
from the 5 red balls in 5 ways.
The 3 remaining balls can be
selected from the 5 white and
black balls in — = 10 ways.
[3 [2
Hence the chance that exactly 1
red ball is drawn is = ^T.
The chance of drawing either no
red ball or exactly 1 is & + -fa
= \\ ; and the chance of draw-
ing more than one red ball is
Exercise CXXIII.
1. The chance that A can solve
a certain problem is £, and the
chance that B can solve it is $.
What is the chance that the
problem will be solved, if both
try?
The chance that A will fail is
£ , and the chance that B will fail
is £. Hence the chance that both
will fail is £ X \ = £ ; and this is
the chance that the problem will
not be solved. The chance that
it will be solved is, therefore,
l-i = *.
2. What is the chance of throw-
ing at least one ace in 2 throws
with one die ?
The chance of not throwing an
ace is J each time. Hence the
chance of not throwing an ace
both times is £ X £ = f f ; and
the chance of throwing at least
one ace is 1 - 1| = \\.
3. If n coins are tossed up,
what is the chance that one, and
only one, will turn up head ?
The chance that any given one
will be head is £, and the chance
that the rest will be tails is (i)n_1.
Hence the chance that a given
one will be head and the others
tails is i X (i)«-i = ($)». And
the chance that some one will be
head and the rest tails is n(£)B.
4. What is the chance of throw-
ing double sixes at least once in
3 throws with 2 dice ?
The chance of throwing double
sixes any given time is J x J = ^
and the chance of not throwing
them any given time is 1 — -fa
= }§. Hence the chance of not
throwing them in any of the 3
throws is (f J)8; and the chance
that they will be thrown at least
once is 1- (||)8 =;&&V
488
ALGEBRA.
5. A copper is tossed 3 times.
Find the odds that it will fail :
(i.) Head and 2 tails without
regard to order.
(ii.) Head, tail, head.
The coin may fall head three
times, which can happen in only
1 way ; or head twice and tail
once, which can happen in 3
ways; or head once and tail
twice, 3 ways; or tails three
times, 1 way. All these 8 ways
are equally likely to happen.
Hence the chance that it will fall
head and two tails is f .
The chance that it will fall
head, tail, head isix^x| = i.
6. If a copper is tossed 4 times,
find the odds that it will fall 2
heads and 2 tails sooner than 4
heads.
The total number of ways it
can fall is 24 = 16. It can fall
head every time in only 1 way.
It can fall head twice and tail
twice in as many ways as 2 selec-
tions can be made from 4; viz.,
Ii
— rr = 6. Hence the odds are 6
L±L£
to 1 that it will fall head twice
and tail twice rather than head
four times.
7. If from a lottery of 30 tick-
ets, marked 1, 2, 3, four
tickets are drawn, what is the
chance that 1 and 2 will be among
them?
Four tickets can be drawn from
130
30 in r~^ ways. If 1 and 2
are to be among them, the other
two can be drawn from the re-
[28
maining 28 in Yj^ ways. Hence
the chance that 1 and 2 will be
drawn is
[2 [26 ' [4 [26 [30 [2 "*•
8. If 2 coppers are tossed 3
times, find the odds that they
will fall 2 heads and 4 tails.
There are 3 ways of getting 2
heads and 4 tails. First, the first
copper may fall head twice, all
the other cases giving tails ; sec-
ond, each copper may fall head
exactly once ; and third, the sec-
ond copper may fall head twice,
all other cases giving tails.
The chance that the first cop-
per will fall head twice and tail
once is 3x}x|x^ = |, since
there are three ways of getting
two heads and one tail. The
chance that the second copper
will fall tail every time is i X £
x £ = £. Hence the chance that
the first copper gives two heads
and a tail, and the second copper
3 tails, is f x £ = ^ ; and the
chance that one of the two cop-
pers gives two heads and a tail,
and the other three tails, is
2xA = TV
TEAUHBKS' EDITION. 489
The chance that each copper 10 are 6, 4, 1 and 5, 3, 2. 5, 4, 1
gives 1 head and 2 tails is f x £ can be drawn in 6 ways. The
= &. chance that 5 will be drawn the
Hence the chance of tossing first time is ^. 5 not being re-
two heads and four tails is • placed, the chance of drawing 4
.3 + 9 — is the second time is J. And the
chance of drawing 1 the third
9. There are 10 tickets, five of time is J. Hence the chance of
which are numbered 1, 2, 3, 4, 5, drawing 5, 4, 1 in this order is
and the other five are blank. ^ X J X J = 7^j. Therefore the
Find the chance that the sum of chance of drawing 5, 4, 1 in any
the numbers on the tickets drawn order is 6 X 7^ = T£7. And
in 3 trials will be 10, one ticket similarly, the chance of drawing
being drawn and then replaced 5, 3, 2 is T^. Hence the chance
at each trial. of drawing a total of 10 is
The numbers drawn must be , , 1 __ .
one of the five sets : T** T3nr " A*
5, 5, 0 ; 5, 4, 1 ; 5, 3, 2 ; 11. A bag contains 4 white and
4, 4, 2 ; 4, 3, 3. 6 red balls. A, B, and C draw
_ . , ^ _ , , each a ball, in order, replacing.
Of ifrese the first can be drawn Fmd the chance ^ th haye
as 5, 5, 0 or 5, 0, 6 or 0, 5, 5, drawn
and the 0 may be any one of the
five blanks, giving 3X6=16 O-) Each a white ball,
ways. The fourth and fifth sete (ii.) A and B white, C red.
can be drawn in 3 ways each. <hi'> Two white «* 1 red'
The other two sets can be drawn (i.) Tne chance that one draws
in 6 ways each, making in all a white ball being T% = §, the
15 + 6 + 12 = 33 ways of draw- chance that they all draw white
ing a sum of 10. Dalls fa (2)8 = ^
The total number of possible
drawings is 10 X 10 X 10 = 1000. (**•) Tne chance that A and B
Hence the chance of drawing a draw white, and C red, is
sum of 10 is T$fo. (2)2 X} = ^.
10. Find the chance in Ex. 9 (iii.) Multiplying the last result
if the tickets are not replaced. ^y 3^ we ^Yq ^ ag tne chance
If the tickets are not replaced, that some one of the three draws
the only combinations which give red and the other white.
490
ALGEBKA.
12. Find the answer to Ex. 11
if the balls are not replaced.
The chance that A, B, and C
successively draw white balls is
now ^ x » x I = 3V-
The chance that A and B draw
white, and C red is T%x fx 5=^.
The chance that A draws white,
B red, and C white is ,4ff x « x }
= ^ ; and the chance that A
draws red, and B and C white
is A x i x I = *0. Hence the
chance that two white balls and
one red are drawn is ^.
13. A draws 4 times from a
bag containing 2 white and 8
black balls, replacing. Find the
chance that he has drawn
(i.) Two white, two black,
(ii.) Not less than two white,
(iii.) Not more than two white,
(iv.) One white, three black,
(i. ) Two white balls and 2 black
can be drawn in — -r = 6 differ-
ent orders. The chance of draw-
ing a white ball is at every trial
T2<j — h an(* tne chance of draw-
ing a black one is f . Hence the
chance of drawing 2 white balls
and 2 black is
« X (i)2 x d)2 = *%.
(ii.) The chance of drawing a
white ball 4 times is (£)*; of
drawing a white ball 3 times and
a black ball once, 4 X ( £)* + f ;
of drawing 2 white balls and 2
black ones, /z65, as just shown.
Hence the chance of drawing not
less than two white balls is
(S)4 + 4x(!)»xf + Vft = iif-
(iii.) The chance of drawing 2
white balls and 2 black is ^*5;
of drawing 1 white and 3 black,
4X JX ( j)8 ; of drawing 4 black,
(J)4. Hence the chance of draw-
ing not more than 2 white balls is
^ + 4XiX (4)8+ (4)4 = MJ.
(iv.) The chance of drawing 1
white and 3 black is
14. Find the odds against throw-
ing one of the two numbers 7 or
11 in a single throw with 2 dice.
To give 7 or 11, the number
thrown must be one of the sets
6, 1; 6, 2; 4, 3; 6, 5;
and each of these can be thrown
in two ways; for example, 6, 1
as 6, 1 or as 1, 6.
The total number of throws
with 2 dice is 36. Hence the
chance of throwing 7 or 11 is
4X2
oa = }, and the odds against
oo
it are 7 to 2.
15. If a copper is tossed 5 times,
what is the chance that it will
fall head either 2 times or else 3
times?
The number of ways it can fall
either head twice and tail three
times, or tail twice and head
teachers' edition.
491
three times is
I* _
10. The
12 I*
chance that it will fall head twice
and tail three times in given
order is (i)6. Hence the chance
that it will fall head twice and
tail three times in any order is
10 X (£)6 = -& ; and the chance
that it will fall head exactly
twice or exactly 3 times is
16. Find the same chance if
the copper is tossed 6 times.
The number of ways the cop-
per can fall head exactly twice is
I?
now .—77 = 15, and the number
[2 [4
of ways it can fall head exactly 3
•timesisi| = 20.
The chance that it will fall
head exactly twice, in one order
or another, is 15 x (£)6, and the
chance that it will fall head
exactly three times, in any order,
is 20 x (*)«.
Hence the chance that it will
fall head exactly twice or exactly
three times is
15x(i)« + 20X(i)« = if.
17. In one bag are 10 balls and
in another 6; and in each bag
the balls are marked 1, 2, 3, etc.
What is the chance that on draw-
ing one ball from each bag the
two balls will have the same
number ?
The number of pairs of balls
that can be drawn is 6 x 10 = 60,
and there are 6 pairs which have
the same number. Hence the
chance of drawing such a pair is
A = TV
18. A bag contains n balls. A
person takes out one ball and
then replaces it. He does this n
times. What is the chance that
he has had in his hand every ball
in the bag ?
The number of sets that can be
drawn in n times is nw. The
number of orders in which the n
balls can be arranged without rep-
etitions is [n. Hence the chance
that there are no repetitions in
\n
the drawing is — •
19. If on an average 9 ships
out of 10 return safe to port,
what is the chance that out of 5
ships expected, at least 3 will
return ?
The chance that all the 5 ships
will return is (-&)*; that 4 will
return and one be lost is 5x (r9ff)4
X T^ ; that 3 will return and 2 be
lost is 10 X (T%)» X (tV)2. Hence
the chance that at least 3 will
return is
x(A)'x(tW2
= (&)>(&*+ Ms +m
— 12893
492 ALGEBRA.
20. What is the chance of the other times is (J)3 x ($)J.
throwing doable sixes at least Hence the chance that it will be
once in 3 throws with a pair of thrown exactly 3 times out of
dice ? the 5 is 10 x (£)» x (j)« = flft.
The chance that they will be (|L) ^ chance ^ m m
thrown any particular time is ^ not ^ t^^ at au jg (f)5.
J x J = A, and the chance that ^ ^ ^ ^ ^ thrown onJy
they will not be thrown any once ^ 6 x (|)4 x , = (s)6. ^
particular time is 1 - *= ft ^ ^ wU1 be thrown exactly
Hence the chance that they will twice k 10 x (|)i x „)2 Hence
not be thrown any time is <ff )», ^ chance ^ ^ ^ wiu be
and the chance that they will be ^^ ^^ timeB or m0M fa
thrown is l-(^)* = AWs-
ML What is the chance of ^x^+^^AV
throwing 15 in one throw with (iii.) The chance that an ace
3 dice ? will be thrown exactly 4 times is
The sum of 15 can be thrown 6 x (i)4 x h and ***** ** wiU ^
as 6, 6, 3 3 ways, thrown 6 times is (J)5. Hence
a k' a a mn„J *he chance that it will be thrown
o, 6, 4 o ways,
5, 5, 5 1 way, more than 3 times te 6 X ®* X *
10 ways. + <*)* = tHj. and *he chance
that an ace will be thrown not
more than 3 times is f f J|.
The 3 dice can fall in 6 X 6 x 6
= 216 ways. Hence the chance
of throwing 15 is ^ = tJt- 23. In a bag are 3 white, 6 red,
22. In 5 throws with a single and 7 black balls, and a person
die what is the chance of throw- draws three times, replacing,
ing an ace Find the chance that he has
(i.) Three times exactly ? drawn :
(ii.) Not less than three times? (i.) A ball of each color,
(iii.) Not more than three times? (ii.) Two white, one red.
(i.) An ace can be thrown 3 ("*•) Three red.
1 5 (iv.) Two red, one black,
times out of 5 in rrrr = 10 ways.
12 1± (i.) In this case the balls can
The chance that it will be thrown be drawn in 6 different orders,
any particular time is £, and the The chance of drawing one ball
chance that it will be thrown 3 of each color in a particular order
particular times and not thrown is f»5 X ^z x T^ = ^|T, and the
493
chance of drawing a ball of each 6 is 5 X (})* x j. Hence his
color in any order is 6 x ^$, = ^f . chance of losing 3 games or less,
(ii.) Here the balls can be tnat k» nis chance of winning 2
drawn in only 3 different orders, games or more, is
and the chance is 1 - [(f)5 + 6 X (»)* X f ]
(iii.) The chance of drawing 3
red balls is (^)8 = jj.
26. The skill of A is double
that of B. Find the odds against
(iv.) The chance of drawing 2 A,g winni 4 Mfm B
red and 1 black is .„ 0 0
wins 2.
In order that A may wm 4
24. A and B play at chess, and games before B wins 2 it is neces-
A wins on an average 2 games gajy and sufficient that A should
out of 3. Find the chance of A's wm at least 4 games out of the
winning exactly 4 games out of first 5.
the first 6, drawn games being The chance that A will lose
disregarded. any particular game is i, and the
A may win 4 games out of the chance that he will lose some one
15 „r TT. of the first five games and win
first 6 in j-tt^ = 15 ways. His .. . . c ._ ° ._ /tX4 -0
[4|2 J the rest is 5 X £ X (f)4= /ft.
chance of winning any particular The chance that he will win all
game is £ , and of winning any 4 the games is (f )5 = /ft. Hence
particular games and losing the the chance that he will win 4
other 2 is (f)* Xtt)2= Jft. Hence . . ., « . . . 80 + 32
i.. ^ * . . a * games out of the first 5 is —rTT—
his chance of winning some 4 of 24d
the 6 games and losing the other 2 = JJf , and the odds against his
is 16 X /ft = /ft. doing it is 131 : 112.
25. A and B engage in a game ^ If B,a gkm ^ a certain
in which A's skill is to B's as game fa equal t0 tnree_fifths of
2:3. Find the chance of A's A^ find A,g chance of w|nning
winning at least 2 games out of fi gameg out of g
the first 5, drawn games not . . , ... . ,
, . . , A's chance of winning a single
bemg counted. . R XT. _ - 7
game is $ . His chance of win-
As chance of losing any par- ning exactly fi gameg QUt of g fa
ticular game is f . His chance of
losing all the games is (f)*, and JL x (<>\5X (§\8 = 55 x 5*x38.
of losing just 4 games out of the [5[3 \8/ \8/ 88
494
ALGEBRA.
His chance of winning exactly
6 games is
His chance of winning exactly
7 games is
o /6\7 3 0 67x3
8X(8)X8 = 8X-8^-
His chance of winning all the
games is /5\8_ —
\s) ~~ 88*
Hence his chance of winning
at least 5 games out of 8 is
6^X3» 6^X^
66X— — + 28X — —
65,
88 ' %s
= j£ (66 X 27 + 28 X 45 + 24
X 25 +126)
56
= g8X3497.
28. A bag contains 4 red balls
and 2 others, each of which is
equally likely to be red or white.
Three times in succession a ball
is drawn and replaced. Find the
chance that all the drawn balls
are red.
The chance that the 2 un-
identified balls are both red or
both white is in each case ^x{
= i ; that one is red and the
other white is £.
If the balls are all red, the
drawn balls are of course all red.
The chance of this is £.
If two of the balls were white,
the chance that the drawn balls
are all red would be (t)8 = ?V
Hence the chance that 2 balls
are white and the drawn balls all
red is J X ^ = fa
If only 1 ball were white, the
chance that the drawn balls are all
red would be ($ )8 = iff. Hence
the chance that there is only 1
white ball and that the drawn
balls are aU red is i X Jff = y|.
Therefore the chance that the
drawn balls are all red, whatever
the 2 unidentified balls may be, is
29. A man has left his umbrella
in one of three shops which he
visited in succession. He is in
the habit of leaving it, on an
average, once in every four times
that he goes to a shop. Find the
chance that he left it in the first,
second, and third shops respec-
tively.
The chance that he would leave
it in the first shop is £.
The chance that he would not
leave it in the first shop, but
would leave it in the second, is
*** = *•
The chance that he would not
leave it in the first or second
shop, but would leave it in the
third, is (f)2xi=TrV
Hence the probability that he
would leave it in one of the three
shops is i + A+A=H-
But as the umbrella was cer-
tainly left in one of the shops, all
teachers' edition. 495
the preceding chances must be he draws a white ball he is to
increased in the ratio of 64 : 37 receive $1, and every time he
(Rule IX.). draws a black ball he is to pay 60
Hence the chance that it was cents. What is his expectation ?
left in the first shop is f 7 x J = J7 ; A's expectation is the same for
in the second shop, §* x A = i* > eacn drawing, viz., T% x $1 — A
in the third shop, 1 1 x & = sV x $$ = — $ A His expectation
Hence, 16 : 12 : 9. from 6 drawings is therefore
30. A bets B $10 to $1 that he "* $h The chances are that he
will throw head at least once in wiU lose 26 cenfcs-
3 trials. What is B's expecta- 32. From a bag containing 2
tion ? What would have been a eagles, 3 dollars, and 3 quarter-
fair bet ? dollars A is to draw one coin and
The chance that A will not then B three coins; and A, B,
throw head at all in 3 trials is and C are to divide equally the
(i)8 = i ; and the chance that he value of the remainder. What
will throw head at least once is are their expectations ?
1 — £ = £. B's expectation is In the final division A, B, and
therefore £ of gaining $10, and C receive equal shares.
i of losing $1 ; that is, his ex- Consider their expectations
pectation is if- — J = $f. from this division.
A fair bet would have been in There are $23.75 in the bag.
the ratio of the chances of win- A and B are to draw out 4
ning and losing, that is, 7 to 1. coins at first.
o-i a j c *.- / i • \ 4 coins can be selected from 8
31. A draws 5 times (replacing)
from a bag containing 3 white jn I— = ^q wavs#
and 7 black balls. Every time [4[*
A and B may draw : wutyT
Amount
remaining
Expectation for the
division.
(1) 2 eagles, 2 dollars A
$1.75
7\x $1.75 = $AV
(2) 2 eagles, 1 dollar, 1 quarter A
$2.50
Ax $2.50 = $^
(3) 2 eagles, 2 quarters A
$3.25
AX $3.25 = $&%
(4) 1 eagle, 3 dollars A
$10.75
A X $10.75 = $TV*
(5) 1 eagle, 2 dollars, 1 quarter 7#
$11.50
7$X $11.50 = $W
(6) 1 eagle, 1 dollar, 2 quarters 7§
$12.25
ft X $12.25 = |ft
(7) 1 eagle, 3 quarters A
$13.00
A X $13.00 = $Jf
(8) 3 dollars, 1 quarter A
$20.50
a x $20.50 = mi
(9) 2 dollars, 2 quarters A
$21.25
^X $21.26= *W
(10) 1 dollar, 3 quarters A
$22.00
AX $22.00 =$H
Total expectation for the division $3AV_ or ^H-
496 ALGEBRA.
Hence the expectation of each from the division is £ of $11 f£,
or $3.862V
Hence C's entire expectation is $3.86^.
Again, A may draw an eagle. The probability of this is }, or £.
The corresponding expectation is $2.50.
Or, he may draw a dollar. The probability of this is |. The cor-
responding expectation of this is 37£ cents.
Or, he may draw a quarter. The probability of this is f . The
corresponding expectation is 9f cents.
Hence A's expectation from his draw is $2.96$.
And A's total expectation is $6.83^.
A and C's expectations together are $10.69^.
Hence B's expectation is $23.76 —$10.69^* = $13.05}|f
33. A, B, and C, staking each and C all fail the first time, and
$5, draw from a bag in which A the second time, and that B
are 4 white and 6 black balls, draws a white ball the second
each drawing in order, and the time is (|)4 x}= (f )* x £ ; and
whole sum is to be received by so on.
him who first draws a white ball. Hence B's chance of drawing
What are their expectations : a white ball first is
(i.) Replacing the balls ? A + A x (!)8 + A x (!)6 +
(ii.) Not replacing the balls ? = A "=■ L1 - (I)8] = tt-
(i.) The chance that A draws c's chance of winning is ther*-
a white ball the first time is £ ; f ore 1 — \ i - it = &•
that A, B, and C fail the first Hence the expectations of A,
time and that A draws a white B> and C respectively are
ball the second time is (J)8 x|; }( x $15 = $7f f ,
that they all fail twice and that Jf X $15 = $4f ? ,
A draws a white ball the third and & X $15 = $2ff
time is ($)« X $ ; and so on. The chances are that A will
Hence A's chance of drawing gam $2f$, that B will lose $\l
a white ball first is and that C will lose $2Jf
_ * + * (!)8 + I (*)6 + (ii.) The chance that A draws
- f "*- C1 - (!)8] - tt- (§ 396-) a white ball the first time is, as
The chance that A fails the first before, j. The chance that A,
time and that B draws a white B, C all draw black balls the
ball is | x -| = ^ ; that A, B, first time, and that A draws a
TEACHERS7 EDITION.
497
white ball the second time, is
f x | x | x $ = fa. The chance
that A, B, C fail each time is
jxjxf xjxf x i = Th. ^d
Amusi then draw a white ball.
Hence A's chance of drawing a
white ball first is
*+A + ii*=t
The chance that A fails the
first time and that B draws a
white ball is $ X J = f5. The
chance that A, B, and C fail the
first time, and A the second time,
and that B draws a white ball
the second time is
|xf x|xfx $=fa.
Hence B's chance of drawing
a white ball first is ^5 4- fa = £j,
and C's chance is 1 — J — ii == t J-
Hence the expectations of A,
B, and C respectively are
ix$15 = $7J,
"X$15=$4*,
and i$ x $15 = $2^.
The chances are that A will
gain $2£, that B will lose $$, and
that C will lose $2^ .
Exercise CXXIV.
Note. — Four-place logarithms do not give very accurate results
in some of the following problems.
1. In how many years will $100
amount to $1050 at 5 per cent
compound interest ?
P=100, 12=1.05, -4 = 1050.
100 X (1.05)»=1050,
(1.06)»=10.5,
nlog 1.05 = log 10.5,
(ii.) B=AR».
.-. n log B = log B — log A,
_ log B — log A
n~~ log*
The time is — ^—. — ^* years.
log-R
_ log 10.5
n log 1.05
_ 1.0212
" 0.0212
= 48, nearly.
The time is 48 years, nearly.
2. In how many years will $-4.
amount to $2? (i.) at simple in-
terest, (ii.) at compound interest,
r and R being used in their usual
sense? •
(i.) —£- years.
3. Find the difference (to five
places of decimals) between the
amount of $1 in 2 years, at 6 per
cent compound interest, accord-
ing as the interest is due yearly
or monthly.
If the interest is due yearly,
A = (1.06)2 X 1
= 1.1236.
If the interest is due monthly,
A - (1.005)» x 1
= 1.12923.
Hence the difference is
$0.00563.
498
ALGEBRA.
4 At 5 per cent, find the
amount of an annuity A which
has been left unpaid for 4 years.
By § 452, the amount due is
S(ifr-l)_^[(1.05)«-l]
r 0.05
= A (0.2155)
0.05
= 4.31.4, nearly.
5. Find the present value of an
annuity of $100 for 5 years,
reckoning interest at 4 per cent.
1
_ 100 1.045
~ 0.04 X 1.04&
= 444.
The present value is $444.
6. A perpetual annuity of §1000
is to be purchased, to begin in 10
years. If interest is reckoned at
3£ per cent, what should be paid
for it ?
8
P =
12* (22-1)
1000
P =
S
12-1
12»-1
2J»
1.035W x 0.035
= 20,270, nearly.
The amount paid should be
$20,270.
7, A debt of $1850 is discharged by two payments of $1000 each,
at the end of one and two years. Find the rate of interest paid.
Amount of $1850 for 2 years = 1850 E2,
Amount of $1000 for 1 year =100012,
Balance due = 1000.
.-. 1850 122 - 1000 B = 1000,
37222-2012 = 20,
__ 10 + V840
*~ 37
= 1.0535.
Hence the rate of interest is 5.35 per cent.
8. Reckoning interest at 4 per __ 80
cent, what annual premium should
be paid for 30 years in order to
secure $2000 to be paid at the
end of that time, the premium
being due at the beginning of
each year ?
Ar
P =
12(22«-1)
2000 X 0.04
1.04 (1.04*> - 1)
1.04 X 2.2426
= 34.402.
The annual premium should be
$34.40.
9. An annual premium of $150
is paid to a life insurance com-
pany for» insuring $5000. If
money is worth 4 per cent, for
how many years must the pre-
TEACHERS7 EDITION.
499
mium be paid in order that the
company may sustain no loss ?
2J»=1 +
(1.04)»=1 +
r
At
T'
6000X0.04
160
= 1 + *^
T150
n log 1.04 = log 7 — log 3,
= log 7 -log 3
71 log 1.04
_ 0.36798
0.01703
= 22 nearly.
The premium must be paid for
22 years.
10. What may be paid for bonds due in 10 years, and bearing
semi-annual coupons of 4 per cent each, in order to realize 3 per
cent semi-annually, if money is worth 3 per cent semi-annually ?
ByS464, PQ+^a+^W + V-1!
P = 9 + r(l + g)» — r
8 q (1 + x)»
_ 0.03 + 0.04 X 1.032Q - 0.04
0.03 X (1.03)«>
0.03 + 0.04 X 1.8063 - 0.04
0.03 X 1.8063
= 0.062262
0.064189
= 1.16, nearly.
The price paid should be 116, nearly.
11. When money is worth 2 per cent semi-annually, if bonds
having 12 years to run and bearing semi-annual coupons of 3i per
cent each are bought at 114|-, what per cent is realized on the
investment ?
1+x= /ag + arq + qy— a-y
i
-(
0.02 + 0.035 (1.022* - i) v A
1.14126 0.02 /
600 ALGEBRA.
-(
1 0.02 + 0.035 X 0.6084 \ A
1.14125 0.02 /
_ / 2.0647 \ A
a.U125/
= 1.025.
The per cent realized is 2£ semi-annually ; that is, 5 per cent.
12. If $126 is paid for bonds due in 12 years and yielding 3j per
cent semi-annually, what per cent is realized on the investment,
provided money is worth 2 per cent semi-annually ?
1+x=/^+5L(L+1)^15rxI
/_1_ 0.02 + 0.035 (1.022* - i) v A
Vl.26 0.02 /
= /_1_ 0.02 + 0.035 X 0.6084 \ A
Vl.26 0.02 /
/2.0647\A
V 126 )
= 1.0207.
The per cent realized is 4.2, nearly.
13. A person borrows $600.25. How much must he pay annually
that the whole debt may be discharged in 35 years, allowing simple
interest at 4 per cent ?
Let P = amount of debt.
S = annual payment.
Then PR, = §I^ZD.
r
P= 600.25,
r=0.04,
B = 1.04,
w=35,
e 600.25 X 0.04 X 1.04*6
o — —
1.0485 - 1
- 2.935" 32*3'
He must pay $32.30 per year.
teachers' edition. 501
14. A perpetual annuity of $100 a year is sold for $2600. At
what rate is the interest reckoned ?
$100 is the interest on $2500 for 1 year. Hence the rate is 4
per cent.
15. A perpetual annuity of $320, to begin 10 years hence, is to be
purchased. If interest is reckoned at 3 J per cent, what should be
paid for it ?
Let P denote the amount paid. Then at the end of 10 years P is
worth P x 1.03210 = P x 1.37026. The interest on this amount at
3 J per cent should be $320. Hence
P X 1.37026 X 0.032 = 320
10000
1.37026
= 7298.
The price paid should be $7298.
16. A sum of $10,000 is loaned at 4 per cent. At the end of the
first year a payment of $400 is made, and at the end of each follow-
ing year a payment is made greater by 30 per cent than the pre-
ceding payment. Find in how many years the debt will be paid.
The amount of the original loan at the end of n years is
10,000 x 1.04".
The amount of the several payments at the end of the n years is
400 X 1.04"-1 + 400 X 1.30 X 1.04»"2
•f 400 x 1.302 X 1.04»-s +
rl.30>
--""[(ST-']
1.30
1.04
0.26
When the debt is paid,
-"•-[(jS)'-']
10,000 X 1.04» = jj^g <
502 ALGEBRA.
10000 X 0.26 /1.30>
400
ll.04j + 400
= 7.5,
n log — = log 7.6,
log 7.5
log 1.30 - log 1.04
_ 0.87506
0.11394 - 0.01703
= 9, nearly.
The debt will be paid in 9 years.
17. A man with a capital of $100,000 spends every year $9000.
If the current rate of interest is 5 per cent, in how many years will
he be ruined ?
The amount of the original capital at the end of n years is
100.000 x 1.06".
The sums spent yearly amount at the end of the n years to
9000 X 1.06^-1 + 9000 X 1.05—2 + 9000 x 1.06*-« +
9000 X LOfrg-ip-Ofr-*— 1).
1.05-1-!
When the money is all spent,
ioo.oooxi.os-^0000^106:^-1),
— 0.06
ioo,ooo = <)000<1-106"<)'
0.00
l-1.06-» = !»
1.06-* = |>
_ log 9 — log 4
n~ log 1.06
= 0.3521
0.0212
= 17t nearly.
He will be ruined in 17 years.
teachers' edition.
503
18. Find the amount of $365
at compound interest for 20 years
at 5 per cent.
The amount is
$365 X 1.05» = |969.
19. In how many years will
§20 amount to $150 at 4 per cent
compound interest ?
20 X 1.04* = 150,
log 150 — log 20
W~~ log 1.04
_ 2.17609 -1.30103
0.01703
= 51, nearly.
The required time is 51 years.
20. At what rate per cent, com-
pound interest, will $2500 amount
to $3450 in 7 years ?
2500 i*7= 3450,
7/3450
/2600
R =
= V1.38
= 1.047.
The rate is 4^ per cent.
21. If the population of a
State increases in 10 years from
2,009,000 to 2,487,000, find the
yearly rate of increase.
2,009,000 Jjw = 2,487,000,
>/2487
/2009
= 1.0216.
The annual rate of increase is
2\ per cent, nearly.
10/2
22. The population of a State
now is 1,918,600, and the yearly
rate of increase is 2.38 per cent.
Determine its population 10 years
hence.
The population 10 years hence
will be
1,918,600 X 1.0238W = 2,428,000.
23. A banker borrows a sum
of money at 3£ per cent, interest
payable annually, and loans the
same at 5 per cent, interest pay-
able quarterly. If his annual
gain is $441, determine the sum
borrowed.
If the sum is A, it will amount
in one year, at 3£ per cent
annually, to
A x 1.035.
At 5 per cent, payable quar-
terly, it will amount to
A x 1.0125* = AX 1.051.
Hence
A (1.051 -
1.035) =
441,
A X 0.016 =
441,
A =
441
0.016
27,563.
The sum borrowed was $27,563.
504 ALGEBRA.
Exercise CXXV.
1. Find continued fractions for \tf; ty; V6; VlT; 4V6;
and And the fifth convergent to each.
r\ !?? = 1 = 1
157 l + £ 1+ *
123 21
3 + 34
1 + 3-v 1 + 1
3 + ^r 3-f1
1 + 2 1+1
21 1 + i-
r13
1 + 1
3 + 1
1 + 1
1+1
1 + 1
1 + 1
1 + 1
Fifth convergent = \\.
... , 169 . , 18
(«•) 4f=3 + 47
= 3 + -*
-1
2 + 1
1 + 1
1 + 1
1 + 1
-I
Fifth convergent = 3^.
teachers' edition. 505
(iii.) Let
then
X
1
X_ V5-2
= Vs + 2.
Let
then
V6
+ 2 = 4 + i>
y
1
""V5-2
= x.
Hence
./7_9 , 1
vo — M -
4+ i
4 + 4 + etc.
-.+*■
Fifth convergent = 2^*-
(iv.) Let
then
Let
then
Let
then
VH=3 + i>
X =
1
Vii-3
Vn + 3
2
2
+ 3_
8 + 1.
y
y-
2
Vn-3
Vn + 3.
Vn
+ 3 =
.+*.
z =
l
Vll-3
X.
606 ALGEBRA.
Hence Vll = 3 +
3-h1-
6 + 3 + etc.
-3 + 1 1-
Fifth convergent = 3j^97.
(v.) Let 4V6=9 + 1
4^6-9
.4V6 + 9,
15
4V6 + 9,., , 1
** — 15--1 + y*
15
4V6-6
2V6 + 3
—2—'
2V6 + 3_0 , 1
Let ———g + j.
.-. z =
2V6-3
4V6 + 6,
15
4V5 + 6,, , 1
** — is- -1+s*
15
4V6-9
= 4V6 + 9
= 18 + i»
x
. m ~ . i ! 1 1
,.4V6 = 9 + I+3 + T+ig;
Fifth convergent is %°-
teachers' edition.
507
2. Find the continued fraction for $T; J$J; UHl *!$£'> and
find the third convergent to each.
(i.)
4T_
257'
5 + -
2065.
4626"
(iii.)
2 + -
2 + -
4 + ^
7 + ;
Third convergent = J j.
6 +
1
8 +
^ = 2 +
204 ^
(u.)
1
Third convergent =
2991
10
568
= 6 +
(iv.)
1
3 + -
4 +
1
1 +
1
6 + ;
3 +
•+s
Third convergent = |JJ. Third convergent = 6^.
3. Find continued fractions for V21; V22; V33; V56.
_ V21 + 3
(i.)
Let V21 = 4 + -»
x
then
Let
then
Let
then
V21-4
3
3 u
V21 + 4 ^en
V21+4 , ,1
K = 1 + -'
6 y
V21-3
V21 + 3
y =
V21-1
V21 + 1
Let ^±3=1 + 1,
4 v
then
V21 + 1 , , 1
= 1 + -»
4 z
V2T-I
V21 + 1
V21-3
Let M±i = 1+i,
5 w
508 * ALGEBRA.
»=^f-7 Let ^ = 2 + i
w V21-4 "" ¥~~"^V
= V21 + 4. t. 3
, then v = — }=
Let V5l + 4 = 8 + l. ^-2
1 _ V22 + 2
• then ' t = —= 6
^-4 T4 V22 + 2 , . 1
Hence 6 w
• ft
-v/oT — ii1 1 * 1 * * then 10 = —7=
V21-4+l + l + 2+l + T+8" ^22-4
= V22 + 4.
(".) _ !
,— j , 1 Let V22 + 4 = 8 + ->
Let V22 = 4 + - » *
x 1
1 then t = —=
then x = —7= V22 — 4
V22-4 =x
= V22 + 4 Hence
r- 6 /— r^.llllli
Lrt V=±i=1 + 3f ^ = 4 + l + 2 + 4 + 2 + l + 8'
6 y
6
then y = —p= (iii.)
V22-2 v '
V22 + 2 Let V33 = 5 + ->
=r • X
3
T V22±_2__,l then x = —rJ
L<* — 3 — = 2 + ;' V33-5
3 V33 + 5
then z = — f= = «
V22-4 8
_V^2 + 4 Let M±*=1 + l,
""2 8 y
T V22 + 4_ A , 1 then y = -pJ
Let — 2 — 4+V V33-3
then «=_L^, = V33 + 3.
V22 — 4 3
V22 4-4 _ . V33 + 3 . , 1
= — A Let — A" = 2 + ^'
509
* = #i -t ^±I-.+i.
8 V65 - 6
^ ^3±3 = 1 + I, -V66 + 5
8 u 6
then «=-7=K Let ^|±-5 = 2 + i,
V33-5 "» — T
z
= V33 + 6.
then
Let V33 + 6=10 + i. ^~5
« _ V66 + 5
then o = -?^ ,— 6
V38-6 Vtt_+5 = 1,
= x. 6 u
Hence 0
i 1 1 1 ^en u — "7=
= V65 + 7.
(iv.) Let V65+7 = 14 + i*
1 W
Let V66=7 + i» t- 1
x tnen v — — =
! V65-7
then x =
VS5 - 7 Hence
6 V0° '^2 + 2 + 2+14
4. Obtain convergents, with only twojigures in the denominator,
that approach nearest to the value of VlO; Vl5; Vl7; Vl8; V20.
(i.)
Let
Vl0 = 3 + i>
X
1
ten
~~ VlO-3
= Vio + 3.
Let
Vl0 + 3 = 6 + i»
then y =
VlO-3
Hence Vl0 = 3 + i»
o
Convergents = }, ^, J#,
The required convergent of VlO
510 ALGEBRA.
(U) Let Vl8 + 4 = 8 + *>
Let Vl6=3 + i'
z
X
then
then
1 Vl8-4
Vl5-3 = x.
^15 + 3 Hence Vl8 = 4+7
6
4 4-8
w77,o , Quotieats =4,8,4,8,
6 = 1 + - ' Convergent* = fc Jft W»
6 The required convergent of Vl8
^ ^Vl5-3 l-W
= Vl6 + 3. (iv<)
Let Vl6 + 3=6 + i> j
2
1
then
Let V20 = 4 + -
x
Vl5-3
= X.
then
V20-4
Hence Vl6 = 3 + i 1. _ V20 + 4,
1 + 6 — 4
Quotient* = 1, 6, 1, 6, -v/on + 4 1
Convergents = f , f, V. ¥» Let -^j^ = 2 + ->
w. -w> /
The required convergent of Vl6 then y = r—
is W-
(iii.)
= V20 + 4.
Let
Vl8 = 4 + -» Let V20 + 4 = 8 + i»
x z
then x = — p= then
Vl8-4 V20-4
_ Vl8 + 4 =*»
2 / — i i
_ Hence V20 = 4 + -, 5»
_ , V18 + 4 . , 1 2 + 8
Let r = 4 + - »
then 2/ =
2 y Quotients = 2, 8, 2, 8,
_2 Convergents = f , f , ff , W>
V 18 — 4 The required convergent of V20
= Vl8 + 4. is W-.
teachers' edition. 511
5. Find the proper fraction 6. Find the next convergent
which, if converted into a con- when the two preceding conver-
tinued fraction, will have quo- gents are ^ and Jf , and the next
tients 1, 7, 5, 2. quotient is 5.
Quotients = 1, 7, 5, 2. Quotients = ,5,
Converges = h h *, H) » jg^T" = ' ** *
The fraction is $$. The next convergent is ¥9/*-
7. If the pound troy is the weight of 22.8157 inches of water,
and the pound avoirdupois of 27.7274 inches, find a fraction with
denominator < 100 which shall differ from their ratio by < 0.0001.
22.8157 = 1
27.7274 1
4 + 1
1 + 1
1 + 1
1 + 1
4 + 1
1 + 1 + etc.
Quotients = 1, 4, 1, 1, 1, 4, 1, 1,
Convergent^ f, {, f, J, A, ih «» it. ltt>
The nearest convergent with denominator < 100 is therefore Jf .
The difference between it and the actual value of the ratio is
^o^TTts0^0-0001-
8. The ratio of the diagonal to the side of a square being V2,
find a fraction with a denominator < 100 which shall differ from
their ratio by < 0.0001.
Let V2 = l+i>
then x =
x
1
V2-1
= V2 + 1.
Let V2 + l = 2 + i>
y
512 ALGEBRA.
then v = — := r- 1
V2-1 Hence V2 = l+±-
= x. 2
Quotients = 2, 2, 2, 2, 2, 2,
Convergent^ J, f, J, H, fi> rh !B.
The nearest convergent with denominator < 100 is therefore fg.
The difference between it and V2 is < .„ w 100 or < 0.0001.
70 X lot*
9. The ratio of the circumference of a circle to its diameter being
3.14150265, find the first three convergent^, and determine to how
many decimal places each may be depended upon as agreeing with
the true value.
3.14150265 = 3 + — Kr
7 + -
15 + 1
1 + -*-
T 288 + etc.
Quotients = 7, 15, 1, 288,
Convergent^ f, ^, flfc, fff. "WW.
The first convergent, ^ , differs from the true value by < 7
or < 0.01; the second, f Jf, by < 1Q6 * ng or < 0.0001 ; the third,
tft. ^ <113x132657 °r < 0-000001-
10. Two scales whose zero-points coincide have the distance
between consecutive divisions of the one to those of the other as
1 : 1.06677. Find what division points most nearly coincide.
1 +
15 + 1
4 + 1
1 + 1
8 + 1
11 + 1
2 + 1
1 + 1
-I
teachers' edition. 513
Quotients = 1, 15, 4, 1, 8, 11, 2,
Convergent^ J, }, ft, ft, Jf, ftf, ft|J,
Hence the 15th, 61st, 76th, divisions of the first scale nearly
coincide with the 16th, 65th, 81st, .... divisions of the other, the
coincidence becoming closer as the series proceeds.
11. Find the surd values of
o+i !. i 1 !. 1+i i i.
M + 6' 3+1 + 6' ^2+3+4
i i 27x + 4x2=7 + x,
(L) x~l + 6 4x2+.26x=7,
1 - 13 + Vl97
1 +
6 + x
4
ill
__6 + x (iii.\ x =
x2 + 6x = 6, = 1
x2 + 6x+9=15, 2 + i
x + 3=±Vl6, __ 3 + _„
x=-3±Vl6. 4 + x
Hence the required value is = 1
3-3±Vl5 = Vl5. 2 + jJf^
1 1 i =13 + 3x,
3+1 + 6 30+7x
x 30x+7x2=13 + 3x,
: i~ 7x2 + 27x=13,
3 + ^
1 Q» L Vl093
T6 + x T¥ 14
. 1 The required value is
"o,6 + x VJ093
3+7 + x ^tt + ^M"
7 + x VJ093-13
'27 + 4x' * 14
The surd values are
A/Tr -13 + VJ97 , V1093-13
V15; ; and -
514 ALGEBRA.
12. Show that the ratio of the diagonal of a cube to its edge may
be nearly expressed by 97 : 56. Find the limit of the error made in
taking this ratio for the true value.
The true ratio is V3.
By§403, V§=1 + I+|"
Quotients = 1, 2, 1, 2, 1, 2, 1, 2,
Convergents= f, |, J, J, A, H» If. tt. iii-
The 7th convergent is If} or §J, and it differs from V3 by
<56xl63°r<^nr'
13. Find a series of fractions converging to the ratio of 5 hours
48 minutes 51 seconds to 24 hours,
6 hrs. 48 min. 51 sec. = 20,931 sec.
24 hrs. = 86,400 sec.
20931 = 1
86400 A , 1
4 +
7 + i
1 + i
4 + i
1 + i
1 + 1
1 + i
1 + i
3+i
2 + i
* + l
Quotients = 4, 7, 1, 4, 1, 1, 1, 1, 3, 2, 1, 2.
Convergent* = f, \, &, &, #T, ^, f&, iff, ftf, #&, Hff,
TEACHEKS' EDITION. 515
14. Find a series of fractions converging to the ratio of a cubic
yard to a cubic meter, if 1 cubic yard = 0.76453 of a cubic meter.
0.76453 = X
' 1+— T
3+—T
4+— T
19 + — j-
1+—
Quotients = 1, 3, 4, 19, 2, 3, 1, 1, 5, 1, 2.
Convergents=J, {, *, ^ fWf m im, m m }m
Exercise CXXVI.
1. Expand - — z— to four terms in ascending powers of x.
2 — oX
Let — ^-r- = A + Bx+Cx* + Dx* + ,
2 — 3x
tken 1 = 24 + (2B-SA)x + (2C-35)x2
+ (2D-3C)x3+
.•.24 = 1, 4 = *,
25-3.4 = 0, B=&
2C-3£ = 0, C=f,
2D-3C=0, D=ih
Hence, 3^ = i+ **+ f *2 + Bx* +
1 + x
2. Expand to four terms in ascending powers of x.
Let -y^" = A + Bx+Cx* +Dx* +
2 + 3x
516 ALGEBRA.
then l + x=2^4 + (2B + ZA)z + (2C + 3B)x2
+ (2D + 3C)xH
.-.2.4 = 1, A = ±,
2B+SA = 1, J5=-i,
2C + 3£ = 0, C=J,
2Z> + 3C=0 D=-^,
Hence' ^p^=i-ia; + i^-A^ +
3. Expand ' __ to four terms in ascending powers of z.
I** Sa ~~ I X = ^ + Bx + Cz* + Dx« +
4 — 3x
then 3-2x = 4.4 + (45-3.4)x
+ (4C-3£)x2 + (4D-3C)x» +
.-.4.4 = 3, -4 = f,
4J5-3^1 = -2, B=T\,
4C-3£=0, C=&>
4D-3C=0, D=jh>
Hence, f^= ! + A*+ &*2 + rf**8 +
1 $
4. Expand ~ — ^— ^ to four terms in ascending powers of x.
Let , 1"~* , = 4 + Bz + Cx2 + Dx* +
1 — X + X2
then l-x = .A + (£-4)x + (C-]3 + 4)x2
+ (D-C+£)x8+
.-.-4 = 1, -4 = 1,
J5T — 4 = -lf 5=0,
C-.B + ^ = 0, C=-l,
D-C+5 = 0, D=-lf
Hence -£- ;=1 — x2 — x» +
1 — x + x2
teachers' edition. 517
5. Expand — 2 to four terms in ascending powers of x.
*** ; — o 1-lq 2 = A + ^x+ Cx2 + Bx +
1 — 2x+ 3x2
then l = A + (B-2A)x + (C-2B+ZA)x*
+ (D-2C+32*)x8 +
.\A = 1, A=l,
B-2A = 0y B = 2,
C-2B + 3A = 0, C = l,
D-2C + 3B = 0, D=-4,
Hence . — — , 0 , = 1 + 2x + x2 — 4x8-
1 — 2 x + 3 x2
5 — 2x
6, Expand , q — 3-^ to four terms in ascending powers of x.
L^t , f o 2X o = ^ + Bx + 0x2 + i)X8 +
1 + 3x — x2
then 6-2x = ^+(B + 34)x+(C + 3J3-4)x2
+ (D+3C-£)x8+
.:A = 6, A = 5,
£ + 34 = -2, £=-17,
C + SB-A = 0, C=56,
2) + 3C-J?=0, D=-185,
Hence tJ*Q 2g .= 5-17x+ 56x2- 185x»+
l + 3x — x2
4 /j. 6 x2
7. Expand __ — 2 to four terms in ascending powers of x.
*** , 4o""?f » = 4 + £s+Cx2 + flx» +
1 — 2 x + 3 x2
then 4x-0x2 = -4 + (B-2A)x+ (C- 2J5+ 3-4) x2
+ (D-2C + 35)x8+
.-.4 = 0, -4 = 0,
5-2.4 = 4, B=4,
C-2JB + 34 = -6, C=2,
2)-2C+3J3=0, D=-8,
HenCe 1lX2g+X3x2 = 4g + 2x2"8g8-
518 ALGEBRA.
8. Revert the series y = x + z2 + xfi +
Le€ x = Ay + By*+ Cy* + Dy* +
then y = Ay + By2 + Cy* + Dy* +
+ -/tV*+ 2ABy*+ (B2 + 24C)y« + -
+ A*y* + 2A*By*+
+ 4*y* +
.:A = 1, A = 1,
B + A2=0, J5= -1,
C+2AB+A* = 0, C=l,
D+£* + 2.AC+342JB+4* = 0, D=-l,
Hence x = y — ^H-y8 — y4-}-
9. Revert the series y = x — 2x" + 3x* —
Let x = ^ly + By«H-Cy»+Dy*H-
then y = Ay + J5y2 + Cy'+Dy4^-
- 24*y2 - 4ABy» - (2 IP + 4 AC) y* +
+ 3-/t*y» + 942By* +
-4^V +
.-.4 = 1, 4=1,
J3-242 = 0, J5=2,
C-44J3 + 34* = 0, C=5,
D-2£*-44C+942J5-44* = 0, 2) =14,
Hence
Let
then
Hence
x = y + 2y*+6y»+14y* +
jvert the series y = x — J x*-h Jx6— |x7+
x = 4y + J3y2 + Cy3 + Dy<+.Ey* +
y = Ay+Btf+Cy* + I>y* + Ey* +
-\A*y*-
-^25y4-^J52y6-
+ i^V+
.-.4 = 1,
4 = 1,
JB=0,
£=0,
~ -4s
c-T = o,
c=h
D-42J3 = 0,
D=0,
^-^452 + ^^ = 0,
J?=-J.
& = y + Jy8 -12^+ -
teachers' edition. 519
X2 X8
11. Revert the series y = x + ^-^ + ^ x 2x g +
Let x = 4y + By*+Cy* + Dy* +
then y = 4y + By2 + Cy8 + ify4 +
+%iA*p+ABp + ($B2 + AC)y*+ ••
+ J^V8 + -i^2By4+
+ A-^V+
.-. il = 1, A = 1,
C + AB+iA* = 0, C=i,
D+iB* + AC+$A*B+v\A* = 0, D=-},
Hence x = y — Jy2+ Jy8 — J^+
12. Find the fractions in the form — ; ; — ; whose expansions
, . . p-r qx-r rx*
produce the series :
l + 3x+2x2-x8-
3 + 2x + 3x2 + 7x8 +
*-A* + H*-Hf* +
(*•) l*hl a=l + 3g + 2x*-x8-
v p + Q'x + rx2
a + 6x = j>+ (3p + g)x + (2p + 3 g + r) X*
+ (-p + 2g + 3r)x8 +
.\a = p, 6 = 3p + g,
0=2p + 3g + r, 0=-p + 2g + 3r.
Eliminating r from the last two equations,
0=7p+7g.
r = P,
o = Pi
b=2p.
Hence a + 6x = p + 2px
p+gx + r p— px + px2
= l + 2x
1-X+X2'
J
520 ALGEBRA.
<**•) fl + ^ = 3 + 2x+3x2 + 7x« +
* ' p + qx + rx*
a + bz = Zp + (2p + 3q)z + (3p + 2 g + 3r)x2
+ (7p + 3g + 2r)x» +
.-. a=3p, .& = 2p + 3g,
0 = 3p + 2g + 3r, 0=7p + 3g + 2r.
Eliminating r from the last two equations,
0= — \bp — 5g.
.-.g=-3p,
r = l>,
a = 3p,
6=-7p.
_ o + te 3p — 7imc
Hence — : — z = — ^ f — i
p + gx + rx2 j> — 3 jix + px*
3-7x
l-3x + x«'
<tt)y+fl*y^-A« + H*-Ht* +
a + te= }P + (~A* + }«)*
+ (HP-A«+fr)«P
+ (-«iP + «9-Ar)x»
+
o = Hp-A« + fr, 0=-HtJ» + ttff-A'-
Eliminating r from the last two equations,
0 = A%l> + fi9.
r=-fp,
a = }p,
6 = 0.
Hence <* + *** - iP
p + qz -f rx2 p + Jpx — Jpx*
_ 3
4 + x— 5x2'
teachers' edition. 521
7x + 1
13. Resolve - — j-tt- ^ **>**> partial fractions.
(x + 4)(x — 5) '
it 7g + l A B
1 (x+4)(x-5) x+4 x-5
then 7x + 1 = A (x- 5) + B(x + 4)
= (,4 + B)x-5A + 4B.
.•.^i + B=7,
-5A + 4B=1,
4 = 3,
5=4.
7x + l _ 3,4
Hence - — . .v . rr = — r~7 +
(x + 4)(x-5) x + 4 x-5
14. Resolve - — r~zn — ttt ^° partial fractions,
(x + 3) (x + 4)
' 6 A B
61 (x + 3)(xX4) x + 3 x + 4'
then 6 = A(x + 4) + B(x + 3)
= (4 + B)x + 4A + 3B.
.-. A + B=0,
4A + 3B = 6,
4 = 6,
B=-6.
6 6 6
Hence
(x + 3)(x + 4) x + 3 x + 4
5X \
15. Resolve — — into partial fractions.
(2 X — 1) (X — O)
Let
5x-l A B
(2x-l)(x-5) 2x-l x-5
then 5x-l = 4(x-5) + B(2x-l)
= (4 + 2B)x-(5A + B).
.-. 4 + 2B=5,
54 + B=l,
A=-h
B=f.
5x-l = — » | t
(2x-l)(x-6) 2x-l x-5
__ 8 1
3 (x-5) 3(2x-l)'
Hence
522 ALGEBRA.
x — 2
16. Resolve t __ g _ 1Q into partial fractions.
Let
x-2 A , B
xa — 3x— 10 x — 6 x + 2
then x — 2= 4 (x + 2) + JS(x — 5)
= (A + B)x+2A-bB.
.:A + B=\,
2A-6£=-2,
*=f
x*-3x-10 x-6 x+2
= 3 + — *
7 (x - 5) T 7 (x + 2)
17. Resolve ,_. into partial fractions.
3 A , Bx+C
Let -r r- = r + "
x»-l x-1 x2 + x+l
then 3 = A (x2 + x + 1) + (Bx + C) (x - 1)
= (4 + £)x2 + (A -B+C)z + A- C.
..A + B = 0, A = l,
A-B+C = 0y B=-l,
^1-C=3, C=-2.
3-4 = 3,
3 __ 1 x + 2
Hence
x8— 1 x— 1 x2+x+l
x9 — x — 3
18. Resolve —r-z — — into partial fractions,
x (x^ — 4)
T«t x2-x-3_;l B C
1M x(x2-4) ~ x x + 2i"x-2'
then x2-x-3 = 4(x2-4) + £x(x-2) + Cx(x+2)
= (J4 + B+C7)x2+(2C-2JB)x-4^l.
.-.4 + B+C=l, 4=|,
2B-2C=1, B=i,
44 = 3, C=-f
x*-x-3 3,3 1
Hence — — — jr = - — h -
x(x2-4) 4x 8(x + 2) 8(x-2)
teachers' edition. 523
19. Resolve — — r-^r into partial fractions.
x2 (x + 5)
3x2-4 A , B , n
Let -r~. — r-=r = — + ~Z + '
X2(X + 5) X X2 X+5
then 3x2-4 = 4x(x+6) + £(x+6) + Cx2
= (4+C)x2 + (64 + B)x + 6B.
.-.4 + 0=8, B=|,
&A + B=0, -A = A.
5D=-4, C=fr
3x2-4 4 4 , 71
Hence -z-t — r-77 = ^ t~z + ;
x2(x + 6)
7x*-x
(X
-l)2(x + 2)
7X2 — x
26x 6x2 ' 25(x + 6)
20. Resolve ^^^a/^o-ox ^^ P*1^ fractions.
.^ ~ A B C
^t (x-l)2(x+2) x-1 ",'(x-l)2't"x+2,
then 7x2-x = 4(x-l)(x + 2) + B(x + 2)
+ C(x-1)2
= (i+Cf)xa+(i + B-2C)x
-24 + 2B+C.
.-.4 + 0=7, 5=2,
A + B-2C=-1, A = ±£,
-2A + 2B+C = 0, C=^.
35 = 6,
7x2-x 11 , 2 10
Hence zrr-t — r^7 = tt; t: + 7 TT* + ;
(x-l)2(x + 2) 3(x-l) (x-1)2 3(x + 2)
O a»2 7 X + 1
21. Resolve ^^r into partial fractions.
T «. 2x*-7x+l A , Bx+C
Let rn = — rr + "
x8+l x+1 x2-x+l
then 2x2- 7x + 1 = A (x2- x + 1) + (Bx + Cj (x + 1)
= (4 + B)x2 + (- -4 + B + C)x + -4 + C.
.-.4 + B=2, il = V,
-4 + B + C=-7, B=-|,
4+C=l, C=-J.
2x2-7x+l 10 4x+7
Hence
x»+l 3(x+l) 3(x2-x + l)
524 ALGEBRA.
Exercise CXXVII.
1. Find the fiftieth term of 1, 3, 8, 20, 43,
Series =1 3 8 20 43
lstdiff. = 2 6 12 23
2d diff. = 3 7 11
3d diff. = 4 4
4th diff. = 0
.\ a = 1, a\ = 2, a8 = 3, a* = 4, a* = 0.
Substituting in formula, we have :
40 x 4ft
Fiftieth term = l + 49x2+gx3
49 X 48 X 47 .
+ 1X2X3 *
= 1 + 98 + 3628 + 73696
= 77323.
2. Find the sum of the series 4, 12, 29, 55, to 20 terms.
Series =4 12 29 55
lstdiff. = 8 17 26
2d diff. = 9 9
3d diff. = 0
.-. a = 4, di = 8, da = 9, as = 0.
Sum to 20 terms
= 20(4+76+613}
= 11860.
3. Find the twelfth term of 4, 11, 28, 55, 92,
Series =4 11 28 55 92
lstdiff. = 7 17 27 37
2d diff. = 10 10 10
3d diff. = 0 0
.-. o = 4, a\ = 7, 02 = 10, a8 = 0.
Twelfth term = 4 + 11x7 + U * 10 x 10
= 4 + 77 + 660
= 631,
teachers' edition. 525
4. Find the sum of the series 43, 27, 14, 4, — 3, to 12 terms.
Series =43 27 14 4 -3
lstdiff. = -16 -13 -10 -7
2d diff. = 3 3 3
3d diff. = 0 0
.-. a = 43, a\ = — 16, a2 = 3, a8 = 0*
Sam to 12 terms
= 12J43-¥X16 + I!I^X3J
= 12(43-88 + 66)
= 120.
5. Find the seventh term of 1, 1.236, 1.471, 1.708,
Series = 1 1.236 1.471 1.708
lstdiff. = 0.236 0.236 0.237
2d diff. = 0.001 0.001
3d diff. = 0
.-. a=l, ai = 0.236, aa = 0.001, a8 = 0.
Seventh term = 1 + 6 x 0.236 + ^-^ x 0.001
1 X L
= 1 + 1.410 + 0.016
= 2.426.
6. Find the sum of the series 70, 66, 62.3, 68.0, to 16 terms.
Series = 70 66 62.3 68.9
lstdiff. = -4 -3.7 -3.4
2d diff. = 0.3 0.3
3d diff. = 8
.-.a =70, ai=— 4, a* = 0.3, a8 = 0.
Sam to 16 terms
= 16J70-¥X4 + rxW3-x0.3}
= 15{70-28 + 9.1}
= 766.5.
526 ALGEBRA.
7. Find the eleventh term of 343, 337, 326, 310,
Series = 343 337 326 310
lstdiff. = -6 -11 -16
2d diff. = -6 -6
3d diff. = 0
.-. a = 343, «i = — 6, 02 = — 5, a* = 0.
10 X 9
Eleventh term = 343 — 10 X 6 - 1 v 0 X 6
1 x &
= 343 -.60-225
= 58.
& Find the sum of the series 7 x 13, 6 x 11, 5 x 9, to 9
terms.
Series =91 66 45 28
lstdiff. = -25 -21 -17
2d diff. = 4 4
3d diff. = 0
.*. a= 91, ai = — 25, a% = 4, as = 0.
Sum to 9 terms = 9 j 91 - f X 25 + ^g X 4 j
= 9(91-100 + 4*)
= 265.
9. Find the sum of n terms of the series 3x8, 6 X 11, 9 X 14,
12 x 17,
Series =24 66 126 204
lstdiff. = 42 60 78
2d diff. = 18 18
3d diff. = 0
.\ a = 24, ai = 42, a* = 18, a8 = 0.
Sum to n terms
= n{24 + 21n-21 + 3n*-9n + 6}
= 3n8+12n2 + 9n
= 3n(n+ l)(n + 3)
teachers' edition. 527
10. Find the sum to n terms of the series 1, 6, 15, 28, 45,
Series =1 6 16 28 45
lstdiff. = 69 13 17
2d diff. = 4 4 4
3d diff. = 0 0
.-. a = 1, a\ = 6, 02 = 4, a8 = 0.
Sam to n terms
-•('♦srJ«+fcri$rH
= n(l + fn-f + fn2-2n + !)
= n(fn2+in-i)
= £(4n2 + 3n-l)
o •
= £(4n-l)(n + l).
Exercise CXXVIII.
1. Determine the number of shot in the side of the base of a
triangular pile which contains 286 shot.
Let n denote the required number ;
then n(n + l)(n + 2)=286t
n(n + l)(n + 2) = 1716
= 11 x 12 X 13.
.-. n=ll.
2. The number of shot in the upper course of a square pile is 169,
and in the lowest course 1089. How many shot are there in the pile ?
In the complete pile, n = 33.
Number of shot in complete pile,
33X34X67
1X2X3 -12'629'
In the part of the pile that is lacking,
n=12.
Number of shot in part lacking,
12 x 13 x 26 _
1X2X3 -650'
12,529 - 660 = 11,879.
There are 11,879 shot in the pile.
528 ALGEBRA.
3. Find the number of shot in a rectangular pile having 17 shot
in one side of the base and 42 in the other.
The required number is
|(n+l)(3n'-n+l)
= ^(17 + 1) (126- 17 + 1)
= 6610.
4. Find the number of shot in five courses of an incomplete tri-
angular pile which has 15 in one side of the base.
Number in complete pile,
15X16X17 •
1X2X3 "~w-
Number in part lacking,
• 10X11X12_^
1X2X3 ~m
680-220 = 460.
The number of shot in the pile is 460.
5. The number of shot in a triangular pile is to the number in a
square pile, of the same number of courses, as 22 : 41. Find the
number of shot in each pile.
The number of courses is in both cases the same as the number of
shot in a side of the base. Hence
n(n+l)(n + 2) n(n+ 1) (2n+ 1)
1X2X3 ' 1X2X3 " - «*
n+2 :2n+l=22 : 41
w = 20.
# w(n+ l)(n+2)_20X 21 X 22
1X2X3 1X2X3
= 1640.
n(n+ 1) (2n+ 1) = 20 X 21 X 41
1X2X3 1X2X3
= 2870.
The number of shot in the triangular pile is 1540 ; in the square
pile, 2870.
529
6. Find the number of shot required to complete a rectangular pile
having 15 and 6 shot, respectively, in the sides of the upper course.
In the first missing course n = 5 and n' — 14. Hence the number
of shot required is j(5+ i} (42 -6 + 1) = 190.
7. How many shot must there be in the lowest course of a tri-
angular pile, so that 10 courses of the pile, beginning at the base,
may contain 37,020 shot ?
Number of shot in the complete pile is
n(n + l)(n + 2)
6
Number in lacking part is
(n - 10) (n - 9) (n - 8)
Hence *<"+ ^n + 2> - <»" 10> <" ^ ^V = 37,020,
n (n + 1) (n + 2) - (n - 10) (n - 9) (n - 8) = 222,120,
n* + 3 n2 + 2 n - n« + 27 n2 - 242 n + 720 = 222,120,
30n2-240n= 221,400,
na-8n = 7380,
n = 90.
Therefore, the number of shot in a side of the lowest course is 90,
and the number in the lowest course is
1 + 2 + 3 + + 90 = 46 X 91 = 4096.
8. Find the number of shot in a complete rectangular pile of 15
courses, which has 20 shot in the longest side of its base.
Here, n = 15, n' = 20 ; and the required number is
if- X 16 X (60 - 15 + 1) = 1840.
9. Find the number of shot in the bottom row of a square pile
which contains 2600 more shot than a triangular pile of the same
number of courses.
n(n+l)(2n+l) n(n+l)(n + 2)_
1X2X3 1X2X3 -^ow»
n(n+l){2n+l-(n + 2)} = 2C00<
»(n+l)(n-l) = 16,600,
(n-l)n(w+l) = 24x 26 X 26,
n = 25.
The required number is 25.
530 ALGEBRA.
10. Find the number of shot in a complete square pile in which
the number of shot in the base and the number in the fifth course
above differ by 225.
n*-(n- 5)2 = 225,
10n-25 = 225,
n=25,
n (n + 1) (2 n + 1 ) = 25 X 26 X 51
1X2X3 1X2X3
= 5525.
The required number is 5525.
1L Find the number of shot in a rectangular pile which has 600
in the lowest course and 11 in the top row.
The difference between the number of shot in the length of any
course and the number in the width is the same for all courses.
Hence, ri — n = 11 — 1
= 10,
nn' = 600.
.-. n' = 30,
n = 20,
£ (n + 1) (3n' - n+ 1) = \° X 21 x 71
= 4970.
The required number is 4970.
Exercise CXXIX.
1X4 2X5
1. Sum to n terms, and to infinity, the series
1
3X6
r+; l • l
1X42X63X6 n (n + 3)
= i/i_l\ + l/l_l\ + 1/1 L_\
8\1 4/^3V2 5/T 3\n n + 3/
= 1/1 + 1 + 1 1 1 1_\
8U 2 8 n+1 n + 2 n + 3/
■.1/11 8n»+12n+ll \
3V6 (n+l)(n+2)(n + 3)>J
lln8 + 48n2 + 49n
18(n+l)(n + 2)(n + 3)
= sum to n terms.
The sum to infinity = i (t + i + *)
teachers' edition. 531
2. Sum ton
1
terms, a
term is
1
md to infinity, the serie
1
1
1
S 1X3X5'
C
+ 4'
4) + Bn(nH
2X4X6'
3X5X71
The general
n(n+2)(n+4)
~A \ B \
iMlt .
7i (n
then
+ 2)(n
+ 4) n ' n + 2 ' n
l=4(n + 2)(n +
M)
+ Cn(n+2)
= (4 + B+C)n2+ (64 + 4B + 2 C)n
+ 84.
/.i + 5+C = 0, 4 = 4,
64 + 4B+2C = 0, ^=~i,
84 = 1, C=J.
XT 1 1 1 1
Hence — : — r-zrz — ttt = s tt — r-^: +
n(n + 2)(n + 4) 8n 4(n+2) 8(n + 4)
= 1/1__ 2
Hence
8vn n + 2
1.1.. 1
n4-4/
1X3X52X4X6 n (n + 2) (n + 4)
= lr/l_2 IX /1_2 lx
8|Al 3^5>/TV2 4T6r
+ (I 2-+-M1
T An » + 2 n + 4/J
= 1(1 + 1 + 1 + + i\
8\V2^3^ n/
~~4V3 + 4 + 5 + + nT2/
+ 1/1 + 1+1 + + _L_\
= l<l + i\-l/i + i\-l/-J^+_L\
8U 2/ 8V3^4/ 8Vn+l n + 2/
8Vn+3Tn + 4/
_ 11 1/ 2n+3 2n+7 \
96 8\.(n+l)(n + 2) (n + 3)(n+4)>>
11 2n2+10n+ll
96 4 (n + 1) (n + 2) (n + 3) (n + 4)
= sum to n terms.
Sum to infinity = i (i + i) ~ i (i + i)
= ii
532 ALGEBRA.
3. Sum to n terms, and to infinity, the series g> . x ,
0 X 8 X 10
The general term is ; —
2n(2n + 2)(2n + 4) 8n(n+l)(n + 2)
Let — - — 7-rrz — r-^7 = — + -
n (n + 1) (n + 2) n n + 1 n + 2
then l = 4(n + l)(n + 2)+Bn(n + 2)+Cn(>i+l)
= (A+B + C)n*+(SA+2B + C)n + 2A.
.•.i + B+C = 0, 4 = *,
34 + 2B+C = 0, B=— lf
24 = 1, C=*.
XT 1 1 1 , 1
Hence — : — r~rrz — r^r = « rr + :
n (n + 1) (n + 2) 2n n + 1 2 (» + 2)
= 1 /I 2 1 \.
2 \n n+1 n + 2/
and 1 = JL/1 2_ + _L\.
8n(n+l)(n + 2) 16\n n+1 n + 2/
XT 1,1, , 1
Hence ftw,v><0+,v>/,w0 + +
X4X64X6X8 2n(2n + 2)(2n+4)
-s[(i-:+i)+(S-i+i)+
\n n+1 n + 2/J
= 1/1 + 1 + 1+ + 1\
WU 2 8T n)
_ 1/1 + 1 + 1 + + _1_\
8 \2T3T4T n+1/
+f(s+H+ *£i)
~ 16 Vl) ~ 16 (2) - 16 (n + l) + 16 Vn+V
1 /l
/l-
)
16 V2 (n+1) (n + 2)
= na + 3n
32 (n + 1) (n + 2)
= sum to n terms.
Sum to infinity =^(i-J)
= A-
teachers' edition. 533
4. Sum to n terms, and to infinity, the series . ^ Q ^ .1 Q w . w _,
1ft iX(JX4 oX4X6
4X6X6
o «. _1_ "I
The general term is
Let
(n + 1) (n + 2) (n + 3)
3n+l 4 , £
(n+l)(n + 2)(n+3) n+1 n+2n+3
then 3n+l = ^4(n+2)(n+3)+£(n + l)(n + 3)
+ C(w+l)(w + 2)
= (A+B + C)n2 + M+ 4B+3C)n
+ 64 + 3B + 2C.
...4+b+c=o, ^ = -1,
6X + 4£+3C=3, 5=5,
64 + 3£ + 2C=l, C=-4.
3n+l -1,6 4
Hence
(n + 1) (n + 2) (n + 3) n + 1 n + 2 n + 3
^ 4,7, , 3n+l
2X3X43X4X6 (n + 1) (n + 2) (n + 3)
V 2^3 4/^V 3^4 h)^
+ f-l + -5 M
V n n+1 n + 2j
=- G+S+i+ +m)
+*(!+H+ +sii)
= _l+4 ^ 4_
2 3 n+2 n + 3
_ 5 3n + 5
6 (n + 2)(n + 3)
_ 6n*+7n
6 (n + 2) (n + 3)
_ n(6n + 7)
6(n + 2)(n + 3)
= sum to n terms.
Sum to infinity = f .
J
534 ALGEBRA.
5. Sum to n terms, and to infinity, the series » »
- lXJXo JXoX4
fxlx 5*
Each term of the series is 8 times the corresponding term in the
series of Ex. 3. Hence the sum to n terms
n2 + 3n
4(n+l)(n + 2)'
and the sum to infinity = £.
Exercise CXXX.
L If 6, 7, 8, 3, 2 are the digits of a number in the scale of r,
beginning from the right, write the algebraical value of the number.
The algebraical value of the number is
2r* + 3r8+8r2+7r + 6.
2. Find the product of 234 and 125 when r is the base of the
scale.
234 X 126= (2r* + 3r + 4) (r«+ 2r + 5)
= 2r* + 7r» + 20 r2 + 23r + 20.
3. In what scale will the common number 756 be expressed by
530?
Let r be the base of the scale ; then
5r* + 3r = 756.
.-. r=12.
4. In what scale will 540 be the square of 23 ?
Let r be the base of the scale ; then
(2r + 3)2 = 5r* + 4r
— r2 + 8r+9 = 0. -
.-. r = 9.
5. Show that 1234321 will, in any scale, be a perfect square, and
find its square root.
In any scale
1,234,321 = r» + 2t* + 3r*+ 4r» + 3r* + 2r + l
= (r3 + r2 + r + l)2
= llll2 in the scale of r.
TEACHERS7 EDITION.
535
6. In what scale will 212, 1101, 1220 be in arithmetical progres-
sion?
Let r be the base of the scale ; then
2 X 2 X 1101 = 212 + 1220,
2(r8 + r2+l) = (2»a + r+2) + r8 + 2r3 + 2r,
r8-2r2-3r = 0,
r(r + l)(r-3) = 0.
,-.r = 3.
7. Multiply 31.24 by 0.31 in the scale of 6.
31.24
0.31
3124
14432
20.2444
8. Find the least multiplier of 13,168 which will make the product
a perfect cube.
13,168 = 16 X 823 = 2* X 823.
Hence the required multiplier is 2* X 8232 = 1646* = 2,709,316.
Exercise CXXXI.
1. Solve the following equations by constructing their loci :
2z + 3y = 8)
3x + 7y = 7j
(i.) Ifx = 0, 1,2, 3, 4, 5, -1,
-2,-3,-4,
y = 2|, 2, 1J, }, 0, -{,
H, 4, 4f , 5}.
(ii.) Ifx = 0, 1, 2, 3, 4, 5, -1,
-2,-3,-4,
y — 1» y» 7» 7» ?»
-1}, If, l?,2f,2?.
The loci are straight lines, as
represented in the figure, and
they meet at the point x = 7,
y=-2.
536
ALGEBRA.
2. Solve the following equations by constructing their loci :
3x — 5y = 2 )
2x+7y=22j
(i.) Kx = 0, 1, 2, 3, 4,5, -1,
-2,-3,-4,
*=-!.*. ti If, 2,2?,
-1, -If, -2J, -2|.
(ii.) Ifx = 0, 1, 2, 3, 4, 5,-1,
-2,-3,-4,
y = 3}, 2f, 2*, 2?, 2, If,
3f, 3^, 4, 4|.
The loci are straight lines, as represented in the figure, and the?
meet at the point x = 4, y = 2.
3. Solve the following equations by constructing their loci :
2x— 9y= 11
3x-12y=15
(i.) Ex = 0, 1, 2, 3, 4,6,-1,
-2,-3,-4,
(ii.) Ifx = 0, 1, 2, 3, 4, 5,-1,
-2,-3,-4,
-i» -o, -f, -i,
The loci are straight lines, as represented in the figure, and they
meet at the point x = 1, y = — 1.
teachers' edition.
537
4. Solve the following equations by constructing their loci :
4x — 2y = 20
6x = 9y
(i.) Ifx = 0, 1, 2, 3, 4,5,-1,
-2,-3,-4,
y=-10, -8,-6,-4,
-2, 0, -12, -14,
- 16, - 18.
(ii.) Ifx = 0, 1, 2, 3, 4, 6,-1,
-2,-3,-4,
y = Q, f, *,M,y, -f,
The loci are straight lines, as represented in the figure, and they
meet at the point x = 7J, y = 5.
5. Solve the following equations by constructing their loci :
2x — 3y = 4
3x + 2y = 3S
(i.) Ifx = 0, 1, 2,3,4,5,-1,
-2,-3,-4,
V = - *• - f 0, f J, 2,
-2,-}, -^,-4.
(ii.) If x = 0, 2, 4, 6, 8, 10, - 2,
-4,-6,-8,
y=16, 13, 10, 7, 4, 1,
19, 22, 25, 28.
The loci are straight lines, as represented in the figure, and they
meet at the point x = 8, y = 4.
538
ALGEBRA.
6b Solve the following equations by constructing their loci :
2x + 3y=7)
4x-5y = 3J
(L) lfx = 0, 1, 2, 3, 4,5,-1,
-2,-3,-4,
y = i, f* i,l, -h -i.
(ii.) If x = 0, 1, 2, 3, 4, 6, - 1,
-2,-3,-4,
*=-!»*» i. {.¥. ¥»
The loci are straight lines, as represented ki the figure, and they
meet at the point x = 2, y = 1.
7. Solve the following equations by constructing their loci :
;-9y=ll)
:-4y=7 )
2x — 9y = 11)
3x-
(i.) If x = 0, 1, 2, 3, 4, 5-1,
-2,-3,-4,
*=-¥>-!. -fc-fc
(U.) If x = 0, 1, 2, 3, 4, 5,-1,
-2,-3,-4,
y=-i,-i,-i,M>2>
The loci are straight lines, as represented in the figure, and they
meet at the point x = 1, y = — 1.
teachers' edition.
539
8. Solve the following equations by constructing their loci :
3x — 4y = — 5
4x — 6y = 1
(i.) Ifx = 0, 1, 2,3,4,6,-1,
-2,-3,-4,
-ii -1, ~b
(ii) If x = 0, 1, 2, 3,4, 5, -1,
-2,-3,-4,
The loci are straight lines, as represented in the figure, and they
meet at the point x = 29, y — 23.
9. Solve the following equations by constructing their loci :
x — 2y = 4
2x — y= 6
(i.) If x = 0, 2, 4, 6, 8,-2,-4,
-6,
y = -2,-l, 0,1, 2,-3,
-4,-6.
fii.) If x = 0, 1, 2, 3, 4, 6, -1,
-2,-3,-4,
y=-5, -3, -1,1,3,6,
-7,-9,-11,-13.
The loci are straight lines, as represented in the figure, and they
meet at the point x = 2, y = — 1.
540
ALGEBRA.
10. Solve the following equations by constructing their loci :
§-* = 5
x y
x y
(i.) Ifx = 0,i,i,i,l,|,2,-},
-*. -t,-l,
y = 0,f 4,-4,-2,-J,
— f > A> — rr> — f >
-*• .
(ii.) If*=0,i,i,|,l,|,2,-},
y=o, i, f, -v, -!,
The loci are hyperbolas, consisting each of two infinite branches,
as represented in the figure ; and they intersect at the two points
s = 0, y = 0, and x = — 1, y = — \.
XL Solve the following equations by constructing their loci :
x y
§_? = 4
x y
•
(i.)lf* = 0,i, J, 1,2, 3, 4,-1,
-2,-3,
y = 0, oo, 1, |, $, £,£,
h J. A-
. (ii.) Ifi6 = 0,$,l,2,3,-l,-2,
-3,-4,
y = 0, oo,-2, -4, -|,
-?.-A.-i.-A-
The loci are hyperbolas, as represented in the figures, and they
intersect at the points x = 0, y = 0, and x = ±, y = 1.
teachers' edition.
541
12. Solve the following equations by constructing their loci :
x2 + 2/*=104)
x + 2/=12 J
(i.) Kx=0,±2,i4,±6,±8,
±10,
y=±10.2, ±10, ±9.4,
±8.2, ±6.3, ±2.
(ii.) If x=0, 2, 4, 6, 8, 10, 12,
y=12, 10,8,6,4,2,0.
The first locus is a circle, the second a straight line, as repre-
sented in the figure, and they intersect at the two points x = 2,
y = 10, and x = 10, y = 2.
13. Solve the following equations by constructing their loci :
x — y = 10
x — y = 10 )
2+y2=178)
(i.) If x = 0, 2, 4, 6, 8, 10, 12, IS,
y=-10, -8, -6, -4,
-2,0,2,3.
(ii.) Ifx = 0, ±2, ±4, ±6, ±8,
± 10, ± 12, ± 13,
y= ±13.3, ±13.2, ±12.7,
±11.9, ±10.7, ±8.8,
±5.5, ±3.
The first locus is a straight line, the second a circle, as repre-
sented in the figure, and they intersect at the two points x = 13,
y = 3, and x = — 3, y = — 13.
542
ALGEBRA.
14. Solve the following equations by constructing their loci :
xy— 12 = 0
x — 2y = 5
(i.) If x = 0, ±2, ±4, ±6,
±8, oo,
y = co , ±6, ±3, ±2,
±|.fc
(ii.) Ifx=0, 1, 3, 6, 7,9,
y = -f,-2,-l,0,l,2.
The first locus is a hyperbola, the second a straight line, as
represented in the figure ; and they intersect at the two points
x = 8, y = J, and x = — 3, y = — 4.
^15. Solve the following equations by constructing their loci :
x + y=13)
xy = I
(i.) If x = 0, 2, 4, 6, 8, 10, 12,
y = 13, 11, 9, 7, 6, 3, 1.
(ii.) Hi = 0, ±2, ±4, ±6, ±9,
± 12, ± 18,
y = co, ±18, ±9, ±6,
±4, ±3, ±2.
The first locus is a straight line, the second a hyperbola, as repre-
sented in the figure ; and they intersect at the two points * = 4,
y = 9, and x = 9, y = 4.
teachers' edition.
543
16. Solve the following equations by constructing their loci :
-4x2=12 )
j + y=_10)
3y*-4x2 = 12
2x-
(i.) Ifx = 0,±2,±4,±6,±8,
±10,
y = ± 2, ± 3.0, ± 5.2,
± 7.2, ± 9.4, ± 11.7.
(ii.) Ifx = 0, 4, -2, -4, -6,
-8,-10,
y= —10, —18,-6, —2,
2, 6, 10.
The first locus is a hyperbola, the second a straight line, as
represented in the figure ; and they intersect at the two points
x = — 3, y = — 4, and x = — 12, y = 14.
17. Solve the following equations by constructing their loci :
4 5 1
5 + y 12 + x L
2x + 5y = 36 J
(i.) Ifx=0, 2, 4, 6, -2,-4,
-6,
y = ¥, V, ¥, ¥. ». *.
-1-
(ii.) If x = 0, 2,4,6, -2, -4,
-6,
The loci are straight lines, as represented in the figure, and they
intersect at the point x = 2, y = ^.
544
ALGEBRA.
18. Solve the following equations by constructing their loci :
x 3y 27
J_ 1 = 11
4x + y 72
(i.) If x = 0, 3, 6, 9, 13.6, 18, 26,
y= 0, 3.2, 9, 22.5, oo,
- 46, 34.6.
(ii) lfx = 0, if, 3, 6, 9, -3,
-6,-9,
y = 0, oo , 14.4, 9, 8, 4.2,
6.1, 5.5.
The loci are both hyperbolas, only one branch of the first being
shown in the figure. They intersect at the two points x=0,
y = 0, and x = 6, y = 9.
Exercise CXXXII.
1. Construct the locus of the
equation,
x* + 3x-10 = 0.
Let x2 + 3x — 10 = y,
thenif x = 0, 1, 2, 4, —2, -4,
-6,-6,
y = - 10, - 6, 0, 18,
— 12, - 6, 0, 8.
The locus is represented in the figure. The roots are 2 and — 5.
TEACHEBS' EDITION.
545
2. Construct the locus of the equation, x8 — 2 x2 + 1=0.
Let x8 — 2x2+l = y,
then if x = 0, 1, 2, 3, - 1, - 2,
y= 1,0, 1,10, -2,-15.
The locus is represented in the
figure. The roots are
1+ V5 J 1- V6
V
2
> and •
3. Construct the locus of the equation, x4 — 20 x* + 04 = 0.
Let x4 — 20x2 + 64 = y,
then if x=0, ±2, ±3, ±4, ±5,
y = 64, 0, - 35, 0, 189.
The locus is represented in the
figure, except that the ordinates
are ^ their computed length.
The roots are ± 2 and ± 4.
4. Construct the locus of the equation, x2 — 4 x 4- 10 = 0.
Let x9 — 4x + 10 = y,
then if x = 0, 1, 2, 3, 4, 5, — 2,
y=10, 7, 6, 7, 10,15,22.
The locus is represented in the
figure. The roots are imaginary.
y
546
ALGEBRA.
5. Construct the locus of the
equation,
x*-6x* + 4=0.
Let x4 — 5x2 + 4=y,
then if x = 0, ± 1, ± 2, ± 3,
y = 4, 0, 0, 40.
The locus is represented in the
figure. The roots are ± 1 and
±2.
Exercise CXXXIII.
1. Determine whether — 5 is a root of the equation
a^ + Cx'-lOx8-- 112x2-2O7x-110 = O.
1 + 6 - 10 - 112 - 207 - 110|-5
-5- 5+ 75+185 + 110
+ 1-15-37-22 + 0
Hence — 5 is a root.
2. Determine whether 1 is a root of the equation
x6 - 8x* + 7x» + x2- 3x + 2 = 0.
1-8+7 +l-3 + 2[l
1-7+0+ 1-2
-7+0+1-2+0
Hence 1 is a root.
3. Determine whether —7 is a root of the equation
x* + 21x+ 7x»+ 147 = 0.
l + 7 + 0 + 21 + 147|-7
- 7 + 0 + 0-147
0 + 0 + 21 + 0
Hence — 7 is a root.
teachers' edition. 547
4. Determine whether — 8 is a root of the equation
x* + 8x4 - 7 x2 - 54x + 16 = 0.
l + 8-7-54+16|-8
-8 + 0 + 56-16
0-7+ 2 + 0
Hence — 8 is a root.
5. Determine whether 2 is a root of the equation
x4-4x8-3x2-2x-8 = 0.
1-4-3-* 2- 8 [2
+ 2-4-14-32
- 2 - 7 - 16 - 40
Hence 2 is not a root.
6. Determine whether —7 is a root of the equation
xs + l4x2 + 65x+ 112 = 0.
l + 14 + 65+112|-7
- 7-49-112
+ 7 + 16 + 0
Hence —7 is a root.
7. Determine whether 6 is a root of the equation
2x* - 4x» - 62 x2+114x- 180 = 0.
2- 4-62+114-18016
+ 12+48- 84 + 180
+ 8-14+ 30 + 0
Hence 6 is a root.
8. Determine whether — 5 is a root of the equation
x*-7x-2x2-15 = 0.
1 + 0- 2- 7- 15|-5
- 5 + 25 - 115 + 610
- 5 + 23 - 122 + 595
Hence — 5 is not a root.
548 AXJ5EBRA.
9. Determine whether — 0.3 is a root of the equation
x* + 2.3x"* + 3.6x2 + 4.9x + 1.2 = 0.
1 + 2.3 + 3.6 + 4.9 + 1.2 1-0.3
-0.3-0.6-0.9-1.2
+ 2 +3 +4 +0
Hence — 0.3 is a root.
10. Determine whether * is a root of the equation
X»-ix2-^X-i = 0.
5 + i+i
+1+ i+°
Hence J is a root.
Exercise CXXXIV.
1. Find the equation whose roots are 2, 6, and — 7.
The equation is (x — 2) (x — 6) (x + 7) = 0
or x8 — x2 — 44x + 84 = 0.
2. Find the equation of which the roots are 1, 4, — 1, and — 3.
The equation is
(x - 1) (x - 4) (x + 1) (x + 3) = 0
or (x2-l)(x2-x-12) = 0
or x*-x«-13x2 + x+12 = 0.
3. Find the equation of which the roots are 2, 3, — 2, — 3, and — 6.
The equation is
(x- 2) (x - 3) (x 4- 2) (x + 3) (x + 6) = 0
or (x2 - 4) (x2- 9) (x + 6) = 0
or x* + 6x* - 13x» - 78x2 + 36x + 216 = 0.
4. Find the equation of which the roots are 0.2, \, and — 0.4.
The equation is
(x - 0.2) (x - 1) (x + 0.4) = 0
or (*-i)(*-i)(x+f) = 0
or (6x - 1) (8x - 1) (5x + 2) = 0
or 200x8+ 15x2 — 21x+ 2 = 0.
teachers' edition. 549
5. Find the equation of which the roots are
5, 3 + V^l, and S-^l.
The equation is
(x - 6) (x - 3 - y/^1) (x - 3 + V^l) = 0
or (x - 5) (x2 - 6 x + 10) = 0
or x8 - 11 x2 + 40x - 50 = 0.
Exercise CXXXV.
1. Form the equation whose roots are 2, 4, and — 3.
2 + 4-3 = 3,
2 X 4 + 2 X (- 3) + 4 (- 3) = - 10,
2 X 4 X (- 3) = - 24.
Hence the equation is
x8-3x2-10x + 24 = 0.
2. Form the equation whose roots are 2,-1, and — 7.
2 - 1 - 7 = - 6,
2 x (- 1) + 2 x (- 7) + (- 1) X (- 7) = - 9',
2 x (- 1) X (- 7) = 14.
Hence the equation is
x« + 6x2-9x-14 = 0.
3. Form the equation whose roots are 2, 0, and — 2.
2 + 0-2 = 0,
2 X 0 + 2 X (- 2) + 0 X (- 2) = - 4,
2 X 0 X (— 2) = 0.
Hence the equation is
oj8 — 4 x = o.
4. lorm the equation whose roots are 6, 6, and 6.
6 + 6 + 6 = 18,
6X6 + 6X6 + 6X6 = 108,
6X6X6 = 216.
H<ice the equation is
x8 - 18x2 + 108x - 216 = 0.
560 ALGEBRA.
5. Form the equation whose roots are 2, 1, — 2, and — 1.
2 + 1-2-1 = 0,
2X 1 + 2 X (-2) + 2X (-1)+1 X (-2)4-1 X(-1)
+ (_2)X(-l)=-5,
2 X 1 X (- 2) + 2 X 1 X (- 1) + 2 X (- 2) X (- 1)
+ 1 x (- 2) x (- 1) = 0.
2 x 1 x (- 2) x (- 1) = 4.
Hence the equation is
x*-5x* + 4 = 0.
6b Form the equation whose roots are 2, $, — 2, — }.
2 + l-2-i = 0,
2 X \ + 2 X (- 2) + 2 X (- i) + i x (- 2) + i x (- i)
+ (-2)x(-i) = -¥,
2 X J X (-2) + 2 X J X (- i) + 2 X (- 2) X (- J)
+ Jx(-2)X(-i) = 0,
2Xi(~2)X(-i) = l.
Hence the equation is
&*-¥«*+ 1 = 0.
Exercise CXXXVI.
1. Find the roots of the equation x2 + 11 x + 24 = 0.
x2 + 1 1 x + 24 = (x + 3) (x + 8) = 0.
Hence x = — 3 or — 8.
2. Find the roots of the equation 7 x2 + 161 x + 714 = 0.
7 x2 + 161 x + 714 = 7 (x2 + 23 x + 102)
= 7(x + 17)(x+6).
Hence x = — 17 or — 6.
3. Find the roots of the equation x* — 4 a2 x2 + 3 a4 = 0.
x*- 4a2 x2 + 3a* = (x2- a2) (x2- 3a2)
= (x + a)(x-a)(x+ VSa)(x- V3a).
Hence x = — a, + a, — V3a, or + V3a.
TEACHERS* EDITION. 551
4. Find the roots of the equation x& + 4 x8 + 8 x2 + 32 = 0.
X6 + 4x8 + 8a:2 + 32 = (x8 + 8) (x2 + 4)
= (x + 2) (x2 - 2 x + 4) (x2 + 4)
Hence x = - 2, 1 ± V11^, ± 2 V-11!.
5. Find the roots of the equation 12 x2 — 5 x — 2 = 0,
12x2-6x-2=(3x-2)(4x+l).
Hence x = J or — ±.
6. Find the roots of the equation 4x* — 9 x2 + 6 x — 1 = 0.
4x* - 9x2 + 6x - 1 = (2x2)2 - (3x - l)2.
= [2x2+(3x-l)][2x2-(3x-l)]
= (2x2 + 3x-l)(2x2-3x+l)
= (2x2+3x-l)(x-l)(2x-l).
„ t , -3± Vl7
Hence x = 1, £, or
4
7. Find the roots of the equation 49 x2 — 112 bx + 64 62 = 0.
49x2- 112 6x + 64 62 = (7x - 86)2.
„ 86 86
Hence x= — »
7 7
8. Find the roots of the equation x6 — 64 = 0.
x8-64=(x8 + 8)(x8-8)
= (x + 2) (x2 - 2x + 4) (x - 2) (x9 + 2x + 4).
Hence x = - 2, 1 ± V11^, 2, or - 1 ± \T=H.
9. Find the roots of the equation 3 x8 — x2 + 3 x — 1 = 0.
3x8 - x2 + 3x - 1 = 3 (x8 + x) - (x2 + 1)
= (x2+l)(3x-l).
Hence x = J or ± V— 1.
10. Find the roots of the equation x — 27 x4 = 0.
x-27x* = x(l-27x8)
= x(l-3x)(l + 3x + 9x2).
„ A , -1± V^
Hence x = 0, J, or
552 ALGEBBA.
Exercise CXXXVII.
L Solve the equation x8 + 3x2 — 25x — 12 = 0.
x« + 3x2- 26x- 12 = 0
-4+3
7-28 -28 + 7 = -4.
Hence x8 + 3x2- 25x- 12 = (x- 4) (x2 + 7x + 3),
-7± V37
and x = 4, or,
2. Solve the equation x8 — 4x2 — 8x + 8 = 0.
x«-4x2- 8x + 8 = 0
2+4
-6 -12 -12 + (-6) =2.
Hence x8- 4x2- 8x + 8 = (x + 2) (x2- 6x + 4),
and x = — 2, or 3 ± sfc.
3. Solve the equation x8 — 7 x2 + 19x — 21 = 0.
x»-7x2+19x-21 = 0
-3+7
-4 +12 12 + (-4) = -3.
Hence x8- 7x2+ 19x-21 = (x-3)(x2-4x + 7),
and x = 3, or 2 ± V^.
4. Solve the equation x8 — 8x2 + 21 x — 18 = 0.
x8-8x2 + 21x-18 = 0
-2+9
-6+12 12 + 6 = 2.
Hence x8-8x2 + 21x- 18= (x -2)(x2-6x + 9)
= (x-2)(x-3)2
and x = 2, 3, 3.
5. Solve the equation x8 — 26 x — 6 = 0.
x8 + 0x2-26x-5 = 0
5-1
- 5 - 25 - 25 + (- 6) = 5.
Hence x8 — 26x — 5 = (x + 5) (x2 — 5x — 1),
5±V29
and x = — 5, or •
teachers' edition. 553
6. Solve the equation x8 — 3 x2 — 54 x — 104 = 0.
x8-3x2-54x-104=0
4 -26
-7 -28 -28-r(-7) = 4.
Hence x8- 3x2-54x - 104= (x + 4) (x2- 7x- 26),
, . 7±Vl53
and x = — 4, or - •
2
7. Solve the equation x8 + 9x2 + 2x — 48 = 0.
x8 + 9x2 + 2x-48 = 0
-2 +24
11 -22 -22-rli=-2.
Hence x8 + 9x2 + 2x-48 = (x-2) (x2+llx + 24)
= (x-2)(x + 3)(x+8),
and x = 2, — 3, or — 8.
8. Solve the equation x8 — 2 x2 — 25 x + 50 = 0.
x8-2x2-25x+50=0
-2 -25
0 0
Hence x8- 2x2-25x + 50= (x-2)(x2-25)
= (x-2)(x-5)(x + 5),
and x = 2, + 5, — 5.
9. Solve the equation x8 — 3 x2 — 61 x + 63 = 0.
x8-3x2-61x + 63 = 0
- 1 - 63
-2+2 2-r(-2) = -l.
Hence x8 - 3 x2 - 61 x + 63 = (x - 1) (x2 - 2 x - 63)
= (x-l)(x + 7)(x-9),
and x = 1, — 7, or 9.
10. Solve the equation x8 — 37 x — 84 = 0.
x8 + Ox2- 37 x- 84 = 0
4 -21
-4 -16 -16-r(-4) = 4.
Hence x8 - 37 x - 84 = (x + 4) (x2 - 4 x - 21)
= (x + 4)(x + 3)(x-7),
and x = — 4, — 3, or 7.
554 ALGEBRA.
Exercise CXXXVIIT.
L Solve the equation x4 — 2x8 — 13x2 + 38x — 24 = 0.
m + p = — 2,
ti + mp 4- 9 = — 13,
np + mq = 38,
Tig = — 24.
By trial it is found that we may take
n = 2, q = — 12, m = — 3, p = 1.
Hence x4- 2x»- 13x2 + 38x- 24
= (x2 - 3 x + 2) (x2 + x - 12)
= (x-l)(x-2)(x-3)(x + 4),
and x=l, 2, 3, or —4.
2. Solve the equation x4 — 6x«— 2x2 + 12x + 8 = 0.
x*_5x8_2x2 + 12x+8
= (x2 - x - 2) (x2 - 4x - 4)
= (x-M)(x-2)(x2-4x-4).
Hence x = — 1, 2, or 2 ± 2 V2.
3. Solve the equation x4 — 4 x8 — 8 x + 32 = 0.
X4_4x8_8x + 32
= (x2-6x + 8)(x2 + 2x + 4)
= (x- 2) (x - 4) (x2 + 2x + 4).
Hence x = 2, 4, or — 1 ± V^.
4. Solve the equation x4 — 12 x* + 50 x2 — 84 x + 49 = 0.
x4 - 12x» + 50 x2 - 84x + 49
= (x2-6x+7)(x2-6x+7).
Hence x=3±V2, 3±V2.
5. Solve the equation x4 — 11 x2 + 18x — 8=0.
x4-llx2 + 18x-8
= x2-2x + l)(x2+2x-8)
= (x - 1) (x - 1) (x - 2) (x + 4).
Hence x = 1, 1, 2, or — 4.
teachers' edition. 555
6. Solve the equation x* — 10x2 — 20x — 16 = 0.
x*-10x2-20x-16
= (x2- 2x - 8) (xa + 2x + 2)
= (x + 2) (x- 4) (x2 + 2x + 2).
Hence x = — 2, 4, or — 1 ± V^T.
7. Solve the equation x* — 7x» + 23 x2 — 47x + 42 = 0.
x*-7x» + 23x2-47x + 42
= (x2-5x + 6)(x2-2x + 7)
= (x-2)(x — 3)(x2-2x + 7).
Hence x = 2, 3, or 1 ± V— 6.
a Solve the equation x* + 2 x8- 9 x2 — 8x + 20 = 0.
x* + 2x8-9x2-8x + 20
= (x2-4)(x2 + 2x-6*
= (x- 2) (x + 2) (x2 + 2x- 6).
Hence x = 2, — 2, or — 1 ± VS.
9. Solve the equation x* - 4x» — 102 x2 - 188x - 91 = 0.
x* - 4x» - 102 x2 - 188x - 91
= (x2 + 8 x + 7) (x2 - 12 x - 13)
= (x + 1) (x + 7) (x + 1) (x - 13)
Hence x = — 1, — 1, — 7, or 13.
10. Solve the equation x* - 11 x* 4- 46x2 - 117 x + 46 = 0.
x* - llx8 + 46X2 - 117x + 45
= (x2 - 7x + 3) (x2 - 7x + 16).
7±V37 2±Vi:li
Hence x = ■
2
Exercise CXXXIX.
1. Determine the signs of, the roots of the equation x* + 4 x8
- 43 x2 — 68 x + 240 = 0, all the roots being real.
+ + - - +
No. of variations, 2.
No. of permanences, 2.
Hence there are two positive and two negative roots.
656 ALGEBRA.
2. Determine the signs of the roots of the equation x8 — 22x*
+ 166 x — 360 = 0, ail the roots being real.
+ - + -
No. of variations, 3.
No. of permanences, 0.
Hence there are 3 positive roots.
3. Determine the signs of the roots of the equation x4 + 4 x3
— 36 X2 — 78 x + 360 = 0, all the roots being real.
+ + - - +
No. of variations, 2.
No. of permanences, 2.
Hence there are 2 positive and 2 negative roots.
4. Determine the signs of the roots of the equation x8 — 12 aJ*
— 43x — 30 = 0, all the roots being real.
+ - - -
No. of variations, 1.
No. of permanences, 2.
Hence there is 1 positive and 2 negative roots.
6. Determine the signs of the roots of the equation x* — 3x*
— 6x8 + 16x* + 4x — 12 = 0, all the roots being real.
+ -- + + -
No. of variations, 3.
No. of permanences, 2.
Hence there are 3 positive and 2 negative roots.
& Determine the signs of the roots of the equation x*-f 12 x2
+ 47 x — 60 = 0, all the roots being real.
+ - + -
No. of variations, 3.
No. of permanences, 0.
Hence there are 3 positive roots.
7. Determine the signs of the roots of the equation x4 — 2x*
— 13 x2 + 38 x — 24 = 0, all the roots being real.
+ - - + -
No. of variations, 3.
No. of permanences, 1.
Hence there are 3 positive and 1 negative roots.
teachers' edition. 557
8. Determine the signs of the roots of the equation x5 — x4
— 187 x8 — 359 x8 + 186 x + 330 = 0, all the roots being real.
+ --- + +
No. of variations, 2.
No. of permanences, 3.
Hence there are 2 positive and 3 negative roots.
9. Determine the signs of the roots of the equation x6 — 10 x5
+ 19 x4 + 110 x3 — 536 x2 + 800 x — 384 = 0, all the roots being
real. + _ + + _ + _
No. of variations, 5.
No. of permanences, 1.
Hence there are 5 positive and 1 negative roots.
JO. Determine the signs of the roots of the equation x7 — 10 x6
+ 22x* + 32 x* - 131 x8 + 50 x2 + 108 x - 72 = 0.
+ - + + - + + -
No. of variations, 5.
No. of permanences, 2.
Hence there are 5 positive and 2 negative roots.
Exercise CXL.
1. Find the successive deriva- 2. Find the successive deriva-
tives of the polynomial x2 + 2 x tives of the polynomial x8 — 3 x2
+ 3. +7x + 25.
F (x) = x2 + 2x + 3, F (x) = x8-3x2+7x + 25,
F« (x) = 2x+2, F' (x) = 3x2-6x+7,
F« (x) = 2, F» (x) = 6x - 6,
Fm(x) = 0. F*"(x) =6,
i?iv (X) = 0.
3. Find the successive derivatives of the polynomial x4 + 2 x3
-5x2 + 64. F (x) = a.4 + 2x8-5x2+ 64,
F* (x) = 4 x8 + 6 x2 - 10 x,
F" (x) = 12 x2 + 12 x,
F»«(x) = 24x+ 12,
F" (x) = 24,
F* (x) = 0.
558 ALGEBRA.
4. Find the successive derivatives of the polynomial x6 + x4 — 6 x8
+ 3x2-4x + 27.
F (x) = xfi + x4-6x8 + 3x2-4x + 27,
F» (x) = 5x4 + 4x8-18x2 + 6x — 4,
F" (x) = 20 x8 + 12 x2 - 36x + 6,
F»" (x) = 60 x2 + 24 x - 36,
F"(x) = 120x+24,
F* (x) = 120,
F" (x) = 0.
5. Find the successive derivatives of the polynomial x4 — 3 ax8
+ 6 6x2 — 9 ex + mn.
F (x) = x* — 3 ax8 + 6 6x2 — 9 ex + mn,
F« (x) = 4x8-9ax2+12te-9c,
F" (x) = 12x2-18ax+12 6,
F»" (x) = 24 x — 18 a,
F*v (x) = 24,
F* (x) = 0.
Exercise CXLL
1. Find all the roots of x8 — 8 x2 + 13 x — 6 = 0.
F (x) = x*-8x2 + 13x-6,
F»(x) = 3x2-16x+13
= (3x-13)(x-l).
<f> (x) = x — 1.
.-. F(x) = (x-l)*(x-6).
The roots are 1, 1, and 6.
2. Find all the roots of x8 — 7x2 + 16 x — 12 = 0.
F (x) = x8 - 7x2 + 16x - 12,
Fi(x) = 3x2-14x+ 16
= (3x-8)(x-2).
0 (x) = x — 2.
.-. F(x) = (x-2)2(x-3).
The roots are 2, 2, and 3.
TEACHERS' EDITION.
559
3. Find all the roots of x4 — 6x* — 8x — 3 = 0.
F (x) = x4 — 6x2 — 8x — 3,
F«(x) =
4 + 0-12-8
4 + 8+ 4
= 4x*-12x-8.
1 + 0- 6- 8- 3
4
- 8 - 16 - 8
- 8 - 16 - 8
4 + 0-24-32-12
4 + 0 - 12 - 8
12)- 12-24-12
-1-2-1 -4x-8
.\ 0(x)=x2 + 2x+l
= (x + l)2.
.-. F(x) = (x + l)«(x-3).
The roots are — 1, — 1, — 1, and 3.
4. Find all the roots of x« — 2x2 — 15x + 36 = 0.
F (x) = x* - 2x2 - 15x + 36,
^(x) = 3x2-4x-15
= (3x + 5)(x-3).
0 (x) = x — 3.
.-. F(x) = (x-3)2(x + 4).
The roots are 3, 3, and — 4.
5. Find all the roots of x* — 7x8 + 9x2 + 27x — 64 = 0.
F (x) = x*-7x8+9x2 + 27x-64,
F'(x) = 4x*-21x2+ 18x + 27.
4-
4-
-21 + 18 + 27
-24 + 36
3-18 + 27
3-18 + 27
.-.*(x) =
.-.F(x) =
The roots are
1-7+ 9+27-64
4
4 - 28 + 36 + 108 - 216
4-21+ 18+ 27
- 7 + 18 + 81-216
4
-28+ 72 + 324-864
- 28 + 147 - 126 - 189
-75)- 75 + 460-676
1- 6+ 9
= x2-6x + 9
= (x-3)2.
= (x - 3)»(x + 2).
3, 3, 3, and — 2.
-7
4x + 3
560
ALGEBRA.
6. Find all the roots of x* — 24x2 + 64x — 48 = 0.
F (x) = ar* - 24x2 + 64x - 48,
Fi(x) = 4x8-48x+64.
4)4 + 0-48 + 64
1 + 0-
1-4 +
12-16
4
4-
4-
16+16
16+16
1-0-14 + 64-48
1 + 0-12 + 16
- 12) -12 + 48 - 48
1- 4+ 4
x + 4
.-. *(x) = x2 — 4x + 4
= (x-2)2.
.'.F(z)= (x-2)»(x + 6).
The roots are 2, 2, 2, and — 6.
7. Find all the roots of x* - 10x8 + 24 x2 + lOx - 25 = 0.
F (x) = x* - 10x8 + 24 x2 + lOx - 25,
F* (x) = 4x* - 30 x2 + 48x + 10.
1 - 10 + 24 + 10 - 25
4
4
9
-30+ 48+ 10
36
36
-270+ 432+ 90
-200+ 100
-70+332+ 90
9
-630 + 2988+ 810
- 630 + 3500 - 1750
-512)- 512 + 2560
4-40+ 96+ 40-
4-30+ 48+ 10
100
10+ 48+ 30-100
-20+ 96+ 60-200
- 20 + 150 - 240 - 50
-6)- 54 + 300-150
9- 50+ 25
-5
4x — 70
.-. 0 (x) = x - 5,
F{x)= (x - 5)2(x2 - 1).
The roots are 6, 5, 1, and — 1.
teachers' edition.
561
a Find all the roots of x6 -11x4+19x8 +115x2-200x- 500 = 0.
F (x) = x* - 11 x* + 19x» + 116x2 - 200x - 500,
F' (x) = 6x* ~ 44x» + 67 xa + 230x - 200.
5-44 + 57 + 230-200
5-40+25 + 260
1-11+ 19+ 115- 200- 600
6
- 4 + 32- 20-200
- 4 + 32- 20-200
5-55+ 96+ 576-1000- 2600
5-44+ 67+ 230- 200
-11+ 38+ 345- 800- 2500
6
-65+190+1725-4000-12500
-65 + 484- 627-2530+ 2200
—294) - 294 + 2362 - 1470 — 14700
1- 8+ 5+ 50
-11
5x-4
.-. 0 (x) = x* - 8x2 + 5x + 50,
0/(x) = 3x2- 16x+ 5.
3
3
-16 + 6
-15
-1 + 5
-1 + 6
.-. H.C.F.
1-8+ 5+50
3
3 - 24 + 15 + 150
3 - 16 + 5
X
2 - 8 + 10+150
3
— 24+ 30 + 450
- 24 + 128 - 40
-8
-48)- 98-490
1- 6
of 0 (x) and 0'(x) = .
3x-l
c-6.
*(x) = (x-5)»(x + 2),
.-. F(x) = (x-5)8(x + 2)«.
The roots are 5, 6, 6, — 2, and — 2.
562
ALGEBRA.
9.
Find all the roots of x6 — 2a* + 3x3 — 7 x2 + 8x — 3 = 0.
F (x) = x*-2x*+3x*- 7x2 + 8x-3,
F> (x) = 6x* - 8x* + 9x2 - 14x + 8,
F»(x) = 20x* - 24 x2 + 18x - 14.
Find H.C.F. of F* (x) and F" (x).
>20-
24 +
18-
14
10-
7
12 +
9-
7
70-
70-
84 +
280 +
63-
220
49
206 —
7
167-
49
1442-
1442-
1099- 343
6974 + 4632
4876)
4875-
4876
1-
5- 8+ 9-
2
14+8
10 - 16 + 18 -
10-12+ 9-
28 + 16
7
- 4+ 9-
6
21 + 16
- 20 + 46 -
- 20 + 24 -
106 + 80
18+14
3)21-
87 + 66
7-
7-
29 + 22
7
—
22 + 22
22 + 22
— 2
lOx-206
7x-22
.-. H.C.F. of F1 (x) and F» (x) = x — 1.
Hence Fl (x) contains (x — l)2 as a factor.
Bat 1 is a root of F(x) = 0.
.-. F(x) = (x - 1)» (x2 + x + 3).
™u x , , , , - 1 ± V- 11
The roots are 1, 1, 1, and
2*
10. Find all the roots of x4 + 6x« + xa — 24x + 16 = 0.
F (x) = x* + 6x« + x2- 24x + 16.
Fi (x) = 4x» + 18x2 + 2x - 24.
2)4+ 18 + 2-24
2 +" 9 + 1-12
2+ 6-8
3 + 9-12
3 + 9-12
1 +
2
6+ 1-24+ 16
2 + 12 +
2+ 9 +
2-48 +
1-12
3+ 1-J
6+ 2-72+ 64
6 + 27+ 3- 36
- 26) -26-76+100
1+3- 4
/. 0(x) = x2 + 3x-4
= (x-l)(x + 4).
.-. F(x) = (x-l)2(x + 4)*.
The roots are 1, 1, — 4 and — 4.
Sx + 3
teachers' edition. 563
Exercise CXLII.
1. Put the equation 2x* + § x2 — x + J = 0 in the form/(x) = 0.
Put - for x ;
o
then the equation becomes
2x» + 4x2- 36s + 36 = 0,
or x«+ 2x2 -18x+ 18 = 0.
2. Put the equation 3x* + 5x2 — \x — 8 = 0 in the form/(x) = 0.
Put ~ for x;
then the equation becomes
3x8 + 30X2 - I26x- 1728 = 0,
or x8 + 10x2- 42x- 576 = 0.
3. Put the equation 5x* — x8 — *f x2 — ^x + 1 = 0 in the form
/(*) = 0.
put Is***;
then the equation becomes
6x* - 30x8 - 6750x2 - 90000x -h 30* = 0,
or x*— 6 x8 - 1350 x2- 18000 x + 162000 = 0.
4. Put the equation x6 + *x* + jx8 - *x2 + x - 3 = 0 in the
form/(x) = 0.
* x
Put g f or x ;
then the equation becomes
x6 + 3x* + 24x8-72x2 + 1296x -23,328 = 0.
5. Put the equation x* — 2x2 + \x — 14 = 0 in the form/(x) = 0.
x
Put - for x ;
then the equation becomes
x*-8x2 + 4x-224 = 0.
564
ALGEBRA.
Exercise CXLIII.
L Diminish the roots of the
equation x* — 11 x2 + 31 x — 12
= 0 by 1.
1-11 + 31-12(1
+ 1-10 + 21
1-10 + 21+ 9
+ 1-9
1- 9+12
+ 1
1- 8
The required equation is
y«-8y2+12y + 9 = 0.
2. Diminish the roots of the
equation x* — 6 x8 + 4 x2 + 18 x
- 5 = 0 by 2.
1-6 + 4 + 18- 5[2
+ 2-8- 8 + 20
3. Diminish the roots of the
equation x8 + 10x2 + 13x — 24
= 0 by - 2.
1 + 10 + 13-241 — 2
- 2 — 16+ 6
1 +
8— 3-18
2 — 12
1+ 6 — 15
- 2
1+ 4
The required equation is
y8 + 4y2-15y-18 = 0.
4 Diminish the roots of the
equation x8 — 9x2+ 22x — 12 = 0
by 3.
1-4-4+10 + 16
+ 2-4-16
1-2-8- 6
+ 2 + 0
1 + 0-8
+ 2
1 + 2
+ 3-18+12
1-6+ 4+ 0
+ 3- 9
1-3- 6
+ 3
1+0
The required equation is
The required equation is
^ + 2^-8^-6^+15 = 0.
1^ — 6^ = 0.
5. Diminish the roots of the equation x* + x8 — 16 x2 — ^x + 48
= 0 by 4.
1+ 1-16- .4 + 48)4
+ 4 + 20+ 16 + 48
1 +
+
6+ 4+ 12 + 96
4 + 36+160
1 +
+
9 + 40 + 172
4 + 62
1 + 13 + 92
+ 4
1 + 17
The required equation is y4 + 17 y3 +
2y2 + 172y + 96 = 0.
teachers' edition. 565
6. Diminish the roots of the equation x4 + 2x8 — 25 x2 — 26 x
+ 120=0 by 0.7.
1+2-25 - 26+ 120|0.7
+ 0.7+ 1.89-16.177-29.5239
1 + 2.7 - 23.11 - 42.177 + 90.4761
+ 0.7+ 2.38-14.511
1 + 3.4-20.73-56.688
+ 0.7+ 2.87
1 + 4.1-17.86
+ 0.7
1 + 4.8
The required equation is
y* + 4.8 y* - 17.86 jfl - 56.688 y + 90.4761 = 0.
7. Diminish the roots of the equation x4 — x2 — 3 x + 4 = 0 by 0.3.
1 + 0 +1-3 +4|0.3
+ 0.3 + 0.09 + 0.327 - 0.8019
1 + 0.3 + 1.09 - 2.673 + 3.1981
' + 0.3 + 0.18 + 0.381
1 + 0.6 + 1.27 - 2.292
+ 0.3 + 0.27
1 + 0.9 + 1.54
+ 0.3
1 + 1.2
The required equation is
y* + 1.2y» + 1.54 y2 - 2.292y + 3.1981 = 0.
# 8. Diminish the roots of the equation x6 + x4 + 3 xa — 2x— 16 = 0
by 0.5. 1+1+0 +3 _2 -16|0.5
+ 0.5 + 0.75 + 0.375 + 1.6875- 0.15625
1 + 1.6 + 0.76 + 3.376 - 0.3125 - 16.16626
0.5+1 +0.875 + 2.125
1 + 2 +1.76 + 4.26 +1.8125
+ 0.5+1.25+1.50
1 + 2.5 + 3 + 5.76
+ 0.5+1.5
1 + 3 +4.6
+ 0.5
1 + 3.5
The required equation is
y6 + 3.6^ + 4.5y» + 5.76y2 + 1.8125y - 16.15626= 0.
5G6 ALGEBRA.
9. Diminish the roots of the equation
a*- 3x* - 2x» + 3x»- 7x + 12 = 0 by - 1.
1-3- 2+ 3-7 + 12J-1
-1+ 4- 2-1+ 8
1
-4 +
2 +
1-
-8 + 20
-1 +
6-
7 + 6
1
-5 +
7-
6-
-2
-1 +
6-
13
1
-6+13-
19
-1 +
7
1
-7 +20
-1
1-8
The required equation isy6 — 8y* + 20y» — 19y«— 2y + 20 = 0.
10. Diminish the roots of the equation
7fi -x* + 2x*- 3z* + 4x*-r- 6x + 6 = 0 by 0.2.
1-1 +2 -3 +4 -6 +6|0.2
+ 0.2 - 0.16 + 0.368 - 0.5264 + 0.69472 - 0.861056
1 - 0.8 + 1.84 - 2.632 + 3.4736 - 4.30528 + 5.138944
+ 0.2 - 0.12 + 0.344 - 0.4676 + 0.60320
1 - 0.6 + 1.72 - 2.288 + 3.0160 -
-3.70208
+ 0.2 - 0.08 + 0.328 - 0.3920
1 - 0.4 + 1.64 - 1.960 + 2.624
+ 0.2 - 0.04 + 0.320
1-0.2+1.6 -1.64
+ 0.2 + 0
1 + 0 +1.6
+ 0.2
1 + 0.2
The required equation is
tf + 0.2 tf + 1.6 y* - 1.64 y« + 2.624 y* - 3.70208y + 6.138944 = 0.
TEACHERS EDITION.
567
Exercise
1. Find the two commen-
surable roots of the equation
a;4-4x8-8x + 32=0.
Try 4 ; then 2.
1-4 + 0-8 + 32
+ 4 + 0 + 0-32
1+0+0-8 0
+2+4+8
1+2+4 0
The commensurable roots are
2 and 4.
2. Find the one commensurable
root of the equation
x8-6x2+10x-8 = 0.
Try 4. *
* 1-6 + 10-8
+ 4- 8 + 8
1-2+ 2 0
The commensurable root is 4.
3. Find the four commen-
surable roots of the equation
x* + 2x8 - 7x2 - 8x + 12 = 0.
Try
1 ; then 2
; then -
-2;
then -
-3.
1 + 2-
7-
8+12
+ 1 +
3-
4-12
1 + 3-
4-
12 0
+ 2+10+12
1 + 6 +
6
0
-2-
6
1 + 3
0
-3
1 0
The
> roots are 1,
2, -2,
and
-3,
CXLIV.
4. Find the one commensurable
root of the equation
x8 + 3x2-30x+36 = 0.
Try 3.
1 + 3-30 + 36
+ 3+12-36
1 + 6-18 0
The commensurable root is 3.
5. Find the two commensurable
roots of the equation
x*- 12x« f 32 x2 + 27x- 18 = 0.
Try 6 ; then — 1.
1-12 + 32 + 27-18
+ 6-36-24+18
1-6-4+3' 0
-1+7-3
1-7+3 0
The commensurable roots are
6 and — 1.
6. Find the two commensurable
roots of the equation
x4-9x8+17x2 + 27x-60 = 0.
Try 4 ; then 6.
1-9+17 + 27-60
+ 4-20-12 + 60
1-6-
+ 6 +
3 + 16
0-16
0
1 + 0- 3 0
The commensurable roots are
4 and 5.
568
ALGEBRA.
7. Find the five commensurable roots of the equation
x* - 6x* + 3x» + 17 x2 - 28x + 12 = 0.
Try 1 ; then 1 ; then 2 ; then 3 ; then — 2.
1-6 + 3+17-28+12
+ 1-4- 1 + 16-12
1-4-1 + 16-12 0
+ 1-3- 4+12
1-3-
+ 2-
-4+12
-2-12
1-1-6
+ 3 + 6
0
1 + 2
-2
0
1 0
The roots are 1, 1, 2, 3, and — 2.
8. Find the four commensurable roots of the equation
x* - 10x8 + 35x2 - 50x + 24 = 0.
Try 1 ; then 2 ; then 3 ; then 4.
+
10 +
1-
35-50 +
9 + 26-
24
24
+
9 + 26-24
2-14 + 24
0
+
7 + 12
3-12
0
+
4
4
0
0
The roots are 1, 2, 3, and 4.
9. Find the three commensurable roots of the equation
x* - 8x* + 11x3 + 29 x2-36x -46=0.
Try 3 ; then 6 ; then — 1.
1-8+ 11 + 29-36-45
+ 3-15-12 + 51 + 46
1-6-
+ 6 +
4+17 + 15 0
0-20-15
1 + 0-
-1 +
4-3 0
1+ 3
1-1- 3 0
The commensurable roots are 3, 5, and
teachers' edition. 569
10. Find the one commensurable root of the equation
x5 — x4 — 6 x3 + 9 x2 + x — 4.
Tryl. 1-1-6 + 9+1-4
+1+0-6+3+4
1+0-6+3+4 0
The commensurable root is 1.
Exebcise CXLV.
1. Compute the value of x4 — 6 x8 + 26 x2 — 4 x + 7 when x = 6.
-6 + 26- 4+ 7|5
+ 5+ 0+130 + 630
+ 0 + 26 + 126 + 637
The required value is 637.
2. Compute the value of x8 — 4 x2 + 6 x — 22 when x = — 7.
- 4+ 5- 22|-7
- 7 + 77-574
-11 + 82-596
The required value is — 596.
3. Compute the value of x5 — 2x4 + 3X3 + x2 — 28 when x = 2.
-2 + 3 + 1+ 0-28(2
+ 2 + 0 + 6+14 + 28
0 + 3 + 7 + 14 + 0
The required value is 0.
4. Compute the value of x5 + 7 x8 — 2 x2 — 49 when x = — 3.
0+7-2+ 0- 49|-3
- 3 + 9-48+150-450
- 3 + 16 - 50 + 150 - 499
The required value is — 499.
5. Compute the value of x5 — 14 x8 + 473 when x = 6.
0-14+ 0+ 0+ 473(6
+ 6 + 36 + 132 + 792 + 4752
+ 6+ 22 + 132 + 792 + 6225
The 'required value is 5225.
570 ALGEBRA.
6.. Compute the value of x« — 2x5 + 3x* + 2x8 + x2 — 7x — 96
when x = — 2.
-2+ 3+ 2+ 1- 7- 96|— 2
-2+ 8-22 + 40-82+178
-4+11-20 + 41-89+ 82
The required value is 82.
7. Compute the value of x6 — x6 — 2x* + x8 — 6x + 14 whenx=3.
-1-2+ 1+ 0- 6+ 14[3
+ 3 + 6+12 + 39+117 + 333
+ 2 + 4+13+39+111 + 347
The required value is 347.
8. Compute the value of x6 — 4x* + 2 x2 — 7 x + 16 when x = 10.
- 4+ 0+ 2- 7+ 16 1 10
+ 10 + 60 + 600 + 6020 + 60130
+ 6 + 60 + 602 + 6013 + 60146
The required value is 60,146.
9. Compute the value of x7 — x6 — 2X6 — 3x* + 2x» + x2 — x + 4
when x = — 2.
-1-2- 3+ 2+ 1- 1+ 4|-2
-2 + 6- 8 + 22-48 + 94-186
-3 + 4-11 + 24-47 + 93-182
The required value is — 182.
10. Compute the value ofx7 — 6x6 + 6x8 + 3x— 1 when x = 4.
0- 6+ 0+ 6+ 0+ 3- 1(4
+ 4 + 16 + 44 + 176 + 728 + 2912 + 11660
+ 4 + 11 + 44 + 182 + 728 + 2915 + 11659
The required value is 11,669.
Exercise CXLVI.
1. Determine the first significant figure of each root of the equa-
tion x8 — x2— 2x+ 1 = 0.
If x = - 2, - 1, 0, + 0.4, + 0.5, + 1, + 2,
/(x) = - 7, + 1, + 1, + 0.104, - 0.125, - 1, + 1.
Hence the roots are — 1. +, 0.4 +, and 1. + .
teachers' edition. 571
2. Determine the first significant figure of each root of the equa-
tion x8 — 5x — 3 = 0.
If x = - 2, - 1 - 0.7, - 0.6, + 0, + 1, + 2, + 3,
f(x) = - 1, + 1, + 0.167, - 0.216, - 3, - 7, - 6, + 9.
Hence the roots are — 1. + , — 0.6+ , and 2.+.
3. Determine the first significant figure of each root of the equa-
tion x8 — 6x2 + 7 = 0.
If x= -2, -1,0,4-1, +2, + 3, + 4, + 6,
/(x) = - 21, + 1, 4- 7, 4- 3, - 5, - 11, - 9, + 7.
Hence the roots are — 1.4- , 1.4-, and 4.+.
4. Determine the first significant figure of each root of the equa-
tion x8— 7x + 7 = 0.
If x = - 4, - 3, 4- 1, + 1.3, 4- 1.4, 4- 1.6, + 1.7,
f(x) = - 29, + 1, 4- 1, + 0.097, - 0.066, - 0.104, + 0.013.
Hence the roots are — 3. +, 1.34- , and 1.6+ .
5. Determine the first significant figure of each root of the equa-
tion x8 + 2xa - 30x + 39 = 0.
If x = - 8, - 7, + 1, + 2, + 3, + 4,
fix) = - 89, +4, + 12, - 6, - 6, + 16.
Hence the roote are — 7. + , 1. + , and 3.+.
6. Determine the first significant figure of each root of the equa-
tion x8 — 6xa + 3x + 5 = 0.
If x= - 1,-0.7,-0.6, + 1, + 2, + 6, + 6,
f(x) = - 6, - 0.383, + 0.824, + 3, - 6, - 6, + 23.
Hence the roote are — 0.6+ , 1.+, and 5.+.
7. Determine the first significant figure of each root of the equa-
tion x8 + 9 x2 + 21 x + 17 = 0.
If x = - 6, - 4, - 3, - 2, - 1,
f(x) = - 3, + 1, - 1, - 3, + 1.
Hence the roots are — 4.+, — 3. + , and — 1. + .
572 ALGEBRA.
8. Determine the first significant figure of each root of the equa-
tion x8 — 15x* + 63x — 50= 0.
If x = 0, + 1, +2, + 6, + 7, + 8,
/(x) = - 50, - 1, + 24, + 4, - 1, + 6.
Hence the roots are 1. + , 6. + , and 7.+.
9L Determine the first significant figure of each root of the equa-
tion x* — 8x8 + 14x2 + 4x — 8 = 0.
If x = - 1, - 0.8, - 0.7, + 0.7, + 0.8, + 2, + 3, + 5, + 6,
/(x) = + 11, + 2.2+, -0.9+, -0.8+, + 0.4+, +8,-5,
- 13, + 88.
Hence the roots are — 0.7+, 0.7+, 2. + , and 5. + .
10. Determine the first significant figure of each root of the equa-
tion x* — 12x2+ 12x — 3.
If x= - 4, - 3, 0, + 0.4, + 0.5, + 0.6, + 0.7, + 2, + 3,
/(x) = + 13, -66, - 3, - 0.09+ , + 0.06+ , 0.0+ , -0.2,
-11, + 6.
Hence the roots are — 3. + , 0.4+ , 0.6+ , and 2. + .
teachers' edition.
573
Exercise CXLVII.
1. Compute to six decimal places the root of the equation
x8 + 10 x2 + 6x — 120 which lies between 2 and 3.
+ 10
+ 2
+ 12
+ 2
+ 14
+ 2
+ 16
+ 0.8
+ 16.8
+ 0.03
+ 18.43
+ 0.03
+ 18.46
+ 0.03
+ 18.49
18
+ 6
+ 24
+ 30
+ 28
+ 68
+ 13.44
+ 71.44
+ 14.08
+ 85.52
+ 0.8
+ 17.6
+ 0.8
+ 18.4
+ 0.5529
+ 86.0729
+ 0.5538
+ 86.6267
+ 86.627
+ 0.055
+ 86.682
+ 0.055
+ 86.837
+ 86.84
+ 86.8
+ 87
-12012.833066+
+ 60
- 60
+ 67.152
- 2.848
+ 2.682187
- 0.266813
+ 0.260046
- 0.005767
""+ 0.005208
- 0.000559
+ 0.000522
- 0.000037
J
674
ALGEBRA.
2. Compute to six decimal places the root of the equation
z ■* + x2 + x — 100 = 0 which lies between 4 and 5.
+ 1
+_§
+ 6
+__§
+ 9
+ J
+ 13
+ 0.2
+ 13.2
+ 0.2
+ 13.4
+ 0.2
+ 13.6
+ 0.06
+ 13.66
+ 0.06
+ 13.72
+ 0.06
+ 13.78
+ 14
+ 1
+ 20
+ 21
+ 36
+ 67
+ 2.64
+ 69.64
+ 2.68
+ 62.32
64.08
64.1
64
-10014.264429 +
+ 84
- 16
+ 11.928
- 4.072
+ 3.788376
- 0.283624
+ 0.256076
- 0.027548
+ 0.8196
+ 63.1396
+ 0.8232
+ 63.9628
+
+
+
0.025632
0.001916
0.001282
0.000634
0.000576
0.000058
+ 63.963
+ 0.056
+ 64.019
+ 0.056
+ 64.075
teachers' edition.
575
3. Compute to six decimal places the root of the equation
x4 — 2 x3 + 21 x — 23 = 0 which lies between 1 and 2.
— 2
+ 1
-1
+ 1
0
+_!
+ i
+ i
+ 2
+ 0.1
+ 2.1
+ 0.1
+ 2.2
+ 0.1
+ 2.3
+ 0.1
+ 2.4
+ 0.05
+ 2.45
+ 0.05
+ 2.50
+ 0.05
+ 2.55
+ 0.05
+ 2.60
+ 0
-1
-1
+ 0
-1
+ 1
0
+ 0.21
+ 0.21
+ 0.22
+ 0.43
+ 0.23
+ 0.66
+ 0.1225
+ 0.7825
+ 0.1250
+ 0.9075
+ 0.1275
+ 1.0350
+ 1.04
+ 1.
+ 0.021
+ 19.021
+ 0.043
+ 19.064
+ 0.039125
+ 19.103125
+ 0.045375
+ 19.151500
+ 19.
+ 0.
15150
00728
+ 19.
+ 0.
15878
00728
+ 19.16606
+ 19.
+ 0.
1661
0004
+ 19.
+ 0.
1665
0004
+ 19.1669
+ 19.167
-23 1 1.157450 +
+ 20
- 3
+ 1.9021
- 1.0979
+ 0.95515625
- 0.14274375
+ 0.13411146
- 0.00863229
+ 0.00766660
- 0.00096569
+ 0.00095835
- 0.00000734
576
ALGEBRA.
4 Compute to six decimal places the root of the equation x* — ox8
+ 3 x* + 36 x — 70 = 0 which lies between 2 and 3.
- 6
+ 2
- 3
+_2
- 1
+_2
+ 1
+ 2
+ 3
+ 0.6
+ 3.6
+ 0.6
+ 4.2
4- 0.6
+ 4.8
+ 0.6
+ 6.4
4- 0.04
+ 6.44
+ 0.04
+ 6.48
+ 0.04
6.62
0.04
+
+
4- 6.66
4-10
+ 3
-6
-3
-2
-6
4-2
-3
4-2.16
-0.84
4-2.62
4-1.68
+ 2.88
+ 4.56
4- 0.2176
4-4.7776
4-0.2192
4-4.9968
4- 0.2208
4- 6.2176
4-6.22
+ 0.05
+ 6.27
+ 0.06
+ 6.32
+ 0.06
+ 5.37
+ 6
+ 36
- 6
+ 29
-10
+ 19
- 0.604
+ 18.496
+ 1.008
+ 19.604
+ 0.191104
+ 19.695104
+ 0.199872
+ 19.894976
+ 19.89498
+ 0.02635
+ 19.92133
+ 0.02660
+ 19.94793
+ 19.9479
+ 0.0036
+ 19.9614
+ 0.0035
+ 19.9649
+ 19.966
+ 19.95
- 7012.645751+
+ 58
-12
+ 11.0976
- 0.9024
+ 0.78780416
- 0.11459584
+ 0.09960665
- 0.01498919
+ 0.01396598
- 0.00102321
+ 0.00099775
- 0.00002646
+ 0.00001995
- 0.00000661
TEACHEB8' EDITION.
577
5. Compute to six decimal places the root of the equation
x4 — 12 x2 + 12 x — 3 which lies between — 3 and — 4.
0 -
12 - 12
- 3 1 3.907378-
h
+ 3 +
+ 3 -
9 - 9
3 - 21
-63
-66
+ 3 +
+ 6 +
18
15
+ 45
+ 24
+ 66.0241
- 0.9769
+ 3
+ 9
+
27
42
+ 48.249
+ 72.249
- 0.97590000
+ 0.92662260
+ 3
+ 12
+
+
11.61
+ 69.427
- 0.05027740
63.61
+ 131.676
+ 0.03984378
+ 0.9
+ 12.9
+ 0.9
+ 13.8
+
+
+
+
12.42
66.03
13.23
79.26
+ 131.67600
+ 0.55580
+ 132.23180
+ 0.55678
- 0.01043362
+ 0.00929908
- 0.00113454
+ 0.00106280
+ 0.9
+
+
+
+
79.26
+ 132.78858
- 0.00007174
+ 14.7
+ 0.9
+ 15.6
0.14
79.40
0.14
79.64*
0.14
79.68
+ 132.7886
+ 0.0240
+ 132.8126
+ 16.60
20
+
+
+
+ 0.0240
+ 132.8366
+ 132.837
+ 80
+ 100
+ 0.007
+ 132.844
+ 0.007
132.851
132.85
378
ALGEBRA.
6. Compute to six places of decimals the root of the equation
x5 + 2 x4 + 3 x8 + 4 x- + 5x — 54321 = 0 which lies between 8 and .9.
+ 2
+ 8
+ 10
+ 8
+ 18
+_8
+26
+ 8
+ 34
+ _8
+42_
+ 0.4
+42.4
+ 0.4
+42.8
+ 0.4
+ 43.2
+ 0.4
+43.6
+ 0.4
3
80
+ 83
+ 144
+ 227
+ 208
+ 435
+272
+ 707
+44.0
+ 4
+ 664
+ 668
+ 1816
+ 2484
+ 3480
+ 5U04
+ 16.90
+ 723.96
+ 17.12
+ 741.08
+ 17.28
+ 289^584
+ 6263.584
+ 296.432
+6550.016
+ 303.344
+6853.360
+ 758.30
+ 17.44
+ 775.80
+6853.4
+ 7.8
+6861.2
+ 7.8
+ 780
+6869.0
+ 7JS
+6876.8
+6880
+ 7000
+ 5
+ 5344
+ 6349
+ 19872
+25221
+ 2501.4336
+ 27722.4336
+ 2620.0064
+30342.4400
+30342.440
+ 68.612
+30411.052
+ 68.690
+30479.742
+30479.74
+ 27.52
+30507.26
+ 27.62
+30534.78
+30534.8
+ 2JS
+30537.6
+ 2JJ
+30540.4
+30540
-5432118.414454 +
+42792
— 11529
+ 11088.97344
— 440.02656
+
+ 304.11052
- 135.91604
+ 122.02904
= 13.88700
12.21504
- 1.67196
. 1.62700
0.14496
0.12216
- 0.02280
TEACHERS7 EDITION.
579
7. Compute to six places of decimals the root of the equation
x4 — 69 x2 + 840 = 0, which lies between 4 and 6.
0
+ 4
+ 4
±_i
+ 8
+ 4
+ 12
+ 4
+ 16
+ 0.8
+ 16.8
+ 0.8
+ 17.6
+ 0.8
+ 18.4
+ 0.8
+ 19.2
+ 0.09
+ 19.29
+ 0.09
+ 19.38
+ 0.09
+ 19.47
+ 0.09
+ 19.56
+ 20
- 59
+ 16
- 43
+ 32
- 11
+ 48
+ 37
+ 13.44
+ 60.44
+ 14.08
+ 64.62
+ 14.72
+ 79.24
+ 1.7361
+ 80.9761
+ 1.7442
+ 82.7203
+ 1.7523
+ 84.4726
+
+
84.47
0.16
+ 84.63
+ 0.16
84.79
0.16
+ 84.95
+ 85.
+ 100.
0
-172
-172
- 44
-216
+ 40.352
- 175.648
+ 51.616
- 124.032
+ 7.287849
- 116.744151
+ 7.444827
- 109.299324
- 109.29932
+ 0.67704
- 108.62228
+ 0.67832
- 107.94396
- 107.9440
+ 0.0765
-107.8675
+ 0.0765
- 107.7910
-107.791
+ 0.008
-107.783
+ 0.008
- 107.775
- 107.78
+ 84014.898989+
-688
+ 162
- 140.5184
+ 11.4816
- 10.50697359
+ 0.97462641
- 0.86897824
+ 0.10664817
- 0.00097002
+ 0.00005259
- 0.09708075
+ 0.00856742
- 0.00754481
+ 0.00102261
580
ALGEBRA.
8. Compute to six places of decimals the real root of the equation
z* - 36499 = 0. »
0 0-
36499|32.865378+
+ 30 +900
+ 30 +900
+ 27000
- 8499
+ 30
+ 60
+ 1800
+ 2700
+
5768
2731
i
+ 30
+ 90
+ 184
+ 2884
+
2519.552 1
211 448
+ 2
+ 92
+ loo
+ 3072
+
194.005666 «
17.442344 \
+ 2
+ 94
+ 2
+ 77.44
+ 3149.44
4- 7ft Oft
+
16.199170
1.243174
+ 96
+ 3227.52
+
0.972108
+ 0.8
4- A QA7«
—
0.271066
+ 96.8
+ 0.8
+ 97.6
+ 0.8
+ 98.4
+ 3233.4270
+ 5.9112
+ 2339.3388
+
+
0.240828
0.030238
0.027520
+ 3239.339
+ 0.495.
0.002718
+ 0.06
+ 98.46
+ 0.06
+ 3239.834
+ 0.495
+ 3240.329
+ 98.52
+ 0.06
+ 98.68
+ 3240.33
+ 0.03
+ 3240.36
+ 99
+ 100
+ 0.03
+ 3440.39
+ 3440.4
+ 3440
TEACHERS' EDITION.
581
9. Compute to six decimal places the real root of the equation
x8- 242970624 = 0.
0
600
+ 600
+ 600
+ 1200
+ 360000
+ 360000
+ 720000
+ 1080000
-242970624[624
+ 216000000
- 26970624
+ 22328000
- 4642624
-t- 6UU
+ 36400
+ 1116400
+ 4642624
o
+ 1800
+ 20
+ 1820
+ 20
+ 1840
+ 20
+ 1860
+ 36800
+ 1163200
+ 7456
+ 1160656
+ 1864
10. Compute to six decimal places the positive real root of the
equation x* — 707281 = 0.
0
+ 20
+ 9
+ 89
0
400
0
8000
-707281|29
+ 160000
+ 20
+ 400
+ 800
+ 1200
+ 1200
+ 2400
+ 8000
+ 24000
+ 32000
- 647281
+ 20
+ 40
-647281
0
+ 20
+ 60
+ 28809
+ 60809
+ 20
+ 80
+ 801
+ 3201
582
ALGEBRA.
11. Compute to six places of decimals the real root of the equation
& - 147008443 = 0.
0
+ 40
+ 40
0
+ 1600
+ 1600
+ 3200
+ 4800
+ 4800
+ 9600
+ 6400
+ 16000
0
+ 64000
+ 64000
+ 192000
+ 256000
+ 384000
+ 640000
0
+ 2560000
+ 2660000
+ 10240000
+ 12800000
-147008443143
+ 102400000
- 44608443
+ 40
+ 80
+ 44608443
0
+ 40
±l5X
+ 2069481
+ 14869481
+ 40
+ 160
+ 49827
+ 689827
+ 40
+ 200
+ 3
+ 609
+ 16609
+ 203
TEACHERS7 EDITION.
583
12. Compute to six places of decimals the positive root of the
equation x2 — 551791 = 0.
0
700
+ 700
+ 700
+ 1400
+ 40
+ 1440
+ 40
+ 1480
+ 2
+ 1482
+ 2
+ 1484
+ 08
+ 1484.8
+ 08
+ 1485.6
0.02
+ 1485.62
+ 0.02
+ 1485.64
0.006
+ 1485.646
+ 0.006
+ 1485.652
+ 1485.65
+ 1495.7
+ 1486
-5517911742.826359 +
+ 490000 ~
- 61791
+ 57600
- 4191
+ 2964
- 1227
1187.
39.
84
16
29.7124
9.4476
8.913876
0.533724
0.445695
088029
074285
013744
013374
0.000370
584
ALGEBRA.
' 13. Compute to six places of decimals the root of the equation
z2 — 17 x + 70.3 = 0 which lies between 7 and 8.
-17
+ 7
-10
+ 7
- 3
+ 0.1
- 2.9
+ 0.1
- 2.8
- 2.793
- 2.79
+ 0.003
- 2.797
+ 0.003
- 2.794
+ 0.0005
- 2.7936
+ 0.0006
- 2.7930
+ 70.3|7.103575+
-70.
+ 0.3
- 0.29
+ 0.01
- 0.008391
+ 0.001609
- 0.00139675
+ 0.00021225
- 0.00019551
+ 0.00001674
- 0.00001395
+ 0.00000279
teachers' edition.
5S5
14. Compute to six places of decimals the root of the equation
x8 + 9 x2 + 24 x + 17 = 0 which lies between — 4 and — 6.
-9 +24 -
17|4.532088+
+ 4 —
-5 +
20
4
+ 16
- 1
+ 4
— 1
4
0
+
0.875
0.125
1
+ 4
+ 3
+
+
1.75
1.76
+
0.116577 1
0.008423 1
X ft Fl
2.00
3.75
"T U.O
+ 3.5
+
+
+
0.008063768 i
0.000359232 |
+ 0.5
+ 4.0
+ 0.5
+ 4.5
T
+
+
+
0. Ioo9
3.8859
0.1368
4.0227
+
+
0.000323288
0.000035944
0.000032328
0.000003616
+ 0.03
+ 4.53
+ 0.03
+ 4.66
+
+
+
+
0.009184
4.031884
0.009188
4.041072
+ O.Od
+ 4.69
+
+
4.0411
4.041
+ 0.002
+ 4.692
+ 0.002
+ 4.694
+ 0.002
+ 4.596
586
ALGEBRA.
16. Compute to six places of decimals the root of the equation
x4 — 8x8 + 14xa + 4x — 8 = 0 which lies between 0 and — 1.
+ 8
+ 0.7
+ 8.7
+ 0.7
+ 9.4
+ 0.7
+ 10.1
+ 0.7
+ 10.8
+ 0.03
+ 10.83
+ 0.03
+ 10.86
+ 0.03
+ 10.89
+ 0.03
+ 10.92
+ 10
+ 14
+ 0.09
+ 20.09
+ 6.58
+ 26.67
+ _7J07
+ 33.74
+ 0.3249
4- 34.0649
+ 0.3258
+ 34.3907
+ 0.3267
+ 34.7174
+ 34.72
+ 0.02
+ 34.74
+ 0.02
+ 34.76
+ 0.02
+ 34.78
- 4
+ 14.
+ 10.
+ 18.
,063
063
+ 28.732
+ 1.021947
+ 29.
+ 1.
753947
031721
+ 30.785668
+ 80.
+ 0.
78567
06948
+ 30.
+ 0.
85615
06952
+ 30.92467
+ 20.
+ 30.
926
9
-810.7320508+
+ 7.0441 I
- 0.9559 i
+ 0.89261841
- 0.06328159
+ 0.06171030
— 0.00157129
+ 0.00154625
- 0.00002504
+ 0.00002472
- 0.00000032
teachers' edition. 587
Exercise CXLVIII.
1. Solve x* + 7x«-7x-l = 0.
x*+7x*-7x-l = 0,
(x2-l)(x2+7x+l) = 0.
-7±3V5
.-.«= ±1, or -
2. Solve x4 + x8 + x2 + x + 1 = 0.
x* + x* + x2 + x + 1 = 0,
x2 + x + l + i+- = 0,
X X2 '
(*a+J0+(*+;)+1=o.
(•+i),+(»+i)-1=a
. i -i±Vs
(1) *»+i-5^x+l = 0,
(2) x« + i^x+l = 0,
5=^pi±*V-2V6-10.
,= -V5-l±iV2Vg_10
3. Solve s«-3!B»+5iB»-5s» + 3x-l = 0.
afi - 3x* + 5x« - 6x» + 3* - 1 = 0,
(x» - 1) (as* - 3x» + 6x* - 3x + 1) = 0.
.-. x = ± 1,
or x* — 8x« + 6x2 — 3x+l = 0,
**-3x + 6-|+± = o,
588 ALGEBRA.
(x+l)»_2_3(x+i) + 6 = 0j
(x+I)'_3(x+I) + 4 = 0.
. 1 Z±^r7
,x+- =
x 2
(1) *»-*±^x + l = 0,
:=3 + VE7± jVeV^-w.
4
x-h 1 =
(2) x2 x+l = 0,
g=3_V-7±i V-6V^7-14.
4. Solve x* - 5x8 + 6x2 — 5x + 1 = 0.
x* — 5 x8 + 6 x2 - 5 x + 1 = 0,
x2-5x + 6-- + - = 0,
X X2 '
(x2+i.)_5(x+l)+6=0,
(x + l)4_2_6(x+i) + 6 = 0,
(x + I)4_6(x+i) + 4 = 0.
.-. x + - = 1, or 4.
x '
(1) x2-x+l = 0,
1±VI:3
X = — 1
(2) x2-4x+l = 0,
x=2±V3.
teachers' edition.
589
5. Solve 2x4-6x3 + 6x2-5x + 2 = 0.
2x*-5x8 + 6x2-6x + 2 = 0,
2x2-5x+6-- + -| = 0,
X X2 '
»(a?+s)-6(«+i)+8=a-
a(x+ly_4-6(*+y+6=0,
2 (x + ~x \2 - 5 (x + i ) + 2 = 0.
(1)
(2)
.-. x + - = 2, or £.
x ' *
xa-2x+l=0,
x=l.
x2-£x + l = 0,
6. Solve x6 — 4x* + x» + x2-4x+l = 0.
x6 — 4x4 + x* + x2 - 4x + 1 = 0,
(x + 1) (x* - 6x8 + 6x2 - 6x + 1) = 0.
.-.x=-l,
r x4 — 6x8 + 6x2 — 6x+l = 0,
*-6x + 6-| + ± = 0,
x2 + l2_5(x+I) + 6 = 0)
(x + l)2-5(x + i)+4 = 0.
(1)
(2)
.-. x + - = 1, or 4.
x '
x2 - x + 1 = 0,
* = —
---4x4-1 = 0,
x = 2 ± V3.
500 ALGEBRA.
7. Solve x* - 10x8 + 26 x* - lOx + 1 = 0.
a-*- 10x8 + 26x*-10x+l = 0,
x2-10x + 26- — +-^ = 0,
x x2
(xJ + la)_10(a!+I) + 26=0)
.-. x + - = 4 or 6.
x
(1) x2-4x+l = 0,
x = 2 ± V5.
(2) x2-6x+l = 0,
x = 3±2V2.
a Solve x8 + mx2 + mx + 1 = 0.
x8 + mx2 + mx + 1 = 0,
(x + 1) {x2 + (m - 1) x + 1} = 0.
.-. x= — 1,
or x2+ (m — l)x+l = 0,
1 — m ± Vto2 — 2 m — 3
x=
9. Solve x5 + 1 = 0.
x6+l = 0,
(x + 1) (x*- x8 + x«- x+ 1) = 0.
.-.x=-l,
r x* — x8 + x2 — x + 1 = 0,
(*+£)-(*+i) + »=«.
, i i±Vs
.-. x+- = —
x 2
(1) x2-1-^x + l = 0,
x =
591
(2) x2-^-~x+l = 0,
x= * A ± 4-V-2V5-10.
4 *
10. Solve 3x*-2x* + 5x8-5x2 + 2x-3 = 0.
3x6 _ 2a;4 + 6x8 - 5x2 + 2x - 3 = 0,
(x- 1) (3x* + x8 4- 6x2 + x + 3) = 0.
.-.x=l,
r 3x* + x8 + 6x2 + x + 3 = 0,
3x2 + x+6 + l + - = 0,
X xa
■(•+s),+ ('+i) = °-
.-. x + - = 0 or - i.
x
(1) X2 + 1 = 0,
X = ± V17!.
(2) x2 + *x+l = 0,
■ 1 ± V- 35
6
Exercise CXLIX.
1. Solve 11* =346.
2. Solve 3* = 10.
11* =346,
3* = 10,
x log 11 = log 346,
x log 3 = log 10,
log 346
log 11
log 10
log 3
2.6391
1.0000
1.0414
0.4771
= 2.438+.
= 2.096 +
592 ALGEBRA.
3. Solve 10* =745.
a Solve 146*= 12984.
10* =745,
146*= 12984,
x log 10 = log 745,
lo?745
x log 10
2.8722
x log 146 = log 12984,
_ log 12984
X log 146
4.1134
""1.0000
2.1644
= 2.8722+.
= 1.900+.
7. Solve 0.2* = 0.4
4. Solve 7* = 324.
0.2*= 0.4,
7*= 324,
xlog0.2 = log0.4,
x log 7 = log 324,
log 0.4
log 324
X log 0.2
X" log7
9.6021 — 10
2.5105
" 9.3010 — 10
0.8451
0.3979
= 2.970 +.
"0.6990
= 0.669+.
5. Solve 4* =3.74.
a Solve 14.74* = 8.64.
4*= 3.74,
14.74* = 8.64,
x log 4 = log 3.74,
x log 14.74 = log 8.64,
log 8.74
log 4
log 8.64
X " log 14.74
0.6729
0.9365
""0.6021
1.1685
= 0.951+.
= 0.801 +.
9. Solve x* = 2.767.
x* =
2.767,
x log X =
log 2.1
'67
0.4420
1 log 1 =
1X0
= 0,
1.7 log 1.7 =
: 1.7 X 0.2304 = 0.3917,
1.77 log 1.77 =
: 1.77X0.2480 = 0.4390,
1.774 log 1.774 =
: 1.774X0.2490 = 0.4417.
.-. x =
: 1.774 +.
teachers' edition.
593
10. Solves* = 23.10.
xx = 23.10,
x log x = log 23.10
= 1.3636.
2 log 2 = 2 X 0.3010 =
0.6020,
2.9 log 2.9 = 2.9 X 0.4624
= 1.3410,
2.92 log 2.92 = 2.92 X 0.4664 = 1.3590,
2.925 log 2.926 = 2.925 X 0.4661 = 1.3636.
.-.x= 2.925+.
11. Given P=750, 4=1797.42,
r = 6%; find*.
f _ log A - log P
log (1 4- r)
4.3383 - 3.7506
0.0294
0.5877
0.0294
= 20.
3.2547 - 2.8751
0.0253
0.3796
0.0253
= 15.
12. Given P= 780,4=1559.!
r=8%; find*.
.log A — log P
log (1 + r)
_ 3.1926 -2.8921
~ 0.0384
0.3005
~~ 0.0334
= 9.
13. Given P = 5630.76, A-
21789.22, r= 7%; find t
.log A — log P
log (1 + r)
14. Given P = 300, A- 615.46,
r=7%; find*.
. _ log A — log P
log (1 + r)
_ 2.7122 -2.4771
" 0.0294
_ 0.2351
0.0294
= 8.
15. Given P =84.65, 4=289.47
r=7±%; find*.
log 4 — log P
log (1 + r)
2.4616 - 1.9276
< = •
0.0314
0.5340
~~ 0.0314
= 17.
594 ALGEBRA.
Exercise CL.
1. x8 + 12x2 + 45x + 50 = 0. +7—49
Letx = y-4. (Cf. § 600.) *- 7+ 12
1 + 12 + 45 +50^ -j^
1+ 8+13- 2 * T y
- 4 - 16 p = 12, q = - 63.
1 + 4-
-4
2= VAf±V64 + i-9^
8i :
l + o = V-6/ ± -*$■
..yS-3y-2 = 0. = -lor4,
p=-B,g=-2. *-2~"3^~3
= -1 + 4+7 or 4 — 1+7
-«/-»«*>/s+$
= 10.
= 1.
p m
= ^1±V^1 + 1 1-21 + 159-490[10
31 1T +10-110
1-11+ 49
x=z~3?~3 x2-llx + 49 = 0.
= 1 + 1-4 llisV^
= -2. .'.x=10, or g
Divide the given equation by ^ x8 — 6x2+ 13x — 10.
x + 2.
1 + 12 + 45 + 60^2 176 + 13710L2
1+10 + 25
Let x = y + 2.
1-6 + 13
+ 2- 8+10
1-4+ 5+ 0
x2+10x + 25 = 0, +2— 4
(3+5)2 = 0. 1-2+ 1
.-.x=-2, -5, -5. +2
1 + 0
2. x8 - 21 x2+159x- 490 = 0. ... ^8 + y = o.
Let x = y + 7. y (y2 + i) - o,
l-21 + 159-490[+7 y = 0, oriV^l,
+ 7-98 + 427 x=y+2
1-14+61-63 = 2, or 2 ± V--1.
TEACHERS' EDITION.
595
4. x8+3x2 + 9x-13 = 0.
Let z = y — 1.
l + 3 + 9-13|-l
-1-2- 7
1 + 2 + 7-20
-1-1
1 + 1 + 6
-1
1 + 0
.-. y* + 6y — 20=0.
p = 6, q = — 20.
2 = VlO ± V8 + 100
= Vio±eV§
= 1±V3,
p___m
X~Z 32 3
= 1 + V3 ^"p-1
1 + V3
= 1 + V3+1-V3-1
= 1.
1 + 3+ 9-1311
+ 1+ 4 + 13
1 + 4 + 13 0
x2+4z+13 = 0.
.-.x=l, or -2±3V=T.
5. y8 + 48y+ 504 = 0.
p = 48, g = 504.
2= V- 252 ± V4096 + 6350^
= V- 252 ± 260
= 2,
y = 2 —
3«
1 + 0 + 48 + 5041-6
-6 + 36-504
1-6 + 84
y2-6y + 84 = 0.
.y=-6, orSisV1^.
6. 2/8- 21 y- 344 = 0.
p = - 21, g = - 344.
z = Vl72 ± V343 + 29584
= V172 ± 173
V
y = Z~Sz
= 1 + 7
= 8.
1 + 0-21-344 [8
+ 8 + 64 + 344
1 + 8 + 43
y2 +8y + 43 = 0.
.•.y = 8, or — 4±3V^3.
7. y8~3y + 2 = 0.
p=-3, g = 2.
3r
= 2-8
2= V- 1 ± V- 1 + 1
= -1,
p
y = Z-S~z
= -1-1
= -2.
1 + 0-3 + 21-2
-2 + 4
1-2+1
2/2-2y+ 1-0.
.-.y=-2, 1, 1.
596
ALGEBRA.
a y»-60y + 071 = 0.
p = - 60, q = 671.
2= V— 4 Ji±V— 8000 +15J^tl
"tf
*/-C71±llV41
_-ii±Vii
2 f
* = *-£
_-ii + Vii .
40
= -ll + V41 11 + V41
2 2
= — 11.
1+0- 60 + 6711-11
-11 + 121-671
1-11+ 61
ya— lly + 61 = 0.
-11+V41 -y=-^or
11 ± V- 123
Exercise CLI.
1. x8 + 3x-6 = 0.
p = 3, g=-6.
= - 0.4000.
log tan e = 9.60206 (n),
0=158° 11' 65",
i 0=79° 6' 58".
log tan ie= 10.71539,
log tan \<p = i log tan * 0
= 10.23846,
i0 = 59°59'39",
0 = 119° 51' 18",
cot 0 = - 0.57709,
csc0= 1.1546,
31 =
COt0
Ip , i —
X2 = ^ £ COt 0 + V — p CSC 0
= -0.57709+ 1.9998 V^.
Xg = -y g cot 0 — V^p esc 0
= - 0.57709 - 1.9998 V1^!.
2. x*+7x + 3.
p=7,g = 3.
tan 6-
sqyls
14 ll
= -1.16418.
log 14= 1.14613
logV7= 0.42255
colog 9= 9.04576 — 10
colog V3 = 9.76144—10
log tan 0 = 10.37688
TEACHERS' EDITION.
597
0 = 67° 10' 36",
1 =33° 35' 18",
2 ,
log tan |=9.82223,
<t>
log tan |=ilogtan£0
= 6.94074,
£=41° 6' 12",
0 = 82° 12' 24",
cot 0 = 0.13686,
esc 0=1.0093,
log 2 = 0.30103
log Vp= 0.42265
colog V3= 9.76144 -10
log cot 0 = 9.13629
log Xi = 9.62131 - 10
xx = - 0.41813.
x2 = -v? cot 0 + V— p CSC 0.
log Vp = 0.42255
log esc 0 = 0.00403
0.42658
X2 = 0.20906 + 2.6704 V^I,
x8 = 0.20906 - 2.6704 V^.
3. x«- 7x+ll = 0.
33\3
log 14=1.14613
log V7 = 0.42255
colog 33=8.48149-10
colog V3 = 9.76144 -10
log sind =9.81161
0=40° 23' 40",
i = 20° 11' 50",
log tan £=9.56570,
0 , 0
log tan -=* log tan -
= 9.85523,
£ = 35° 37' 21",
0 = 71° 14' 42".
log cot 0 = 9.53091,
log esc 0 = 0.02369.
*=-Vi
CSC0.
sin 0-
log 2 = 0.30103
logVp = 0.42255
colog V3 = 9.76144 -10
log esc 0 = 0.02369
log xi = 0.50871
Xx = - 3.2264.
«2 = \/| CSC 0 + V— p COt 0.
logVp = 0.42255
log cot 0 = 9.53091
9.95346
Xa = 1.6132 + 0.89838 i,
x8 = 1.6132 - 0.89838 i.
598
ALGEBRA.
4 x«-4x-5=0.
»•(£<?>
sin0 =
2|> J>
3g \3
16
15 V§
log 16 =
1.20412
colog 15 =
8.82391
colog V3 =
9.76144 -
10
log sin 6 =
9.78947-
10
0 =
38° 0/ 48"
»
0 _
19° 0/ 24"
.
log tan - = 9.53713,
log tan | = - log tan |
= 9.84571,
£ = 36°1'48",
<t> = 70° 3' 36".
log cot 0 = 9.55965,
log esc 0 = 0.02685.
*i= ~ 2 \f csc * = ~ Taj086*"
log 4 = 0.60206
colog V3 = 9.76144 -10
log csc 4> = 0.02685
log Xi = 0.39035
Xi = - 2.4567.
x2 = -v/^ csc 0 4- V— p cot 0.
log Vp= 0.30103
log cot 0 = 9.55965
9.86068
Xa = 1.2283 + 0.72557 V^lf
x8 = 1.2283 - 0.72557 V^l.
5. x*-5x+4 = 0.
»- — • «?>?>
2j> \p
6 /3
= 5\5'
log 6 = 0.77815
logV3 = 0.23856
colog 5 = 9.30103
colog V5 = 9.65051
log sin $ -
= 9.96825
0 =
= 68° 21' 24",
e _
3"
= 22° 47' 8",
eop-f=
= 37° 12' 52",
60o+| =
= 82° 47' 8".
Xi =
=Wl-i-
log 2 =
logV5 =
colog V3 =
: 0.30103
= 0.34949
= 9.76144
U«2^ =
= 0.41196
log sin - =
: 9.58803
log Xi =
Xi =
: 9.99999
1.
teachers' edition. 699
*=2Vf8to(6oo-f>
log2>/f =
■ 0.41196
logsii
,(60°-!) =
10gX2 =
= 9.78161
= 0.19367
X2 =
: 1.6616.
Xz-
=-2>lh*
(60O + I). *
log2iJf =
= 0.41196
logsii
log x8 =
: 9.99665
0.40851
Xz =
-2.5616,
1
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