r ^^
*\ ^^p^ / \ A
\. J"
AN
ELEMENTARY ALGEBRA :
DESIGI^ED AS
I
AN INTRODUCTION TO A THOROUGH KNOWLEDGE
OF ALGEBRAIC LANGUAGE, AND TO GIVE
BEGINNERS FACILITY IN THE USE OF
ALGEBRAIC SYMBOLS.
CHARLES S. VENABLE, LLD.,
PfiOPESSOR OF Mathematics in the University o? Yirginia ; Author op " First
Lessons in Numbers," "Mental Arithmetic," "Practical
Arithmetic," and "Higher Arithmetic."
UNIVERSITY PUBLISHING COMPANY,
NEW YORK AND BALTIMORE.
1872.
or THE
VNIVERRITY
urn
V4.
/^
Entered according to Act of Congress, in the year 186f),
By the university .^L'BLISHING COMPANY,
m the Clerk's ofScQ rf the District Co'irt of the United States for the Southern
i;:slrict of New \ ork.
PREFACE.
The present Elementary Algebra has been prepared with
a view to enable the beginner to obtain a thorough knowl-
edge of Algebraic Language, and to acquire an early facility
in the use of Algebraic Symbols. The translation of Ex g-
lish into the symbolical language of Algebra, and the inter-
pretation of Algebraic Symbols by arithmetical operations
are made prominent from the beginning. Throughout the
work I have endeavored, in the Algebraic operations and
solutions of problems, to present examples of elegance and
conciseness in the transformation of Algebraic expressions.
I am convinced by long observation that the difficulties of
students in their more advanced mathematical studies are
greatly enhanced by their want of knowledge of Algebra as
a Language, and their want of facility m the transformation
and combination of expressions in the solution of problems.
These tM7igs form tlie dasis of any thorough Icnowledge of
Algebraic Analysis, and should he learned ivellin the begin-
ning. The demonstrations are, I think, clear and easily
rntelligible to the young student. The examples for exercise
are numerous.
In addition to the fundamental Algebraic operations on
Entire Quantities and Fractions, Evolution, Surds, Equa-
4 PEEFACE.
tions, Arithmetical and Geometrical Progressions, and Pro-
portion, I have treated in an elementary manner the subjects
of Fractional Exponents, Permutations and Combinations,
the Binomial Theorem for whole-number exponents, Har-
monical Progression, Theory of dotation, and Logarithms.
I am convinced by long experience that it is important to
present these subjects to the young student in a simple and
practical manner before he comes in contact with them in
their greater extensions and more difficult applications. In
the preparation of this book I have consulted many of those
works which give a view of the progress and improvement in
elementary instruction in Algebra. But three English works
— Todhunter's Algebra for Beginners, Colenso's Algebra,
and Lund^s Wood's Algebra — are made the basis of the
worK. The demonstrations of Wood, (a standard of more
than half a century,) are singularly clear and simple, while
those of Colenso are models of elegance and brevity. Tod-
hunter's illustrations are clear and copious. The examples
have been selected mainly from the above authors, many of
them having been taken by them from the Cambridge Ex-
amination papers. I have also used Lund's Easy Algebra,
Bobillier's "Principes d'Algebre," Kitt's Problemes d'Al-
gebre, and Wrigley's Collection of Problems.
University of Virginia,
Aug. I, 1869.
CONTENTS.
Pagb
I. Principal Signs 7
II. Factor — Coefficient — Power — Terms . ii
III. Remaining Signs— Brackets . . . i6
IV. Change of Order of Terms— Like Terms 20
V. Addition 24
VI. Subtraction 28
VII. Brackets 33
VIII. Multiplication 38
IX. General Results of Multiplication . 45
X. Division 52
yXI. Factors 61
XII. Greatest Common Divisor . • • . (^^
XIII. Least Common Multiple .... "](>
XIV. Fractions 80
XV. Reduction of Fractions .... 84
XVI. Addition and Subtraction of Fractions 88
XVII. Multiplication of Fractions ... 94
XVIII. Division of Fractions 98
XIX. Complex Fractions, and other Results ioi
XX. Involution 105
XXI. Evolution . . ' . . . . . no
XXII. Simple Equations 128
XXIII. Simple Equations — continued . . 137
XXIV. Problems Solved by Simple Equations . 145
XXV. Problems — continued . . . . 153
6 CO]SrTEis-TS.
XXVI. Simultaneous Equations of the First
Degree 164
XXVII. Problems Solved by Simultaneous Equa-
tions OF THE First Degree . . 176
XXVIII. Indices 184
XXIX. Surds ...'..... 191
XXX. Quadratic Equations 200
XXXI. Equations which may be solved like
Quadratic Equations . . . . 212
XXXII. Problems which lead to Quadratic
Equations containing One Unknown
Quantity .... ... 217
XXXIII. Simultaneous Equations involving Quad-
ratics . . . . . . .221
XXXIV. Ratio 230
XXXV. Proportion 233
XXXVI. Arithmetical Progression .... 240
XXXVII. Geometrical Progression ... 245
XXXVIII. Harmonical Progression .... 252
XXXIX. Permutations and Combinations . . 254
XL. Binomial Theorem . . . . . . 263
XLI. Scales of Notation 269
XLII. Logarithms . . 275
Answers to Examples . . . • 284
ELEMENTARY ALGEBRA.
I. The Peincipal Signs.
1. Algebra is the science in which Ave reason about num-
bers with the aid of letters to denote the numbers, and of
certain signs to denote the operations performed on the
numbers, and the relations of the numbers to each other.
These letters and signs are called Algehraic Symbols,
2. Quantity signifies anything which admits of increase,
or diminution. The word quantity is often used with the
same meaning as numler,
3. The sign + placed before a number denotes that this
number is to be added. Thus a ^1) denotes that the num-
ber represented by h is to be added to the number repre-
sented by a. If a represent 9 and t) represent 3, then a + 5
represents 12. The sign -f is called the ])lu8 sigti, and
a + Z> is read " a plus b."
4. The sign — placed before a number, denotes that the
number is to be subtracted. Thus a — b denotes that the
number represented by Z>, is to be subtracted from the num-
ber represented by a. If a represent 9 and b represent 3,
then a — b represents 6. The sign — is called the minus
sign, and a ~ ^ is read thus, " a minus b."
5. Similarly, a + b ■\- c denotes that we are to add b to a,
and then add c to the result ; a-\-b — c denotes that we
Define Algebra ; Algebraic Symbols ; Quantity, How is Addition indicated ?
Subtraction ?
8 ELEMENTARY ALGEBRA
are to add h to a, and then subtract c from the result ] a — l)
+ c denotes that we are to subtract 1) from a, and then add c
to the result ; a — h — c denotes that we are to subtract h
from a, and then subtract c from the result.
6. The sign = denotes that the numbers between which it
is placed are equal. Thus a = l denotes that the number
represented by a is equal to the number represented by l.
And a -{-h — c denotes that the sum of the numbers repre-
sented by a and h is equal to the number represented by c;
so that if a represent 9 and ^ represent 3, then c must
represent 12. The sign — is called the sign of equality,
and a = Z> is read thus, " a equals b," or a is equal to h,
7. The sign X denotes that the numbers between which it
stands are to be multiplied together. Thus a Xl) denotes
that the number represented by a is to be multiplied by the
number represented by b. If a represent 9 and b represent
d, then a X b represents 27. The sign X is called the sign
of multiplication, and ^ X ^ is read thus, " a into b/^ or " a
multiplied ly b.^^ Similarly, aXbX c denotes the product
of the numbers represented by a, h, and c,
8. Sometimes a point is used instead of the sign X . Thus
a. I) instead of a X ^. Both of these signs are, however, often
omitted for the sake of brevity ; thus a h is used instead of
aXh, and has the same meaning. So also, ahc i^ used in-
stead of aXb X c, or a.b.c, and has the same meaning,
Nor is either the point or the sign X necessary between a
numbei expressed by a figure and a number expressed by a
letter ; so that 3 a is used instead of 3X a, and has the same
meaning. The sign of multiplication must not be omitted
between numbers expressed in the ordinary way by figures.
Thus 45 cannot be used to represent the product of 4 and 5,
because the meaning, forty-five, has already been assigned to
45. Nor can the point be used between figures to express a
How is Equality indicated ? Multiplication ?
PKIIS^CIPAL SIGKS. 9
product, witliout prodiiciDg confusion, as, for example, to
4 . 5 has already, in Arithmetic, been assigned the meaning
4 + Y^Q- ; still, 4 . 5 is sometimes written for 4x5.
0. The sign -i- denotes that the quantity which stands
defore it is to be divided by the quantity which follows it.
Thus a-^h denotes that the number represented by a is to
be divided by the number represented by I. If a represent
12, and 1) represent 4, then a -^l represents 3. The sign -^
is called the sign of division, and « -f- Z> is read thus, " a
divided hy b," or briefly, " a ly b."
But most frequently to express division the dividend is
placed over the divisor, with a line betw^een them. Thus -^
is used for a-^l, and has the same meaning.
10. A number or quantity expressed by Algebraic Sym-
bols is called an Algebraic exinession, or briefly, an expres-
sion,
11. We shall now give some examples as an exercise in the
use of the symbols which have been explained ; these exam-
ples consist in finding the numerical values of certain Alge-
braic expressions, and in finding the Algebraic expression
for certain quantities expressed in ordinary language.
Suppose a = 1, l) — 2, c — 'd,d — 6, e = 6, f= 0. Then
7« + 3Z'-2 6?+/=74-6-10 + = 13-10 = 3.
2 ad + Shc-ae + df =4: + 4.8 -6 + = 62-6 = 4^6.
4:ac , lObe de 12 ,120 30 ^ , _ ,^
b cd ac 2 lo 3
14 - 10 = 4.
4:c + 6e _ 12 + 30 _ 42__
d-b "" 5-2 ~ 3 ~ '
How is Division indicated? What is an Algebraic Expression ?
10 eleme:n^tary algebra.
Examples — 1.
li a=l, b^ Vf c=3, cl=4:, e=5,f = 0y find the numerical
values of the jllowing expressions :
1. 9a+2I)+3c-2f. 2. 4:e-3a-dI}-\-5c.
3. 7ae-{-3I)c+dcl-af. 4. 8adc-hcd+9cde-def.
5. ahcd-habce-habde + acde+dcde, 6. -7- h h-r .
c d e
4:ac 8bc 6cd _ 12a 6b 20c
7. —J j J . O. -y 1 ---H —•
ode be cd de
^ cde , hbcd 6ade ^. ^.77- ^^de
9. — 7-H = — . 10. 7e-hbcd — ^ — .
ab ae be 2ao
2a+6b Sb-h2c a+b+c + d b+c-{-3e
c d 2e e+c—d
-Q a + c b-\-d . c+e ^. a+b+c + d-\-e
c — a d—b e—c e—d + c—b-\-a
15. What is the difference between ~-\ — and -j-' when
be be
a = 4, 6 = 5, c = 10 ? Ans. 5^.
16. A person who possessed a fortune, expressed by x, re-
ceives by inheritance two sums, a and b. Express his whole
property.
17. An estate is divided among three heirs: the second ob-
tains a dollars more than the first, and the third b dollars
more than the second. Express the value of the estate, tlie
share of the first heir being x.
18. A man who possessed x dollars,* lost a of them. How
many has he left ?
FACTOR. — COEFFICIEKT. — POWEK. — TERMS. 1 1
19. The Slim of two quantities being represented by s, if
one of these quantities is expressed by x, what will be the
other ?
20. A debtor set out to pay his creditor a part, a^ of his
debt X ; but on the way he met another person, to whom he
gave a sum Z> and carried the remainder to his creditor. Ex-
press what remains of the debt.
21. The three figures of a number are such that the tens
figure exceeds the units figure by 2, and the hundreds figure
exceeds the tens figure by 3. What is the sum of the figures,
X being the units figure ?
22. Express 45.6 by means of algebraic signs.
23. The figures of a number in the hundreds, tens, and
units place respectively are x, y, z. Express the number.
Express the number with the figures reversed.
24. A w^orkman whose wages were x dollars a day, receives
during n days an increase of wages amounting to h dollars a
day. Express the amount of his wages for this time.
25. Express the number which divided by 11 gives 5 for a
remainder, x being the quotient.
26. A man travels uniformly a distance s in a number
t of hours. AYhat is his rate of travel per hour ?
27. Two fountains fill a reservoir, one in a number of
hours represented by x, the other in 3 hours more. What
part of the reservoir does each one fill in one hour ?
28. A number N divided by d gave for remainder the
number r. Express the quotient.
II. Factor — Coefficient — Power — Terms.
12. When one number consists of the product of two or
more numbers, each of the latter is called a factor of the
product. Thus, for example, 2 X 3 X 5 = 30 ; and each of
Define Factor.
1^ ELEMENTARY ALGEBRA.
the numbers 2, 3, and 5 is a factor of the product 30. Or we
may regard 30 as the product of the two factors 2 and 15, or
as the product of the two factors 3 and 10. So also we may
consider 4 a Z> as the product of the three factors 4, a, and ^,
or of the two factors 4 a and b; or as the product of the two
factors 4 and ad; or as the product of the two factors 4 d
and a,
13. When a number consists of the product of two fac-
iTors, each factor is called the coefficient of the other factor ;
so that coefficient means co-factor. Thus, considering 4: ad
as the product of 4 and a d, we call 4 the coefficient of a h,
and ab the coefficient of 4; and considering 4a^ as the
product of 4: a and d, we call 4 a the coefficient of b, and I?
the coefficient of 4 a. In practice, the name coefficient is ap-
plied to the factor which precedes the other, and usually the
first factor is called the coefficient. When this first factor is
a number expressed by a figure, it is called the oiumerical
coefficient. Thus 4 is the numerical coefficie7it of ab in the
expression 4:al). Since 1 is a factor of every product, when
no numerical coefficient is written before an algebraic ex-
pression 1 is always understood as its 7iumerical coefficient.
Thus 1 is the numerical coefficient of the expressions, a h, a,
abc, Wlien we use one of the letters of a product as a coef-
ficient it is called a literal coefficient. Thus in the expres-
sion a X, a is the literal coefficient of x,
14. When all the factors of a product are equal, the
product is called a poiver of that factor. Thus 7 X 7 is
called the second poiver of 7; 7 X 7 X 7 is called the third
foioer of 7 ; 7 X 7 X 7 X 7 is called the fourtli power of 7,
and so on. In like manner, « X <^ is called the second poiver
of « ; aX aX a\s called the third power ofa; aXaXaXa
is called ihQ fourth power of a, and so on. And a itself is
called the first p)0wer of a.
Coefficient, numerical and literal. Power.
FACTOE — COEFFICIEXT — POWEE — TEEMS. 1 3
15. Instead of writing all the equal factors, we express a
power more briefly by writing the factor once, and placing
over it, and a little to the right, the number which indicates
how often the factor is to be repeated. Thus c^ is used to de-
note ^ X « ; ci^ denotes a X aX a] a^ denotes a X a X a X a;
3a^ b^ c^ d denotes oaaaahl)h ccd, ^ndi so on. And a^ may
be used to denote the first power of a, that is, a itself; so
that a^ has the same ^meaning as a. The number thus placed
over another to indicate how many times the latter occurs as
a factor in. a power, is called an index of the jpoiver, or an ex-
ponent of the i^oiver^ or briefly an index or exponent. Tlius,
for example, in a^ the exponent is 3 ; in a"" the exponent is
n ; in 2"" the exponent is m,
16. The second power of a, that is a^, is usually called the
square oi a oy a squared ; and a^ is often called the cuhe of
a or a cubed. For the powers higher than these tliere are no
similar words in use ; a'^ is read thus, " a to the fourth
potver/' or " a to the fourth f a"^ is read " a to the 7nth."
17. The student must distinguish carefully between a
coefficient and an exponent. Thus 3 c means three times c,
and m a stands for m times a. Here 3 and m are coefficients.
But c^ means c times c times c ; and c"^ stands for c times o
times c times c and so on ... . times c.
18. If an expression contain no parts connected by the
signs + and — , it is called a simple expression or monomial.
If an expression contain parts connected by the signs +
and — , it is called a compound expression ; and the parts
connected by the signs + and — are called terms of the ex-
pression. Thus ax, 4: be, and 6 a"" c"^ are simple expressions;
a^-^-F—c^ a^-{-?ta^b'\-c^, are compound expressions, of which
a^, F, and c\ and ct\ 3 «^ b, and c^ are the tenns respectively.
19. Let the student distinguish carefully between tenns
Index or Exponent, a^ . ^3 • ^.,1 Difference between Coefficient and Exponent.
Monomial Terms. Terms and Factors.
14 ELEMEKTARY ALGEBRA.
and factoids, recollecting that factors are those parts of an
expression which are connected by Multiplication, and terms
are those parts of a compound expression which are connected
by Addition and Subtraction.
Thus 5, a^y d, and c are factors of the expression 6a'^bc;
while 6cr, h, and c are terms of the expression ba^ -^-1)— c»
20. When an expression consists of tivo terms it is called
a Uiiomial expression, or briefly a binomial ; when it consists
of three terms it is called a triyiomial ; any expression con-
sisting of several terms is called Si multinomial or jwli/nomial
Thus 2«^ -\-dabc is a binomial expression ; a — 2b -h 6c
is a trinomial expression ; and a"^— ¥+ c^— 4tab — e is a
polynomial.
21. Each of the letters or literal factors of a term is called
a dimension of the term, and the number of these literal fac-
tors is called the degree of the term. Thus 2a^b^c on: 2 a abb be
has six dimensions and is said to be of the sixth degree.
The numerical coefficient is not counted; thus 9a^^* and
a^ V are of the same dimensions, namely, seven dimensions.
The degree of a term, that is, the number of its dimensions,
is evidently the sum of the exponents of its literal factors,
provided we remember that when no exponent is expressed
the exponent 1 must be understood.
22. An expression is said to be Jiomogeneous when all its
terms are of the same degree.
Thus '7 a^ -hSa^ -i- 4:abci8 homogeneous, for each term is
of three dimensions, that is, of the third degree.
We shall now give some more examples of finding the nu-
merical values of algebraic expressions.
Suppose a = 1, b = 2, c = d, d = 4:, e = 5, /= 0. Then
b' =4, b' = 8, h* = 16, b' = 32.
Binomial. Trinomial, Polynomial. Dimension and Degree of Term. When is
an expression homogeneous?
VN{Y£RSrTY I
15.
a'^^^z^lxS^S, 3^V=3X4X9 = 108.
^ + c'-7aZ»-f-/'=: 64 + 9-14 + = 59.
3c^-4^-10
27-12-10 5 .
— = 0.
c'-2c' + 5c-23 27-18 + 15-23 1
6^ + rr g^-a^ _ 125+ 64 27-l _189 26_ _
e+^ c-a~ 5 + 4 3-1 ~ 9 2~ ■^^~^'
Examples — 2.
If a = 1, Z> = 2, c = 3, rf = 4, e = 5, / = 0, find the numer-
ical A'alues of the following expressions.
1. «'+Z/' + c'+^+e'+/'.
5. a'-\-?>a^l^Zah^^l\
^V ^_32
9 ^' + ^' ^' + ^' e'-cl"
6.
8.
2. e'-J^ + c'-JHa'.
4. c^-"2c^+4c-13.
e*-4e'^+6e*J'-4eJH^*.
2g + 2 3g-9 e--l
e-d
e—2 e+3
8a^+3^^ 4:c''+6b\ c' + ^
A v. o , 7 9 "1
11.
a' + b'
28
1 +
12
+ -
a^+^^ + 6''^rf-^^-^*^^V + 6^-c'-6?'*
12.
a'-V^fn-V^crb^-V^ab^-\-b''
a^-^?>d'b-\-Zab^V '
13.
d'
14.
^ + Z>''
16 ELEMENTARY ALGEBRA.
III. Kemaining Signs — Brackets.
23. Tlie sign > stands for is greater tlian, and the sign <
denotes is less than, Tims a> h denotes that the quantity a
is greater than the quantity h, and h <a denotes that the
quantity h is less than the quantity a. In this sign the
opening is always turned toward the greater quantity.
24. The sign .*. denotes then or therefore; the sign •.* de-
notes since or because,
25. The square root of a quantity is that quantity whose
square or second potver is equal to the giyen quantity.
' The cube, fourth, &c. root of a given quantity is that
quantity whose cube, fourth, &c. power is equal to the given
quantity. Thus since 49 == T, the square root of 49 is 7;
also the square root of a^ is a. In like manner, since
125 =: 5^ the cube root of 125 is 5 ; and so if a = c^y the
cube root of a is c,
26. The symbol used to denote a root is \/ (a corruption
of r, the first letter of the word radix), which with the proper
number as index on the left side of it, a little above, is set
before the quantity whose root is expressed.
Thus V«' ^ a, V64 = 4, V3125 = 5, VI = 1, VI = 1, &c.
The index, however, is generally omitted in denoting the
square root ; thus v^a is written instead of V«.
Examples — 3.
1. v/4"+2v/25 + 3v/49-v/ 64 = 25.
2. 3n/i6-4v/36 + 2v/9'-\/8T=-15.
3. V8"+2Vi25-4Vr4-V64 = 12.
4. vr+3Vi6-2V32+3vr=6.
If a=2D, b=9, c=4:, d=l, then
5. y/a-{-2^/l; + 3^/'^ + 4:^/d=2l.
Explain the signs > < .*. •.• Square root of a quantity ? Cube root? Th«
Bymbol used to denote a root.
KEilAlXIXG SIGXS. — BRACKETS. 17
7. 3Va+2VAb-4:Vrc+Vm=7.
8. V5^+2V3b-V2c-h4:Vd=13.
10. Vbc+dVacd — 4:Wd+V^^ — 4:,
27. |/^ means that the square root of the fraction % is to
be taken ; but — — means that the square root of a is to be
divided by t.
Examples.
1. What is the difference between 2\/'^ and 2+ \/^^
when X is 100 ? . Ans. 8.
2. What is the difference between 3 \/"^, and V^, when x
is 64 ? Ans. 20.
3. What is the difference between \/ a-^h and v/"^ + Z>,
when a stands for 1, and 1) for 8 ? Ans. 6.
4. What is the difference between \/%- and -— ^, when
a stands for 16 and Z> for 4? Ans. 1.
5. What is the difference between v/ « + \/y and >/^+^^
when a = 16 and Z> = 9? Ans. 2.
28. Brackets, (),{},[], are employed to show that all the
quantities within them are to be treated as though forming
but one quantity. It is of great importance to notice care-
fully the effect of using them.
Thus a—{b—c) is not the same as a—b—c; for, in this
last, both b and c are subtracted, whereas in the former it is
only the quantity b — c which is subtracted.
The use of Brackets.
18 ^ ELEMEKTAEY ALGEBRA.
Hence, if a = 4:, b = 3, c = 1, we liaye
a — d — = 4.-3 — 1 = 0, a—{b — c)=4 — 2 = 2;
2a-'3h + 2c=S-9+2 = l,2a-{3b + 2c) = S-ll = -3,
2a-^b-c=S+3-l = 10,2{a+b)-c=U-l = 13,
2{a + I)-c)=12.
If we wisli to denote that the sum of a and d is to be mul-
tiplied by Cy we write {a+b)Xc or {a + I)}Xc, or simply
{a-}-I))c or {a + d}c; here we mean that the whole of a-\-b is
to be multiplied by c. Now if the brackets were omitted we
would have a + be, which denotes that b only is to be multi-
plied by c and the result added to a. Similarly {a + b—c)d
denotes that the result expressed by a + Z* — c is to be multi-
plied by d.
So also {a — b + c)x{d + e) denotes that the result ex-
pressed by a — J + c is to be multiplied by the result ex-
pressed by d + e. This may also be denoted briefly thus,
(a — b + c) {d-\- e); just as a X ^ is shortened into a b.
So also V{a + b-\-c) stands for the square root of the result
expressed hj a + b-\-c.
So also \/{a+b-\-c) denotes that we are to obtain the re-
sult expressed hj a+b+c, and then take the square root of
this result.
So also {aby denotes abX ab; and (a by denotes
abXabXab.
So also {a+b—c)-^{d-\-e) denotes that the result expressed
by a -\- b — c is to be divided by the result expressed by
d-\-e.
29. Sometimes instead of using brackets a line is drawn
oyer the numbers which are to be treated as forming one
number. Thus a—b-\'CXd+e is used with the same mean-
Vinculum.
kemai:n'ikg sigxs — brackets. 19
ing as {a—bi-c)x{d+e), A line used for this purpose is
called a vincuhun. So also {a+b—c)-^{d-{-e) may be de-
noted thus, — 9— — ; and here the line between a -\- b — c
and d-{- eh really a vinculum used in a particular sense.
Thus, too, a vinculum from the top of a radical sign is fre-
quently used, and \/ a + h + c has the same meaning as
V{a-\-b-\'c).
30. We have now explained most of the signs used in
Algebra. It is well to observe that the word sign is applied
specially to the two signs + and — . Thus the expressions
"changing the signs^^ "like signs," and "unlike signs" refer
exclusively to + and — .
31. We shall now give some more examples of finding the
numerical values of expressions.
Suppose ^ = 1, 5 = 2, c = 3, ^ = 5, e = 8. Then
V(4c-2^>)=V(12-4)=V(8):ir:2.
eN/(25+46')--(26Z-^)V(4c-2^)=:::8x4~8X 2=32-16 = 16.
v/{(6-^)(26-5Z^)}-v/{(8-2)(16-10)} = v/(6x6)3=6.
{(e-^)(^4-c)-(c?-c)(c+a)}(fl^ + ^)^{3x5-2x4}6
= 7X6=42.
V(c'' + 3c'^5 + 3c5'^ + &')-^v/(^^+Z>^-2a^)
= V(27 + 54 + 36 + 8)--v/(l + 4-4)=V(125)-^l=5.
Examples — 4.
If a = 1, 5 = 2, c = 3, J = 5, e = 8, find the numerical
values of the following expressions.
The word Sign^ how applied?
20 ELEMEXTARY ALGEBRA.
1. a{h + c). 2. b{c + d): 3. c{e~d).
4. b^a'+e'-c'). 5. c'ie'-F-c'), 6.
cr+b'
7. ^"^^-. 8. n/(3&c^). 9. V{2b+U+6e).
10. (r^+25 + 3c+56— 4<^)(6e-5tZ-4c~36 + 2a).
11. {a'' + b' + c''){e'-d'-c'). 12. (3^-7c^)^
13. ex/(rr-36) + ^v/(^+3e).
14. e-{V{e+l)+2}+{e-Ve)V{e-'4:).
If a = 5, 5 = 3, c = 1, show that the numerical values are
equal.
15. Of a'-b' and (a-b) {a'+ab + b').
16. Of b'-c' and (b + c) {b-c) {b' + c').
17. Of a' + a'b'+b' and (a^+a5 + ^'^) (a'-^&4-^>').
18. Of b'+4.c' and {Z>^+2(^ + ^)4{^'-2(5-c)4.
lY. Change of the Order of Terms — Like Terms.
32. The terms in an expression which are preceded by the
sign + are called positive terms, and the terms which are
preceded by the sign — are called 7iegative terms.
33. We must now extend the meaning and use of the
sign — beyond the strict application of ordinary arith-
metical notions. If in the expression a — b -\-c, « =4, 5=7,
and c = 8, then by our first definition of the sign — we
should have to subtract 7 from 4, which is impossible. In
this case we subtract the 4 from the 7 and write the remain-
der with the sign -. Thus -7 + 4= -(7 -4) = -3. Then
Positive and Negative Terms. Algebraic meaning and use of the Sign — .
CHANGE OF THE ORDER OF TERMS — LIKE TERMS. 21
we consider —3 + 8 to be the same as 8 — 3 = 5, and 5 is the
numerical value of the expression a—b-{-c=7—4z-\-8,
34. It is then indifferent in what order the terms of an
Algebraic expression be written. This is clear from the com-
mon notions of Arithmetic, and from the convention that
— 5 + ^ is the same as a — b. Hence, if a term is preceded
' hy no sign, the sign + is to be under stood, and such a term is
counted loith the positive terms, ^
Thus, 7+8-2-3 = 8 + 7-2-3=:~3+8-2+7, &c.
a-]rb—c=b-\-a—c=b—c-\-a=—c-\-b-\'a, &c.
35. Terms are said to be like when they do not differ at all
or differ only in their numerical coefficients ; otherwise they
are said to be unlike. Thus a, 4a, and 7«^ are like terms;
a^b c, 5 a^b c, and 7 a^b c are like terms ; ba^, 6a b, and 6b^
are unlike terms ; 4a* and b c are unlike terms.
36. An expression which contains like terms may be sim-
plified. For example, consider the expression
6a—a+3b + 6c—b + Sc—2a.
This expression, by Art. 34, is equivalent to
6a—a—2a+db—b + 6c^dc.
JSTow 6a— a— 2a— da. For a from 6a leaves 5a; and taking
2a from 6a we have da left. Similarly db — b = 2b ; and
6c-\- dc = Sc. Thus the expression may be put in the form
3a + 25 + 8c,
Again, consider the expression a— 3^— 4^. This is equal
to a — 7^. For if we have first to subtract 35 from any
number a and then to subtract Ab from the remainder, we
shall obtain the required result in one operation by subtract-
riie Terms, in what order to be written. Like and unlike terms. Simplification
of expressions containing like terms.
^2 ELEMENTARY ALGEBRA.
ing ^ib from a; this follows from the common notions of
Arithmetic. Thus
« - 35 - 4^ = a - 75.
37. The statement — 35 — 45 = — 75 is explained thns :
if in the course of an Algebraic operation we have to sub-
tract 35 from a number and then to subtract 45 from the
remainder, we may subtract 75 at once instead. It will be
seen that by an easy extension of this, the expression — 75
has a meaning when standing by itself.
38. It may be noticed (as we have proved in Arithmetic)
that it is immaterial in what order the factors of a quantity
are arranged. Thus 7X5 is the same as 5x7; 2x6x9 the
same as 6X2X9 or 9x2x6, &c.; abc the same as cab or
cba, &c. ; 6 x^a and a x^ are like terms. It is usual, however,
to arrange literal factors and terms as much as possible in
the order of the letters of the alphabet.
39. The simplifying of expressions by collecting like terms
is the essential part of the processes of Addition and Sub-
traction in Algebra.
Ex. 1. Group together like quantities, with their proper
signs, from 5a — 35, 4a + 75, and — 8a — 55.
Ans. + 5a
+ 4a
~8a
—35 Here the quantities in each column
+ 75 are lilce, but the two columns are
—55 unlike.
Ex. 2. Group together like quantities, with their proper
signs, from
a'+ 3a'5 + 3a5^+ 2a'+ 25^4- 5a5^- Sac''- a^b - b\
-Sac'
Ans. + a' I -f-3a'5
+ 2a^ I - a^b
+ 5a5'
+25'
— b^
Ex. 3. Group together lilce quantities, with their proper
signs, from 2a - 35 + 75c + 5'c - 5a5c + 2.Ty -3:c' + 55" +
Order of arraK<rement cf Factors.
CIIAKGE OF THE ORDER OF ' TERMS. — LIKE TERMS. 23
Wc - 9a - W + 6^ 4- 10a - hx'^ - xy ^ x^ -\- ahc - 2bc + c'
-b- dc\
Ans.
+ 2a
-3b -{•7bc\+ Fc
—oabc\-\-2xy
-^x"
■VW
— 9a
-4-6^
-2bc -\'Wc
+ abc
— ^y
-bx'
-2b'
+ 10a
- b
+ x"
40. We shall close this chapter with some more examples
of the conversion of ordinary language into Algebraic lan-
guage.
Ex. 1. A person makes a mixture of three sorts of wine.
The second costs a dollars more per gallon than the first ;
and the third b dollars more per gallon than the second.
There are m gallons of the first sort, n of the second, and
p of the third. What is the price of the mixture, x being
the price of the first sort ?
Ans. mx ■{- n (x -\- a) ■\- ;p {x ■\- a ■\- b),
Ex. 2. Express algebraically a number of five figures a, b,
c, d, e, taken in their order from left to right in the decimal
system.
Ans. lO'Xa + 10'X^ + lO'Xc + lO^+e.
Ex. 3. Three fountains run successively into a reservoir,
the first during a hours, the second during b hours, the
third during c hours ; the second fountain supplies m gallons
per hour more than the first, the third n gallons more than
the second : how many gallons of water did the three foun-
tains yield, x being the number of gallons per hour which
the first yields ?
Ans. ax + b (x + m) -\- c{x -\- m + n).
Ex. 4. A merchant sells a certain number of yards of cloth
for a dollars per yard. A second merchant sells b more
yards of the same cloth at c dollars more per yard. If a; is
the number of yards sold by the first, express the difference
in the amounts received by the two.
Ans. {x-\-b){a-\-c) — ax.
24 ELEMENTARY ALGEBRA.
Ex. 5. A mixture is made of 4 substances, A, B, C, D. It
is composed of a gallons of A, costing m dollars per gallon,
of h gallons of B at n dollars a gallon, of c gallons of C at
2) dollars a gallon, and of d gallons of D at g dollars a gal
Ion. Express tlie price of a gallon of the mixture.
, am + bn -\- cp + da
Ans. • ;-, — 7-—.
a + -\- c + a
Ex. 6. Three laborers paid at the same rate worked, the
first, m days, the second n days, and the third q days. They
received altogether a dollars. Express the daily wages of
each.
Ans. — ■ -— .
m+n+q
Ex. 7. A sum a produced in b years c dollars at simple in-
terest. Express the rate per cent.
100c
Ans. -7 — .
ba
Ex. 8. A sum a placed at simple interest amounts in n
years to b dollars. What is the rate per cent. ?
Ans. m^-^.
an
V. Addition.
41. It is conyenient to make three cases in Addition,
namely : I. When the terms are all like terms and have the
same sign. II. When the terms are all like terms, but have
not all the same sign. III. When the quantities to be added
consist of both like and unlike terms.
■ 42. I. To add like terms which have the same sign.
Add the numerical coefficients, prefix tJie common sign to
the sum, and annex the commo7i literal factors.
For example :
6a + 3a + 7a-{-6a = 21a.
- 3b'c - bb'c -10b' c = - ISb'c.
Three cases in Addition, Rule for Case I.
ADDITION". 25
43. II. To add like terms which haye not all the same
sign. Add separately the positive numei^ical coefficients, and
the negative numericcd coefficients ; take the difference of these
tivo sums, prefix the sign of the greater to this difference, and
annex the common liter cd factors.
For example :
7rr - 3^f + 11«' +a' - oa' - 2a' = 19a' - 10a' =9a'.
%l)0 _ ^ic - Zlc + ^hc + hhc - Q>hc = llhc—l^hc = - 6bc.
44. III. To add expressions which consist of both like
and nnlike terms. Add together the like terms by the rule in
Case II, Affix to the sums thus obtained the unlike terms^
each preceded by its proper sign,
For example : add together
4a-f 5^-7c + 3f/, 3a-& + 2c+5^, ^a-^b-c-d,
and —a + 'db-\-4:C—M^-e,
It is conyenient to arrange the terms in columns^ so that
like terms shall stand in the same column; thus we
haye
4a+5^-7c+3f?
3a- b-\-2c-VM
9a— 2b— c— d
-a-^Zb^4.c-M-\-e
lDa-\-bb—2c^^d-\-e.
Here the terms 4a, 3a, 9a, and —a are all like terms; the
sum of the positiye coefficients is 16 ; there is one term with.
a negative coefficient, namely —a, of which i\\Q coefficient
is 1. The difference of 16 and 1 is 15 ; so that we obtain
-f- 15a from these like terms : the sign + may, however, be
omitted. Similarly we have bb — b — 2b -\- ^b = 6b. And
so on.
Rule for Case II ; for Case III.
2
26
ELEMENTARY ALGEBRA.
45. In the following examples the terms are arranged suit-
ably in columns.
4:x' + '7x''+ x-9
— 2x'+ x'- dx + S
-3x'- a;' + 10.T-l
dx'- x-1
a'-h ab+ b'-c
a^-'Zab-W
In the first example, we have in the first column
x^ + 4zX^ —2x^ —Zx^ , that is, bx^—bx^, that is, nothing; this is
usually expressed by saying the terms ivhicii involve x^ cancel
each other.
Similarly, in the second example, the terms which involve
ab cancel each other; and so also do the terms which in-
volve z>^
Ex. Add together a + ^b — c, a—6e-h2c, and x + y + 3e.
Here a and a are UJce,
— 6e and +3e
— c and +2c
the rest are imlike.
a + 2b— c
a—5e-\-2c
3e + x + y
Sum =: 2a + 2b + c—2e + x-\-i/,
Ex. Add together Za^ — bc, 2b^—ac, 4cG^ — ab, and a^ + b'^
^c\
Here 3a^ and a^ are lihe,
2^^ and ^-b^
4c^ and — c'
the rest are unlike.
3a^-bc
2b' -ac
^&-ab
a'+ y- c'
Sum = 4ca^ + 3b' -{-3c'-ab-ac - be.
ADDITIOX. 27
Ex. Add together xy—1, vc'-f 2, and ^'' + 3.
Here the terms are all x^ + 2
icnlilce, except —1, +2, and
+ 3.
^^ + 3
Sum = x^ + xi/-^y^ + 4:.
46. The Kules aboye given for the Addition of like and
imliJce algebraical quantities are in no wise different from
those employed in Arithmetic. For, suppose we have to add
together 3 hundreds and 4 hundreds, we combine these lihe
quantities by taking the sum of the coefficients 3 and 4, so as
to make 7 hundreds. But if we have to add together 3
hundreds, 5 tens, and 6 units, these, being unlike quantities^
cannot be added in the same sense, but are merely collected
together in one line, 3 hundreds +5 tens +6 units, which,
for convenience, is written shortly 356. It will be observed,
however, that algebraical Addition involves the processes
both of arithmetical Addition and Subtraction.
Examples — 5.
Add together
1. a + 5 and a^-h, ^, a^h and a—h,
3. G^— 6 and a— ^. 4. « — S + c and a4-^— f-.
5. a—h-\-c^VL^a-\-h-^c, 6. 1 — 2m + 37^ and3m— 2;i + l.
7. 5m + 3 and %m —4. 8. Zxy—%x and xy^Qx,
9. 4;9-25' + l and 7-3^?-!-^.
10. 5aZ> — 2^c and ah -{-be,
11. 3a~2^, 4a-55, 7a-115, a+95.
12. 4a;' -3?/', 2x^-by\ ~x^-Vy\ -22:'-^4y^
28 ELEMEXTARY ALGEBRA.
13. 5a-i-3I) + Cy da + 3b + 3c, a + db + oc.
14. 3X + 22J-Z, 2x-2ij + 2z, -x-\-2y-^3z.
15. "/a-U + c, 6a + 3b-6c, -12a + 4.c.
16. x—4:a + b, 3x + 2I), a—x—hl,
17. a-\-l—c, h + c—ay c^-a—l, a^l—c.
18. a4-2Z> + 3c, 2a— I— 2c, l—a—c, c—a—K
19. a— 2J + 3c— M ?>l}—4.c-\-Dd—2a, 5c-6^+3a— 45,
20. x'-^x''-\-bx-Z,2x'-W-li:X-\-^, -x'-^-^x^'-Vx-^S.
21. a;'-2:^' + 3z',ix;' + a;'^ + :?;, 4:^;' + 5:c^ 2a;'^ + 3:?;-4,
-32:^ -2.-^-5.
22. a'-'^cn-{-'daV-d\ 2a' + 6a'I)-6ad'-7b\
a'-ab' + 2I?\
23. x'--2ax' + a''x + a% x' + dax\ 2a^-ax^-2x\
24. 2a'b—Zax^ ^2(j^x, \2db-\-V)adi^ — ^(^Xy
— Sal? + ax^—5a^x,
25. 2;^ + y* + ;^^ -4.x'-6z% Sx'-lif + lOz', 6t/-6z\
26. 32;' - 4:?;?/ 4- ^' + 2:^; + 3?/ -7, 22;'-4?/'-h32;-5y + 8,
lOa;?^' 4- 8^' + 9y, 5a;' - 62;?/ + 3/ + 72^-7^ + 11.
VI. Subtraction.
47. Suppose we have to take 5 + 3 from 14. The result is
the same as if we first take 5 from 14, and 3 from the re-
mainder. This result is denoted by 14 — 5 — 3.
SUBTRACTION^. 29
That is, 14 -(5 + 3) = 14 — 5 —3. The brackets mean-
in 2: that the whole of the 5 + 3 is to be taken from 14.
In like manner, suppose we have to take 'b-{-c-{-d from a.
The result is the same as if we first take h from a^ and then
take c from the remainder, and then d from that remainder;
that is, the result is denoted by
a—h—c—d.
Thus
a~i^^c^-d)=a—'b—c—d.
We see in these cases the positive terms of the expression
to be subtracted have all been changed to negative terms in
the result.
48. l^ext suppose we have to take 5 — 3 from 14. If we
take 5 from 14 we get 14—5; but we have taken too much
from 14, for we had to take, not 5, but 5 diminished by 3.
Hence we must increase the result by 3. The result is then
denoted by 14—5 + 3.
Thus
14-(5-3) = 14-5 + 3.
In like manner, suppose we have to take h—c from a. If
we take l from a, we obtain a—h'^ but we have thus taken
too much from a, for we had to take, not Z>, but Z> diminished
by c. Hence we must increase the result by c. Thus we
obtain
a— 1)^-0,
That is
a—{h—c)—a-''b^-c.
By the same reasoning
a—{h—c—d)=a—l)-\-c-\-d.
Here the positive term of the expression to be subtracted
is negative in the result, and the negative terms are posi-
tive.
80 ELEMEKTAKY ALGEBRA.
49. Hence, we have tlie following rule for Subtraction :
Change the signs of all the terms in the ex^jression to be
stihtracted, and then proceed as in Addition,
For example: from 4:X—?>y -\-2z subtract 2x—y-\-z,
Change the signs of all the terms to be subtracted ; thus
we obtain —^x + y—z'y then collect the terms, and simplify
as in Addition. Thus
4:X—'dy + 2z—'dx + y—z=x—'^y-\-z.
From ^x' + bx'-Qx^-'lx + b, take 2.x*-2aj' + 5cc'-6a;-7.
Change the signs of all the terms to be subtracted, and
proceed as in Addition. We thus have
3aj' + 5cc'-6x'-7cc + 5
-2a3^ + 2aj'-5x' + 6aj-f7
«j' + 7i«'~llaj'-cc + 12.
The beginner will find it best at first to go through the
process fully as above ; but he will soon learn to put down
the result without actually changing all the signs, but
merely doing it mentally.
50. We often have a single negative term to be subtracted
from another term or expression. Thus, from a subtract
— c. Here we can reason thus: Since a=a + 'b—'b, if we sub-
tract — h from a, the result is a + h, the same as if we add + h
to it. Or we can apply our rule, at once considering the
result to have a meaning in connection with some other
parts of an algebraical operation.
Examples.
1. From ^a 2. From la 3. From a
take a take 6a take a
Ans. 2a a
Rule for Subtraction.
SUBTRACTION.
31
4.
From
3a
5.
From 7a
6.
From a
take
—a
take —6a
take —a
Ans.
4a
13a
2a
7.
From
~3a
8.
From —7a
9.
From —a
take
Ans.
a
~4.a
•
take 6a
take a
-13a
-2a
10.
From
—3a
11.
From —7a
12.
From —a
take
Ans.
— a
take —6a
— a
take —a
-2a
13.
From
a-hb
14.
From a—b
15
. From y + ax
take
Ans.
a-b
2b
take a + b
-2b
take y—ax
2ax
16.
From 3a-U + 6c
17.
From
7a-
-2b-\-4:C-2
take
Ans.
a— 2Z> + 9c
take
6a'
-6b-\-4.c-l
2a-2b-
-3g
a + ^b-1
18. From 2a— 6a^— ac + 5 19. From 3cc?/— cc'— y^' + a
take 5a~8aJ— 2ac— 1 take 2xy + x^-\-2y'^ — b
An s. — 3 a + 2a^ + ac + 6
xy—2x'' — 3y'' + a + b
20. From a^ + 2aJ-36''
take 2a'^ — Dab—7c^
Ans. -a' + 7a2>+4c'
21. From bx"^- xy+ y^
take — a3^ + 4i^^ + 3^^
6x'-6xy-2y^
32 ELEMENTARY ALUt:i3EA.
22. 'FiomSa'+x'-bb'-Dc'' 23. From ^'- 3a;' + 6:^;- 10
take x'-{-2F-6c'' tske x'-4.x'-^8x- 9
Ans. 8a' -n' x^-%x-\
24. From a-\-\h-\-\ 25. From \x^—\xij^\if
take 2^ + ^ + 2 ' ^^^^ '~3^'~i^^~2^'
111
Ans. -ir^-17^4-^ x^—xy-\rm^
z z z
Examples — 6.
1. From 7a + 14^ subtract 4a + 10Z>.
2. From 6a--2J— c subtract 2a— 2^— 3(?.
3. From 3a— 2Z> + 3c subtract ^a—lh—c—d.
4. From "Ix^—^x—l subtract bx'—Qxi-3.
5. From 4.x' -2>x' -2x^-^1 x-^^
subtract x' — 2x^ — 2x^ + 7x—9,
6. From 2x^ — 2ax+3a^ subtract x^—ax + a^.
7. From x'—3xy—y''-]-yz—2z'
subtract x^ + 2xy + bxz — 3 ?/' — 2^;'.
8. From hx'' + Qxy-12xz-4.y''-^2jz-bz^
subtract 2a;' - 7a;?/ + 4.T;a;- 3?/' + ^yz— ^z\
9. Froma'-3a'Z^ + 3aZ>'-Z^''
subtract -a' ^-^cn-^alf + 1)\
10. From 7a;' - 2a;' + 2a; +2 subtract 4a;' -2a;' -2a' -14,
and from the remainder subtract 2a;' — 8a;'-f-4:c+16.
VII. Brackets.
51. On account of the extensive use of brackets in the
algebraic language, it is necessary that the student should
observe very carefully the rules respecting them.
Since the sign + or — preceding a bracket means that
the whole included quantity is to be added or subtracted, if
we wish to remove the bracket, we must actually perform
the operation indicated by means of it; i. e., we must add or
subtract the quantity in question. ISTow when a quantity is
added, the signs of its terms are not altered ; but when it is
subtracted, the signs of its terms are changed.
Hence,
When an expression is ivitliin a pair of brackets preceded
ly the sign -\-, the hrachets may he removed, the signs of the
included terms 'being unchanged.
When an expression is ivithin a pair of hracJcets preceded
ly the sign — , the hraclcets may le removed if the sign of
every term luithin the hraclcets he changed.
Thus, for example :
a—'b-\-{c'-d-\-e)=^a—'b-\-c—d-\-e
a—I? — {c—d-\-e) = a—h—c-\-d—e,
Kemember, that if the first term within the brackets has
no sign, the + sign is understood before it.
^2, In particular, the student must notice such statements
as i\\Q following:
These are immediate consequences of what we have said
of the addition and subtraction of single terms.
Eules for removing Brackets.
2^
34 ELEMENTARY ALGEBRA.
53. Expressions may occur with more than one pair of
brackets ; these may be remoyed in succession by the pre-
ceding rules, beginning with the inside pair.
Thus, for example :
a+{l)-{-{c—d)}=a+{b + c—d]=a + 'b-\-c—d,
a-v{b—{c—d)]=a^-{'b—c-^d]—a^-'b—c-\-d,
a— ' {!)-{- {c—d)}=a—{'b^- c—d]-=ia—'b — c-{- d,
a— {l)—{c—d)} ^a— {l)—c-[- d) =a—d -{- c—d.
Similarly,
a—[b—{c—(d—e)}'\=a—[h—{c—d+e]'\
=a—[b—c-\-d—'e]=a—I) + c—d + e,
It will be seen in these examples that, to prevent confu-
sion between various pairs of brackets, we use brackets of
different shapes ; we might distinguish by using brackets of
the same shape but of different sizes.
A vinculum is equivalent to a bracket ; see Art. 30. Thus,
for example :
a-\h-{c-{d-^)]l^=a-\l-{c-{d-e-^f)]\
=a-\h—{G-d-\-e-f]^=a-\l-c + d-e+fl^
^a—l + c—d-^-e—f.
54. The beginner is recommended to remove brackets in
the order shown in the preceding article, {i. e.) by removing
first the innermost pair, next the innermost pair of those
which remain, and so on. We may, however, vary the
order, by removing first the outermost pair, next the outer-
most pair of those which remain, and so on.
Thus, for example :
a-\'{h-\-{c-d)}=a-hh-\-{c-d)=a + b + c—dy
BEACKETS. 35
a+ [b— {c— d)} = a + b— {c— d) -—a + I)—c + d,
a—{b + {c — d)}=a—I) — {c—d)=a — I) — c + d,
a—{b — {c—d)}=a—b-{-{c—d) = a--b-\- c—d.
Also,
a—[b-{c-{d—e)}]^a-b+{c—{d-e)}
=a—b + c—{d—e)~a—b-{-c—d+e,
55. It is often convenient to take up in brackets any given
terms of an expression. Tlie rules for thus introducing
brackets follow immediately from those of removing
brackets.
Any oiiwiber of terms 'i?i an expression may be put loitJmi
a pair of braclcets, and the sign + pUtced before the brachet,
the signs uf the terms being unchanged.
Any number of terms in an expression may be put icithm
a pair of braclcets, and the sign — placed before the brachet,
'provided the sign of every term put loithin the brackets be
changed.
In applying this rule, we shall for convenience take the
sign of whatever term we choose to set as first term within
the brackets, as the sign to be placed before the bracket.
Thus -\-a—b — c, collected in a bracket with -\-a SiS first
term, will be +{a—b—c); but, with —b as first term, it
will be —{b — a + c), and with —c as first term, it will be
— {c'-a + b); and now, if we resolve again these last two
brackets, the sign ( — ), preceding each of them, will correct
the changes we have made, and the quantities will be repro-
duced, as at first, —b + a—c, —c-i-a—b.
So also we might use an inner bracket, and write the
quantity -\r {{a~b)—c}, or + {a—{b + c)}, or —{{b — a) + c},
or —{b—{a—c)}, &c.
Rules for intv^d'^clng Bracketg.
'56 elementary algebra.
Examples — 7.
Eeduce to their simplest forms :
1. {a-x)-{2x-a)~{2-2a) + {3~2x)-{l-x).
2. {a' - 2a' c + 3ac') - {a'o - 2a' + 2ac') + {a' - ac' - a'c).
3. {2x'~2jf-z')-{3if + 2x'-z')-{3z'-2tf-x').
4. {x' + ax' + a'x)-{i/-I?y'-hh'i/) + {z' + cz' + c'z)
- i^'-y" +-^") + {cix' + hf + cz') - {a'x-Fij + c'z).
5. a"- {W-c') - {b'- {c'-a')] + [c'- ifi'-a')}.
6. {2a'-{dad-F)}-{a'-{4.ad + W)]-\-{2h'-{a'-al))].
7. {x'^f-{dx\j-^3xif)}-{{x'-3xhj) - Cdxif-if)}.
8. {'^x-{3y-z)}-{y-\-{2x-z)}-\-{3z-{x-2y)]
^{2x-{y-z)}.
9. l_{l-(l-4r^)} + {2a;^(3-5:r)}-{2~(-44-5a;)}.
10. {2a-{3I)-\-c-2cl)]-{{2a-3h)-{-{c-2d)]
> '\-{2a-{3h-\-c)-2d)}-{{2a-3'b-\-c)-2d}.
Express by brackets, taking the terms (i) tivo together, (ii)
three together : —
11. a-2l-{-3c—d-\-2e—f. 12. —2'b-\-3c—d-^2e—f-\-a.
13. 3c-d^2e-f-\-a-21). 14. -d-^2e-f-\-a-2'b-Y3c,
15. 2e-f-]-a-21)-\-3c-d. 16. ^f^a-21)^-3c-d-\-2e
56. In Addition and Subtraction we have spoken hith-
erto only of numerical coefficients; but as any one of the
factors of which a term is composed may be considered a
coefficient, we often have to apply the rules to these literal
Rules for Brackets, as applied to literal coefficients.
BRACKETS. 37
coefficients. Thus, when any terms of a quantity contain
some common factor, brackets are often employed to collect
tlie other factors considered as its literal coefficients into one
expression, which is set before or after the common factor.
Thus, just as dx-{-2x—x=4:X, that is, ={3-\-2—l)x,
so, likewise, ax-]-dx—x={a+I)—l)Xf
2a-4.ax + 6ay=2a{l-2x-\-3i/),
{a+2h)x'-{2d-c)x'={{a+2I))-{2d^c)}x'={a+c)x\
Ex. 1. Add {a-2jo)x''+{2c-3r)x
{22J-{-a)x^ —x
'—{p—a)x^ —{c^l)x
^x^ —{c—2r)x
Ans. {^a—p—Vjx^ —rx
Ex. 2. From ax' -M +x
take —px' ^qx^ -\-rx
Ans. (a-\-p))x^ — (b—q)x^ + {l—7')x
Examples— 8.
1. Collect coefficients in ax^ — lx'^ — cX'—lx^-\-cx^~-clx-\-cx^
—dx^—ex,
2. Add together ax—ly, ^-\-y, and {a'~l)x—{l)-\-l)y.
Jl. Add together {a-\-c)x^—3{a—'b)xy-\-{ib—c)y'', and
{l)-c)x'-\-2{a^l)xy + (a-'b)y\
4. Add together {a + h) x + {I) + c)y and {a—'b)x—{h~c)y,
and subtract the latter from the former.
38 ELEMEKTAEY ALGEBRA.
5. Add together (i) the first two, (ii) the last two, and
(iii) all four together, of '^{a+'b)x-^^{l) + c)y,
-'3{a-I))x + 2{a-c)t/, —(2b-^c)x+{a-2J))y, and
{a-2l)x-{h + %c)ij.
6. In (5) (i) subtract the second quantity from the first,
and (ii) the fourth from the third, and (iii) add the
two results together.
YIII. Multiplication.
57. It is convenient to make four cases in Multiplication.
I. To multiply one positive single term by another ; II. To
multiply a quantity consisting of two or more terms by a
positive single term ; III. To multiply one quantity by an-
other when both consist of two or more terms ; IV. To mul-
tiply one negative single term by another, or by a positive
single term.
58. I. Suppose we have to multiply da by 4Z>. The prod-
uct may be written thus, ZaX^l)] or, since the product of
any number of factors is the same in whatever order the fac-
tors may be taken, we may write it dX^XaXh', and it is
therefore equal to V^ah, Here we multi2:)ly together the nu-
merical coefficients and put the literal factors after this
product.
Thus, for example :
laXdhc^^lalc.
Similarly,
4.aXhl)X'dc=ma'bc,
59. Poivers of the saine numler are multiplied together hy
adding the exponents.
Four Cases in Multiplicatiou. Rule for Case I. Powers of the same luim
ber, how multiplied?
MULTIPLICATION. 39
Thus, a'^Xa^—a''] for a'—aa, and a^~aaa, ^'.a'Xa^
=aaX ciaa = aaaaa, or a^
In the same manner it maybe shown that a^Xa^"=a'";
and so on for other powers, always taking the sum of the
exponents. To prove this generally, viz., that
oJ^XdP'^dr-^'^, whatever positive whole-numbers m and n
may stand for, we have, by definition,
oJ^=za.aM, &c. to m factors,
and a^'^aM.a. &c. to n factors,
:, a'''Xa''=a.a.a, &c. to m factors x a.a, &c. to n factors,
—a, a, a. &c. to 7n + n factors,
— a!^^'^, by definition.
The reasoning and the rule are the same, if for a we wnte
a-^l), or a-\-'b^c, or any other quantity ; that is, the poioers
of such quantities are multiplied together by adding the
exponents of the powers together. Thus the 2d power of
a^h multiplied by the 3d power of the same quantity will
produce the 5th poAver of that quantity.
Ex. 1. 2vT'x3.^•'=2x3X.^•V=:6.^^
Ex. 2. "^ax X ^axy == 7 X 2 X aaxxy — UaVy.
Ex. 3. ha%Xa'bc = oa\iUc=^^a''h''c,
Ex. 4. ?>x'y'z' X 4.xYz = 3 X 4 X xVyYz'z = 12xyz\
Ex. 5. 7n7ix^yXpy=ninpx^yy—mnpx^y^,
Ex. 6. 2r/"' X 3a' = 2 X 3 X a'^a' = 6a"^+^
Ex. 7. ax'^Xix'' = abx'''x''=aI)x'''+\
Ey 8. ax'"Xbx''Xcx''=:abcx'^x''x'' = ccI)cx''''^''+^.
60. 11. Suppose we have to multiply a-f Z> by 3. We have,
3[a + b)=a + t?-\-a + b+a+b=:3a-{-Sb.
40 ELEMENTARY ALGEBRA.
Similarly,
In the same manner we have,
3{a—b) — da—3b c{a—l) — ca—cb
Tims, to mnltiply an expression consisting of two or more
terms by a single positive term: Multiply each term of the
expression hy the single term, and put 'before each product the
sign of the term tvhich produced it j then collect these results
to form the complete product.
Ex.
1.
a-^-b—c multipl]
Led by 2=2a + 2J-26*.
Ex.
2.
a—b-^-c
d=ad—bd + cd.
Ex.
3.
ax-\-by
c—acx-\-bcy.
Ex.
4.
ax-\-by-'Cz
2p = 2apx + 2 b^jy — 2 cpz.
Ex.
5.
2a + db—4:C ..
2x=4:ax+6bx—Scx,
Ex.
6.
ax-\-by
ax=aV-{-abxy,
Ex.
7.
ax^by
by=abxy-\-Fy^,
Ex.
8.
7a;-4y+6 ...
3:r = 2l2;'- 12^:^ + 18:*.
Ex.
9.
^X^'-\?>X-\-\ ...
6 = 30x'-Gox+5.
Ex.
10.
x^—px-\-q^
px —px^ —p^x^ -\-pgx.
Ex.
11.
\ah^\cd
. 4.ab=2a'V-VGal)cd.
61. III. Let it be required to mnltiply a-\-b by c-\-d\
this means that a-\-b \^ to be taken c-\-d times, that is, g
times and d times. Kow a-\-b taken c times produces, by
rule of Art. 60, ac-\-bc', and a-\-b taken d times produces,
Rule for Case IL
MULTIPLICATIOIS". 41
by tlie same rule, ad-^-hd; .*. a-{-b taken c times aiid d
times, that is, c-{-d times, produces ac-^-hc-^-ad-^-ld, wliicli is
the product required.
Or, if the quantities ho, a-\-'b and c—d, a+h multiplied by
c—d means that a+i is to be taken d times less than c
times. Now a+b taken c times produces ac-\-bc; but this
is too much by d times a+b, that is, by ad+hd; .*. ad+M
is to be subtracted from ac-\-bc. Hence the product re-
quired is ac+bc—ad—bd, following the rule of Subtrac-
tion.
Or, if the quantities be a—b, and c—d, the product of
these is, as in the last case, c times a—b wanting d times
a—b, that is, ad—bd siMracted from ac—bc^ which leaves
ac—bc—ad-\-bd (changing the signs in the quantity to be
subtracted, according to rule).
Collecting these results, we have,
(a + b) (c-\-d)—ac-\-bc-\-ad+ bd
{a + b){c—d) = ac + bc—ad—bd
{a—b){c—d) = ac—bc—ad^-bd.
Considering these results, we see, for example, that cor-
responding to + « in the multiplicand, and + c in the multi-
plier, there is a term -\-ac in the product; corresponding to
the terms +a and —d there is a term —ad in the product;
corresponding to the terms —b and +c, there is a term —bo
in the product; and corresponding to the terms —b and — ^
there is a term + bd in the product.
These observations are briefly collected in the following
important rule in Multiplication: Lihe signs produce +,
and unlike sig7is — . This rule is called the Enle of Signs.
62. IV. Let it be proposed to multiply 2a by —4zb, or
The Rule of Si^us.
42 ELEMENTARY ALGEBRA.
— 4:C by da, or -4c by -4^. We apply the Eule of Signs,
aboye establislied, to these single terms. Thus, we have,
2aX-^h='-Sab
-4cX 3a=—12ac
'-4:cX—4:b=+16bc,
We attach a meaning to these operations on single terms,
after the same manner as in Addition and Subtraction.
Thus, the statement — 4cX~45= + 16^c, means that if —ic
occur among the terms of a multiplicand, and —4^ among
the terms of a multiplier, there will be a term i-lQic in the
product corresponding to them.
As particular cases of examples of this sort, we haye,
2aX-4=-^8a, 2X-4=~8, -2X-l=+2.
Eemark. — If several single terms are to be multiplied together the
product will be -f or — , according as the number of negative factors is
even or odd.
Thus, 4aX-25x Zc is -^idbe
AaX-2hX—oc is -\-24:abo
4aX-2bX-ScX-2d is -iSabcd.
63. The rules for Multiplication may now be conveniently
presented thus :
To multiply single terms: Multiply together the numeri"
cal coefficie7its, init the literal factors after this product, and
determine the sign hy the Rule of Signs,
To multiply quantities of two or more terms: Multijjly
each term of the multiplicand hy each term of the multiplier,
according to the ride for single terms, and the sum of tJicse
separate products will be the product required.
The process is generally conducted as in the following
examples.
Two General Rules of Multiplication.
EXAMPLES. 48
Ex. 1. Multiply -2a'b''+ 5ab^ - 7b*
by — 4«&
Prod. 8a'b''-20a''b^+2Sab\
Ex. 2. Multiply 2a+db-ie
by a+ 5— c
Prod, by a = 2a''+Sab-4:ac
by +5= +2ab + Sb''^Abc
by -c = —2ac-dbc-\-4:G'^
Whole prod. = 2a^ + 5a6 - 6ac + db'' - 7bc + 4c^
rix. 3.
a + b Ex. 4. « + & Ex. 5. cJ-5
a+b a—b a-b
a' + ab a' + ab a'-ab
+ ab + b'' -ab-b^ -ab + P
a'' + 2ab + b\ «' ^^ -b\ a^~2ab + b\
Ex. 6.
x + a Ex. 7. x^ + {a + b)x + ab
x + b x + c y^
x^ + ax x^ + {a + b)x^ + abx
+ bx + ab + cx'' + (ac + bc)x + abc
Ans,
x^ + {a + b)x + ab x^ + {a+b-\- c) x^ -^{cib^roc-^ be) x + abc
Ex. 8,
x^—ax'^+ bx — c
x^ + mx + n
x^—ax'^+ bx^ — cx^
+ mx'^ — amx^ + bwtx^ — cmx
+ ?zrt^ — aiix^ +bnx—cn
Ans. ic'^ — (<x — m) x"^ + {b— am + ?i)«""* — (c — 6//i + a») a?^ — {cm—bn)x — en .
64. We arrange the terms of the partial products so that
like terms may stand in the same column. This enables us
to collect the terms easily, in order to get the final result.
With the view of bringing the like terms of the product into
The order of Arrant^jcment of Terms of Multiplicand and Multiplier.
44 ELEMENTAEY ALGEBKA.
the same column, we arrange the terms of the multiplicand
and multiplier in a certain order. "We fix on some letter
which occurs in many of the terms, and arrange the terms
according to the poivers of that letter.
Thus, taking the last example, we fix on the letter x ; we
put first in the multiplicand the term containing the third
power of X ; next we put the term which contains the next
power of X ; next the term which contains the first power of
x\ and last we put the term which does not contain x at all.
The multiplicand is said now to be arranged according to de-
scending poicers of x. We arrange the multiplier always in
the same way as the multiplicand. It would haye done as
well to arrange them both according to asce7iding poivers
of X.
Examples — 9.
1. Multiply ax?\f by Ixy ; mx^ by —nx^ ; —acx by —2axi/;
ale by Ic, —ale by —ac] x^y by —xy"^,
2. Multiply x^—xy-^-y"^ by x, and a^ — ax-{-x^ by —ax]
x^ — ax-\-l) by — alx ; x^ — ox^y + ^xy"" — %f by xy, .
3. Multiply 2« + Z' by a-^-U, and "la-l by c—M,
4. Multiply "^x-V^ by 2:^+3?/, and 3^5+4^' by 2al-U\
5. Multiply x' + ^x—^ by ^+3, and x''—4:X-\-^ by x—2,
6. Multiply a'^-^a-l by a'-a-\-l, and by a'-3a-L
7. Multiply 2W-\-^x\j-\-^xy'' + y'hjdx-y,
8. Multiply a'-2a'l-^^a''l''-%al)' + Ul'hy a-^^l.
9. Multiply x"" -\-%ax-{-'^a^ hj x'^—2ax-\-a'.
10. Multiply 9a'-3«J + Z?'-6a-2^+4by 3r«+J + 2.
GE]!s^EHAL EESULTS IX MULTIPLICATION. 45
1 1 . Multiply x'^-\-y'^+z'^ + xy — xz + yz by x—y-]-z.
12. Multiply a'-\-2a''-h2a + lhy a'-2a''-{-2a-l.
13. Multiply a' + W-i-9o''+2aI) + dac-6I)chj a—2b-3c,
14. Multiply a'-2a'b + 3a'b'-2aF + I)'hja' + 2ad + I?\
15. Multiply x^ — ax + b hj x—c, and by x^ + ax—c,
16. Multiply l — ax={-I?x^ — cx''hjl+x—x''.
IX. General Eesults in Multiplication.
65. N.B. — The rules for the management of BracTcets,
given in YII., apply only to the addition and siiMractmi of
quantities so enclosed. If a collection of quantities within
brackets is to be ^nultijolied or divided by any quantity or
collection of quantities, the brackets must not be struck out
until the multiplication or division is actually performed.
Thus {a-\-'b)x{c-\'d) signifies that « + Z> is to be taken c-\-d
times, and is obviously not the same as either a-\-'b{c-\-d), or
{a-\-'b)c-{- d. Again, [a+'b)-^{c-{- d) is not equivalent to either
a^-'b^{c-{-d), OY\a-{-'b)-~c-{-d] but it may be written -^,
the line which separates the numerator and denominator
serving as a vinculum to hoth.
The learner would do well to practise multiplication of
quantities by means of braclcets as early as possible.
Thus,
Ex. 1. {a—l){c—d)^{a—'b)c—{a—'b)dy
=ac—'bc—{ad—'bd),
- ac —be— ad + bd.
The Management of Brackets In Multiplication.
4G ELEMENTARY ALGEBRA.
Ex. 2. (x-\-a){x+b)=^{x + a)x+{x-\-a)I?,
= x^ + ax + Ix + ad,
^x' + {a + b)x + ah*
Ex. 3. {x+l){x-\-2){x+3) = {x' + 2 + l.x + 2){x+3),
= {x' + dx+2)x+{x' + dx-\-2)3,
=x' + 3x' + 2x + 3x^ + 9x + 6,
=x' + 6x'' + llx + Q,
Ex. 4.
{a-]-b—c){a + b—c) = {a + h—c)a + {a-^h—c)h—{a + I?—c)c,
=a'^ + ah—ac+al) + d^—l?c--ac—dc + c^,
=:o^-\-2al)-\-V-2ac-21)c-\-c\
66. The student should notice some results in Multiplica-
tion, so as to be able to apply them when similar cases occur,
and write down at once the corresponding products.
Ex. 3, Art. 63/giyes {a-\-'b){a-\-'b) or {a^-iy
=:a'+2a$4-Z>Xi). Thus,
The square of the siwi of two qicantities is equal to the
sum of the squares of the two quantities ijicreased ly tiuice
their product,
Ex. 5, Art. 63, gives {a—l){a—l), or {a—iy
=.a^-2db^V (ii). Thus,
Hie square of the difference of two quantities is equal to
the sum of the squares of the two quantities diminished dy
twice their product,
Ex. 4, Art. 63, gives {a + d){a-i)=a'-'b' (iii). Thus,
The product of the sum. and difference of two quantities is
equal to the difference of their squares.
The sqnare of the sum of two quantities. The pquare of the difference of two
quantities. The product of the difference of two quantities.
MULTIPLICATION. 47
67. General results expressed by symbols, as in the equa-
tions (i), (ii), and (iii), are called formulas.
In these formulas, a and h indicate any quantities or ex-
pressions whateyer.
Bemark. — We may express the two formulas,
{a + hf=a'' + Uh + ¥ ; and {a-hf-=a'' -2al) + b\
in one formula. Thus,
{a±bf=a''±2ab-{-b\
where ± indicates that we may take either the sign + or the sign —
keeping throughout the vpper sign or the lotcer sign. a±b is read thu»
" a plus or minus Z>." ± is called the double sign.
As applications of these rules or formulas, we have,
(x + yy=x'-]-2xij-\-y% {x-2y=x'-4.x + 4:,
{2x + yy=4:x'' + 4.xy+y\
{2ax-3hjy=4:aV-12aI?xy+9by,
{29y={30-iy = 900-60 + l,
(54)^^= (50 + 4:)^=:2500 +400 -f 16=2916,
{x-]-2){x-2)=x'-4:,
{2ax-3hy) {2ax-\-Uy) =4aV-9Z>y,
(127)'-(123)^=:(127 + 123)(127-123)=250X4
==1000.
68. Ex. 6, Art. 63, gives
(x + a) {x -{■l)=:x^-\-{a-\- h)x + ab (iv),
where the coefficient of x is the sum of the two latter terms
of the factors, x-ha, x-\-b, and the last term, -i-ah^ is their
product. In like manner, we shall have,
(x^a) (x—h) =x^ — (a + d)x+ abf
(x—a) {x + h) —x^ -f {l)—a)x—ab,
The use of Fo!*mulas. The double sign.
48 ELEMENTARY ALGEBRA.
Thus,
{x-5){x + 2)=x'-{-{2-d)x-10=x'- 3.T-10,
{x-Q){x-6)=x'-{6 + 6)x-hdO=:x'--llx + 30,
{xi-2){x-2){x + d){x-d)=z{x'-4:){x'-9),
=x'-{9 + ^)x'' + 36=x'-ldx'-\-3(j,
69. By a little ingenuity the formulas (i), (ii), (iii), and
(iv), may be extensively applied to lighten the labor of Mul-
tiplication.
Suppose we require the square of x + y + z. Denote x -{-y
by a.
Then x + y-\-z=a-\-z] and by the use of (1) we haye,
(a^- zy = 0" + 2az+ z^=:{x + ijy + 2{x-\-^j)z^- z"
^x'^ 2xy +rf + 2xz + 2yz + 2;^
Thus, {x-^ y + zy ^x" -^ if + z" + 2xy + 2yz + 2xz,
Suppose we require the square oi p — q + r—s. Denote p—q
by a, and r—s by l] then^ — ^ + r— 5 = a + Z>.
By the use of (i) we have,
{a^iy:=:a' + 2al) + ¥={p-qy + 2{p-q){r-s)-\-{:r-sy.
Then by the use of (ii) we express {p—qy and {r—sy.
Thus, {p-q + r-sy
=p'' -'2p)q^- q" -\-2{pr—ps—qr + qs) -\-r'^ — 2rs-\- s"
—p" + g^ + r"" + / + 2pr + 2qs —2pq—2ps—2qr— 2rs.
Suppose we require the product of p—q-\-r—s and
p—q—r + s,
ljeip—q=a, and r—s^h; then
p—q-^-r—s — a + h, and p — q—r + s=a—b,
Wliat are the Formulas (i), (ii), (iii), and (iv)?
multiplicatio:n'. 41)
Then by the use of (iii) we have,
and by the use of (ii) we have,
{p—q-\-r—s) {2y—q — r-\-s)=p'' — ^pq + q^—{r^—2rs-[-s^)
=2o'' + q'-r'-s'-22Jq + 2rs,
As the student becomes more familiar with the subject,
he may dispense with some of the work. Thus, in the last
example, he will be able to omit that part relating to a and
Z>, and simply put down the following process :
(p — q + r—s) {p — q—r + s={p — q + {r—s)) {p—q—{r—s)),
=:{p~qy-{r-sy
—p"" — 2pq -\-q^— (r- — 2rs + s-)
—p^ — 2pq + (^^ — r- + 2r5 — 5I
70. Ex. 1. {ax + l) -^ cy + dy={ax-\-h -\- cy + ciy,
= {ax-^I)y+{cy + dy + 2{cfx-{-b){cy + d),
= a^x^ + ^' + 2al?x -f- (fy^ + <^ + 2cdy
+ 2acxy + 2adx + 2I)cy + 2M.
Ex. 2. (a^—ax-hx^) {a'^—ax — x'^)=^hj (ii) {a^—cfxy-^x*
= a^— 2a^x + a-x^ — x*.
Ex. 3. {a^-i-ax—x-){a'^—ax—x'^)
= { («^ — x") -f ax} { (a^ — x"^) — ax}
=z{a'^— ^-) - — a^x^ =a^— 2a^x" -\-x^~ a'x^
= a'* — 3a'x'^ + x\
N. B. — The formula here em])]oyed, {a-{-b)X{a—b)z=a'^ — b'^, may be
alwaj^s applied, whenever it is seen that the two quantities to be mul-
tiplied consist of tirms which differ only (some of them) in sign, by
taking for a those terms which are found with their signs unaltered in
each of the given quantities, and the others for b. Thus, in Ex. 3, a?
-■x^ appear in both the given quantities, whereas in the one we havo
-i-ax, in the other —ax\ hence, the product required is {a^—x^f — a\-;^,
as a Dove.
3
V ele:.iextary algebra.
Ex. 4. (a' + ax + x') {a'-ax+x') = {cr+xy-a'x'
= a^ + a^x^ -f x^,
Ex. 5. {a'^+ax—x-){a-—ax + x^) = a*~{ax—xy
=^a*--a'x'^+2ax^—x\
Ex. 6. {ar—ax-\-x^^) {ax + x^—a")=x^ — {a^—axy
=x^—a^-\-2a^x—a^af,
Ex. 7. (a + b + c + d) {a + b-c-cl) = {a + by-{c-{-dy
=za' + 2al) + ¥- & - 2cd- d\
Ex.8. {a^2h-?>c-d){ci-%b-^U-d)
= {a-dy-{:2b-^cy
=a''-2ad+d'-4:b' + 12bc-9c\
Examples — 10.
1. Write down the squares of a—x, l + 2ic'', 2a* + 3
3x—4:i/,
2. Write down the squares of 3 + 2x, 2x—3y, a^—3axw
bx^ — cxy,
3. Write down th© products of {2a-\-l)x{2a—l),
{3ax-{-b)x{3ax-b), {x-~l) {x^l) {x'-^l).
4. Write down the products of (a: + 3) X (^ + 1),
{x''-\-4)x(x'-l), {ab-3) {ab-\-2y
(2ax—3b) {2ax-b),
6. Find the continued product ofx-{-a, x—a, x+2a,
and x—2a.
EXAMPLES. 51
n. Obtain the product of mx + 2ny, mx~27iy, mx—Sny,
and mx-\-3ny,
7. Simplify 3{a-2Q:y + 2{a-2x) {a + 2x)
+ {3x—a) {3x + a) — {2a-3xy,
8. Multiply x'' + 2xy + 2^' by x^ — 2xy + 2y\
and 2a^ - 3ah + 6' by 2a' 4- 3ab + Z>'.
9. Multiply a-\-h-{-c by a-^h — c^ by a— 5 + c,
and by a—h—c.
10. Multiply a— ^ + c by a— 5--C, by 5 + c— flj,
and by c—h—a.
11. Multiply 2a+5— 3c by 2a— ^ + 3c, and by h + 'dc~2a.
12. Multiply 2a — h—3c by 2a + ^> + 3c, and by Z* — 3c— 2a.
13. Multiply a + 5 + c+c? by a—h-^c—d, by a—h—c-\-d,
and by b-^c—d—a,
14. Multiply a— 2^ + 3c + 6^by a + 2^-3c+<^,
by 25 — a + 3.c + <^, and by a + 25 + 3c— ^.
71. There are other results in Multiplication which are of
less importance than the four formulas given in Art. QQ, but
which are deserving of attention. We place them here in
order that the student may be able to refer to them when
they are wanted ; they can be easily verified by actual multi-
plication.
{a'-b){a'-\-ab-hb')=a'-b%
{a+by=z{a + b) (d'+2ab + ¥) = a''\-3a'b-i-3ab' + b%
(a-by={a-b){a'-'2ab-{-b')^a'-3a'b-{-3ab'-b\
other results in Multiplication.
52
ELEMEXTAIIY ALGEBKA.
X. DlYi:,ION.
72. Division, as in Arithmetic, is the inverse of Multipli-
cation. In Division we have given the product and one of
the factors, and we have to determine the other factor. The
factor to be determined is the quotient,
73. I. The rule for the division of simple expressions
follows at once from the corresponding case in Multiplica-
tion.
For example, we have.
therefore — -
therefore
therefore
therefore
l^ahc
4.^^"'^
12ahc
■Aab.
Also ^abX —3c = — 12al)c;
-12al?c
12abc
— = — dc,
4,ah
Also — 4:011) X
-VZahc _
Also --4a^X— 3cj=
^iat.
=^-4.ab.
I2abc_
-iab
= -3c,
12ahc_
-3c
= -^ab.
Hence, we have the following rule for dividing one singte
term by another : Divide resjjedively the coefficient and literal
parts of the dividend ly those of the divisor ; and then, if the
tivo quantities have like signs, prefix to the quotient the sign
+, if different, the sign — .
What is Division? Rale for dividing one single tenn by another.
DIVISION. 53
This division is the familiar process of cancelling like fac-
tors in Arithmetic. Hence the rule may be given briefly
thus :
Strike out from the dividend the factors which occur in the
divisor ; the rest of the dividend is the quotient, ivhose sign is
deternmied hy the Rule of Signs, Viz, \ Like sigjis give +.
unlike sig7is — .
Thus,
-7^-v-5=-7, -ax-^a^x, Uab-^n=2a, 7b^n=l,
abc~ab=c.
74. One poiuer of a quantity is divided by another i)oioer
of the same quantity by subtracting the index of the latter
from that of the former.
For example, suppose we have to divide a^ by a^
a' a\a' .
'=a'-\
Or we may
■ show the truth of the rule thus :
a'Xa'.
=a\
Therefore,
a' 3
And generally, if
7n and n be
positive
integers.
and m > n
fl^^'-f-a"-
^a'''-^
Similarly,
Za'l)'
= 2a'b%
xhfz'
z^x'yzK
xyz
ha'b\
d'bc
(=5a'b'c%
a^'b'
_^m-pyn-q^
Division of one power of a quantity by another power of the same quantity.
54 ELEMENTARY ALGEBRA.
75. It may happen that the factors of the divisor do not
occur in the dividend. In this case we can only indicate
the division. Thus, if 6a is to be divided by 3c, the quo-
tient is indicated by 6a-^3c, or by — .
oC
Again, it may happen that some of the factors of the di-
visor occur in the dividend, but not all of them ; or that a
power of a quantity occurs in the dividend, and a higher
power of the same quantity in the divisor. In this case the
indicated quotient is a fraction, which can be simplified by
striking out common factors, as in Arithmetic.
Suppose, for example, 16a^b is to be divided hjQbc; we have,
loa'b _ 5a'xSb _5a^
6bc ~ 2cX3b ~ 2g'
striking out the common factor 3b.
Again, if 4tab'^ is to be divided by 3cb% the quotient is indi-
cated by ^r-n.- Eemove the factor b"^ which occurs in both
dividend and divisor,
^aW _ ia
'3cb'~'3cb^'
76. II. To divide a quantity consisting of two or moro
terms by a single term :
Divide each term of the dividend by the divisor, and collect
the results to form the complete quotient.
For, since fl^+^— c + &c. multiplied by m produces
ma-hmb—mc-{-&c.,
/. ma-\-mb—7nc+&o. divided by m, gives
a + b—c-i &c.
Hence, the rule is as above stated.
Rale for Case n.
DIYISIOX. 55
Ex. 1. — h— =4:a'-3Z>c4-a.
a a a a
a^x'^ — 6aJ)x^ + 6ax^_a^x'^ 6abx^ (5ax*
aa; ax ax ax
= a'-6I)x + 6x\
Ex. 3. (a+I)+c)-~abc=:-T-+-Y~-\--j-^Y--h — h- 1^.
^ ^ aoc ado abc be ac ab
^ , a'c'-2abc' + dac' a'c' 2abc' 3ac'
Ex. 4 T-r-a = — 7-T-2 + -
-4:abc^ ^abc^ 4:abc^ 4iabc^*
__^ l_3c
77. III. To divide one expression by another when tne
divisor consists of two or more terms^ we must proceed as in
the operation called Long Division in Arithmetic. The fol-
lowing rule may be given :
Arra7ige both dividend and divisor aecording to ascending
poiuers of some common letter, or both according to descending
poiuers of some common letter. Divide the first term of the
dividend by the first term of the divisor, and ][tut the result
for the first term of the quotient ; multiply the luhole divisor
by this term and subtract the product from the dividend. To
the remainder join as many terms of the dividend, tahen in
order, as may be required, and repeat the whole operation.
Continue the process until all the terms of the dividend have
been tahen doivn.
The reason for this rule is the same as that for the rule of
Long Division in Arithmetic, namely, that we may break
the dividend up into parts and find how often the divisor is
contained in each part, and then the aggregate of these re-
sults is the complete quotient.
Rule for Case III.
56 ELEMENTARY ALGEBRA.
78. We shall now give some examples of Division arranged
in a convenient form.
Ex. L 1— .t) l-2x-i-x' {l~x
1— X
— x + x^
— x + x^
Ex. 2. 3x ^ %) Gx' - 1 7x'ij + 1 (jy\2x' - dxi/ - 4.f
- 9x'y-j-l2x?/
-12xf-{-lGy'
Ex. 3. a—x) a^—x'' {a^^ax^x^
a^—ct^x
d\v-
-x'
cc'x-
- ax^
a/jy —
■x^
ax'-^
x'
Ex. 4. a-^x) d-\-'^ {a^—ax + x'^
a^ + a'^x
—cC'x-^x^
^a/x—ax^
ax^ + x^
aaf + x^
Ex. 5.
-2^^4-3^=) na*-10a'b-22aW + 22aF-{-lDl?' {da' -4.ab + 5d'
3a'- 6a'b+ 9a'b'
- 4.a'b + rda'b'-22ab'
- 4a'b+ 8aW-12ab'
baW-lOab' + lob'
da'b'-lOab'i-lDb'
Consider the last example. The dividend and divisor are both ar-
ranged according to descending powers of a. Tiie first term in the
dividend is 3a*, and the first term in the divisor is a"^ ; dividing the
former by the latter we obtain Sa^ for the first term of tlie quotient.
We then multiply the whole divisor by 3a- , and place the result so
DIVISION. 57
lliat each term comes below the term of the dividend which contains
the same power of a; we subtract, and obtain — 4a^b + lda^b'^ ; and wo
bring down the next term of the dividend, namely, — 22a&^. We di'
vide the first term, —Aa'^b^ by the first term of the clivisor, a'^ ; thus we
obtain — 4:ab for the next term in the quotient. We then multiply the
whole divisor by — 4«Z> and place the result in order under those terms
of the dividend with which we are now occupied ; we subtract, and
obtain 5a'^b'^—10ab^ ; and we bring down the next term of the dividend,
namely, 15^^. We divide 6a'^b'^ by a\ and thus we obtain 5^^ for the
next term in the quotient. We then multiply the whole divisor by 55^,
and place the terms as before ; we subtract, and there is no remainder.
As all the terms in the dividend have been brought down, the opera-
tion is completed; and the quotient is Sa"^ — 4:ab-hob'\
It is of great iinj^ortance to arrange ioth dividend and di-
visor according to the same order of some conamon letter ; and
to attend to this order in every part of the operation,
79. It may happen that the division cannot he exactly
performed. Thus, for example, if we divide <^^+2«^-f2Z>^ by
a-^1) we shall obtain a+J in the quotient, and there will
then le a remainder, V, This result we place, as in Arith-
metic, in the quotient over the divisor, in the form of a
fraction, thus indicating that If remains still to be divided
by a+h. Thus,
-T — =a + Z>H — -y.
Ex. 6. a+x) c^^rx^ {a—x^
a+x
a^ +' ax
—ax + x
—ax—x^
2x'
2x^
Ex. 7. a-x) a^^x^ {a^x-\-
a—x
-ax
ax+ x^
ax— x^
2?
Important principle in Division. Division with a remainder.
58 ELEMENTARY ALGEBRA.
80. We give some more examples :
Ex, 8. 1-^)1 {li-x+x^ + x^ + &c,+ -^~ ^
-X
+ x
+ x—x^
+ X''
+ x''-x'
+ x'-x*
+ x' &c.
Ex. 9. Divide x'-5x' + '7x'-{-2x'-6x-2 by l + 2x—3x' + x\
Arrange both dividend and divisor according to descend-
ing powers of x.
x'-3x''-\-2x-]-l) x'-bx' +7x'-^2x'-6x-2 (x''^2x--2
x''-3x' + 2x'+ x'
--2x'-2x' + 6x' + 2x'-Gx
—2x' +6x'-4:x'-2x
-2x' +6x'-4.x-
-2x' +6x''-4:X-
-2
-2
Ex. 10. Divide 64-^" by 2-
-a.
64-32a
33a-
-a'
32a-
-16a'
16a'-
16a'-
-a'
-8a'
8a'-
8a'-
-a'
-ia'
4a*-
4a'-
-a'
-2a'
2a'-
2a'-
-a'
-a'
DIYISIOIS^ 59
Ex. 11. Divide a^ ^h^-^-c^—^^aU by a + ^ + c.
Arrange the diyidend according to descending powers of «,
a + 'b + c)a' -?>abc-\-¥ ^c\a' -ab-ac-^F -U-^G^
a^-^a'h + a'c
—a^h—a^G —?>abG
—c^G ■\-a¥ — 1abG
—a^G — obG—aG^
aW— abc + aG^ + h^
a^ +h' + l)'G
— abc^a& —Fg
— abc —Wg—W
a? '\-M-\-g''
ac^ +Ig''-\-g'
The above is the easier method in such a case, but the fol-
lowing, in which the coefficients of the different powers of a
are collected in brackets, is the neater and more compendi-
ous:
— {b + G)a'-dI)Ga
-{b + G)a'-{b'+2bc+G')a
+ {b'- bG+G')a-^{b' + G')
+ {b'- bG+G')a+{b' + G')
Examples — 11.
Divide
1. 16x' by ^x\ 2. 24a° by -Sa\ 3. 182:y by 6<y.
4. 2^.a'b'G' by -da'b'G\ 5. 20a'b'xY by 5Z»Vy.
^^ ELE31EOTAKY ALGEBRA.
6. 4^^- 8:^^ + 16:. by ^x, 7. 3^^-12^^ + 15a^ by ~-U\
8. xhj-3xY + 4:xy'hjxy,
9. -15^^Z*^-3a^Z^^+12a^ by ^3a^.
10. 60r.^^V-48a^^V + 36«T.^~20«5^^ by ^ahc\
11. ^^^-7^ + 12 by :.-3, 12. ^^ + r.^72 by .'. + 9.
13. 2x'-x' + ^x-^hj2x~d.
14. Qx' ^Ux'^ 4.x + 24 by 2x + 6.
15. 9x' + 3x' + x-~lhj3x~l.
^^' 7x'~24.x'-{-58x~21hj7x~3.
17. ^«-l by x~l. 18. a«-2a^^+^^ by a~^.
19. ^'^-81^^by:r-3y
20. x'~-2x\j-\-2x\f-xf by x~y.
21. ^^_y^ by a;-y. 22. a^+32^^ by a + 25.
23. 2a'-V%7aF-%Wh^a^3-b.
24. ^^ + :^V + ^y + 2;y + :^3/^ + ,y^ by r^^/.
^h. x' + 2x^y+3xY^xY-2xf-~3f by o.^-.^^
26. ^'-5a5^ + lla3^-12a; + 6byi«^-3^ + 3.
27. x' + x'^^x''-^Ux-4.hjx'!-\-4.x + L
28. aj*-13a3H36 hj x'-^^x-\-Q.
29. i«* + 64by^H4.T + 8.
30. x' + 10x' + 3bx' + b0x + 2i by ^^^ + 5^ + 4.
31. ^'-(a + 5 + £?)a3^ + (a5-l-^^ + ^^)^_^^^^
32. ^'~{a+p)x''+{qj^ap)x-aqhj x-a,
33. v' - ^?z?/^ + ny"" — ny^ ^my-lhj y-^1.
FACTOKS. 61
34. a^—lf + c^ + ^ahc by a—h-{-c, and a'—F—c^-dadc
by a—h—c.
35. a by 1 + 33, and l-f2a; by 1— 3;^', each to four terms in
the quotient.
36. 1 by 1— 2i;c+.T^, to four terms.
XI. Factors.
81. We shall now notice some general results in Division,
m connection with those already given in Multiplication;
and we shall apply some of these to find what expressions
will divide a given expression, or, in other words, to resolve
algebraic exjjressions into their factors,
82. For example, by the use of formula (iii) of Art. 66,
we have,
a'-Jj'-^^a' + h') {a'-h'')^{a'' + h'') {a-^h) (a-h)
a'^I?'=--{a' + h') {a'-b') = {a' + b') {a' + b') (a^b) {a-b).
Hence, Ave see that a^—b^ is thQ product of the four fac-
tors a^-\-b\ a^-^y, a + b, and a — b. Thus, a^ — b^ is divisible
by any of these factors, or by the product of any two of
them, or by the product of any three of them.
Again, in Art. 70, we have,
{a'^ab + b'){a'-a])^b') = {a'^-by^{aby=a'-^a''b'-Vb\
Thus, a^^a^b'^-^b" is the product of the two factors
€^-\-ab^-W and a''—ab-\rb'', and is therefore divisible by
either of them.
83. The following results in Division may be easily veri-
fied, and will enable us to write out with ease the quotients
in many similar cases.
x—y
-y
-1,
How to resolve Algebraic quantities into their Factors.
62 ELEME^^TARY ALGEBRA.
X^ — lt
x-y ^'
Also,
x-y
x'-y'
x-\-y ^'
— =x — X y + xy — y\
x + y ^ iJ J ^
— ^ —x''—x^y-^x^if—x^y^-\-xy''—y^j
x-\-y
and so on.
Also,
x-\-y '
"^^^.x'-x'y + xY-xf^f,
and so on.
The student can carry on these operations as far as \w
pleases, and he will thus gain confidence in the truth of the
statements which we shall now make, and which are strictly
demonstrated in larger works on Algebra. The following
are the statements :
x'^^y^ is divisible by x—y if n be any whole number;
jc"— ?/" is divisible by cc + ^ if w be any even whole number;
iK" + ^" is divisible by a; +^ if ^ be any odd whole number.
FACTORS. G3
"VVe might also put into words a statement of the forms of
the quotient in the three cases; but the student will most
readily learn these forms by looking at the above examples,
and, if necessary, carrying the operations still farther.
We may add that x'^^'if is never divisible by x-\-y or x—y^
when n is an even whole number.
84. The student will be assisted in remembering the re-
sults of the preceding Article by noticing the simplest case
in each of the four results, and referring other cases to it.
For example, suppose we wish to consider whether x^—y'^i'^
divisible by x—y or \i^ x^y\ the index 7 is an 06?^ whole
number, and the simplest case of this kind is a:— y, which is
divisible by x—y, but not by x-{-y\ so we infer that x'—y''
is divisible by x—y and not by x + y. Again, take x*—y^\
the index 8 is an even whole number, and the simplest case
of this kind is x'—y'', which is divisible both by x—y and
x-{-y\ so we infer that x^—y^ is divisible both by x—y and
%-\-y,
Now, in every case the quotient will consist (as above.
Art. 83) of terms in which the exponents of x decrease, and
of y increase continually by 1 ; but when the divisor is x—y,
these terms are all plus; when it is x-^-y, they are alter-
nately + and — .
We shall now apply these results to some examples.
Thus,
'lax — l
x + oy
x'-lQ
r=x'-3xy+9y\
x-2
= x'-2x^-h^x-
Law for the esponents and signs of the quotient
64 ELEMEXTAliy ALGEBRA.
Examples — 12.
Divide
1. a^—x^ by a-\-x, a^—x^ by a—x, and a^—x^ by a-^x.
2. 9^'-- 1 by Zx-1, 252:'-l by hx-\-\, and 4a;'-9
by 2:^+3.
3. 9wV — 25 by 3??^?^+5, and 16m*— 7^* by Am^ + n^.
4. 1 + 82;' by 1 + 2^;, 27^'-l by 3:r-l, and l-16a;*
by 1+2:?:.
5. :r*-8l3/' by x-Sy, a''+32h' by «+2Z>, and x''~-t/^
by :^;' + ^^
6. ia^ + h^ by i^+5, and x*y*—z' by 0^^ + ;^.
7. (« + &)' — <?' by a + h—c, and a^—(l)'-cY by ^— Zi + (7.
8. (^+^)'+^^ by .':c + y+^, and x^ — {ij—zf by a;— ?/+^.
85. The above results and those of {^^) may also be ap-
plied to resolve algebraical quantities into their elementary
factors, a process which is often required.
Ex. 1. ^x^-if^{%x^y) {^x-y),
Ex.2. .TH8 = (a; + 2)0'?;'-2:r + 4).
Ex. 3. {2a-'by-{a-21)Y=^{U-'b^a-'21)){2a-l-a + 2h)
Ex.4. x'-a'={x' + a'){x'-a')
=z{x-{-a) (x^ — ax-ha^) (x—a) (x^ + ax + a^),
Ex. 5. {a'-xy={{a-x) {a'+ax + x')y
=:{a-xyx{a'i-ax + xy.
FACTORS. Go
Examples — 13.
Eesolve into elementary factors
1. l-4a;^ a'-dx', 9r}f-hi\ 25aV-4< 16xy~2Dxy.
2. x' + y% x'-y\ l + x'y\ x'-l, a'xy'-x'y, 2a'b'c-SabV.
3. 25x'-a'x% a'-9a'b% 8^^-27, a'-Sb% aVy-{ 27a;y.
4. x' + 32, aV + 27x% 8x'-{-y% a'h''-c\ a'bc + 2a'I)c'' + abG'
5. 81:?;*-1, x'-G4:, x'-2baf + bV, x'-2a'x'-\-a'x\
6. (3.T-2)^~(2;-3)^ (« + ^)'-4Z^', (4a3 + 3^)^-(32: + 4^)^
7. {x' + yy-^xY, c'-ia-by, {2a + by-{2a-by.
8. a;' + '?/' + 3a:?/ (x + y), m'— if —in {m^—n'')+ 72 {m—iiy,
a'-ab + 2 {b'-ab)+3 {a'-b')-4. {a-by.
9. 5 (x'-y')-\-'^ {x + y)\ 3 {x'-y')-b {x-yy,
(^4.^)^ + 2 (:.^+:r^)-3(a:^-y^).
10. 2{a' + a:'b-{-ab')-{a'-F), a'-b'-3ab {a-b),
a'-b' + {a'-by-2a' + 2a'b\
86. So, too, Ave may often apply (68, iv) to resolve a trino-
mial into factors when it is of the form ax"^ -{- bx + c. We
repeat formulas (iv) :
x''-\-{a + b)x-\-ab = {x + a){x-^b);
x'^—{a-\-b)x + ab—{x—a){x—b);
x'^ + {a—b)x—ab={x + a){x—b).
Ex. 1. x''+7x+12 = { x + 3){ X+4-).
Ex. 2. x'-dx + U^i x-2){ x-^iy
Ex. 3. x'-DX-U^i x-'7){ .T+2).
Ex.4. 6.r'4.^-12 = (3.7;-4)(2:^-f3).
Resolution of Trinomials into factors.
66 ELEMENTARY ALGEBIIA.
The student may notice that, if the lasfc term of the given
trinomial be positive (Ex. 1, 2), then the last terms of the
two factors will have the same sign as the middle term of
the trinomial ; but if negative (Ex. 3, 4), they will have, one
the sign +, the other — .
In Ex. 4, it is clear that the first terms of the two factors
might be 6x and x, or '^x and 2.^•, since the product of either
of these pairs is 6x^, and so the last two terms might be 12
and 1, 6 and 2, or 4 and 3 ; it is easily seen on trial which
are to be taken, that is, which serve also to produce the
middle term of the trinomial.
Examples — 14.
Eesolve into elementary factors
1. a;' + 6:?: + 5, 2;' + 9:r + 20, x'-hx-^Q, x'-^x-Vl^,
x'' + ^x + 'l,x'-l^x-\-^.
2. ir'+^-6, ^'-a;-6, a;'~22;-3, x^-\-2x-lb, x'^-lx-^,
x'-^x-^.
\ 3. 4a;' + 82; + 3, 42;' + 13a: + 3, 42;' + lla;-3, 4:x''-4:X-3,
dx' + 4^x-4., 6x'' + 6x-4c.
4. 12^-'-5:z;-2, 12:i;'-14:z; + 2, 12a;'-a;-l, a;'-f ^-12,
dx^-2x-6,
6. aV-3a'x + 2a\ a'-a^x-6ax\ 3a*I)+a''b^-2ad\
12a' + aV-x\
6. 2x'y-^bxy-j-2xf, 9xYSxf-6y\ 6aV + a'a;-a%
(jb'x'-7bx'-dx\
87. We shall now give a few examples of multiplication
and division of expressions in which factors or terms occur
with letters for their exponents.
GREATEST COMMOX DIVISOR. 07
Ex. 1. ]\Iultiply a'-'^'F^+'o' by w"''-'¥c,
Ans. a'"'Z>'^+V.
Ex. 2. Multipljof^ + fl^ii;'^ -a':^^ by a"^^^^.
Ans. a"^a;'^ + a"^+'a;'^--a"^+V»
Ex. 3. Multiply a'-'l-d!'--lf^aV'-^ by a J.
Ans. oJ'W-aJ'-^l^^a^lf.
Ex. 4. Find the continued product of
Ans. a"+^Z»^^+V+^(^"+^.
Ex. 5. Multiply a"*— 2c'* by a"*— c^
Ans. a-'**— 3a"'c"4-2c-".
Ex. 6. Divide 115a'"^V^+'^J-'-' by -69a"^>V+'^d
Ans. -\dr-''c''dr'-\
Ex. 7. Divide a"'5-^"^-*^- + a"^--^^'-«'^-'^' + a"^-'Z>' by fl^Z».
Ans. oJ^-^ - a^'-l + oT-^V' - oT'-'W + ^'"-'Z^'.
Ex. 8. Divide aj'^+^ + a:"^^^.^"^^ + /'*+« by a;" + ?/^
Ans. a;"'^-?/".
Ex. 9. Divide c^-"^-3a'"c^ + 2r« by fl^'^-c^
Ans. a'"-2c".
XII. Greatest Common Divisoe,
88. In Arithmetic, a whole number which divides another
whole number exactly is said to be a divisor or measure of it,
or to divide or measure it. A whole number which divides
two or more whole numbers exactly is said to be a co7nmon
divisor or commmi measure of them.
Divisor or Measure. Common Divisor or Measure.
CS ELEMENTARY ALGEBllA.
In Algebra, an expression wliicli divides another expres-
sion exactly is said to be a divisor or 77ieasicre of it. An ex-
pression which divides two or more expressions exactly is
said to be a common measure or common divisor of them.
ISToTE. — The English use the word measure ; the French, the words
divisor and divide in the same sense. We shall use the latter, as they
have been generally adopted in this country.
- 89. In Arithmetic, the greatest common divisor of two or
more whole numbers is the greatest whole number which
will divide them all. The expression Greatest Common Di-
visor, in Algebra, must be understood as applying, not to
the numericcd magnitude of the quantity, but to its dimen-
sions only; on which account it is sometimes called the
Highest Common Divisor,
The expression Greatest Common Divisor is, however, re-
tained in accordance with established usage, and we shall
use the letters g.c.d. for shortness, to indicate it.
90. The following is the rule for finding the g.c.d. of mo-
nomials :
Find ly Aritlimetic the g.c.d. of the Numerical Coeffi-
cients ; after this number put every letter which is common
to cdl the monomials, giving to each letter respectively the least
exponent which it has in the monomials,
91. T'or example : required the g.c.d. of IQa^Vc and 20a^b\L
Here the numerical coefficients are 16 and 20, and their g.c.d.
is 4. The letters common to both the expressions are a and
I) ; the least index of a is 3, and the least- index of b is 2.
Thus we obtain 4a^^^ as the required g.c.d.
Again : required the g.c.d. of Sa^Fc^'x^yz^, 12a'^l?cx^y% and
16a^cVy\ Here the numerical coefficients are 8, 12, and 16 ;
and their g.c.d. is 4. The letters common to all the expres-
Rule for the Greatest Common Divisor.
GREATEST COMMON DIVISOR. 69
sions are a^ c, x, and y ; and their least indices are respect-
ively 2, 1, 2, and. 1. Thus we obtain ^a^cx'y as the required
G.C.D.
92. Tlie folloAving statement gives the best practical idea
of what is meant by the term greatest common divisor, in
Algebra, as it shows the sense of the word greatest lic^e.
When ttuo or more exjjressions are divided dy their greatest
common divisor, the quotients have no common divisor.
Take the first example of Art. 91, and divide the expres-
sions by their g.c.d. ; the quotients are 4ac and. bhd, and
these quotients have no common measure.
Again, take the second example of Art. 91, and divide the
expressions by their g.c.d. ; the quotients are 2Fcx^z^, Zc^ly''^
and 4«cy, and these quotients have no common measure.
93. The idea which is supplied by the preceding Article,
with the aid of the Chapter on Factors, will enable the
student to determine in many cases the g.c.d. of compound
expressions. For example: required the g.c.d. of AiC^(ci-\-l)Y
and ^ab {a^ — lf). Here 2a is the g.c.d. of the factors ^a"^ and
Q>ah ; and 6^+^ is a factor of {a-\-hy and of a^—lf, and is the
only common factor. The product 2a{a + I)) is then the
G.C.D. of the given expressions.
The rule in this case is similar to that given in Art. 90 :
Put doto7i every factor common to cdl the expressiojis, giving
to each factor respectively the least exponent luhich it has in
the exp)ressions. The product of these factors iviU be the Great-
est Common Divisor of the expressions,
Ex. 1. The G.C.D. of 15.?;' and l^x\ is ^xJ",
Ex. 2. The g.c.d. of SG.r'^V and A:%x'y'z\ is 12x\fz\
Ex. 3. The g.c.d. of SoaWx't/ and ^^a^Vx'y\ is Ha'b'xy.
Ex. 4. The G.C.D. of oax'' — 2a"x and a'^x^—dabx, is ax.
The G.C.D. of Compound Expressions.
70 ELEMENTARY ALGEBRA.
Ex. 5. The G.c.T>.of 6x'y-12xY-\'3xy'
and 4:ax'^ -}- 4:axy + 4:a^x, is x.
Ex. 6. The g.c.d. of 6aV {a'-x') and 4^\r (a+^)\
is 2a'^x {a + x),
Ex. 7. The G.C.D. of a\a'x' -Sax' + 2x') and x^a'-iaV)^
that is, of 0^x^(0^— 3ax + 2x'^) or aV(a— 2a;) (a— a;) and
aV(a'--4a;''), is aV(a~ 2:c).
Examples — 15.
Find by inspection the g.c.d. of
1. Ax' {a+xy and 10 {a'x-x'y.
2. x' {a'-xy and {a'x+axy.
3. {a'b-aby and a^ (a^-5^)^
4. 6 (:^;'-l) and 8 (2;^-3.r+2).
5. {x^+xy and a;' (:r'-2:~2).
6. 4 (ic'+a') and 6 {x'-2ax-3a'').
7. a' (^^+12^+11) and a'x'-lla'x-12a\
8. 9 (aV-4) and 12 (^V + 4(^:?;+4.
9. :?;' — 9a;+14 and a:'— lla;4-28.
10. a;'' + 8a;+15 and ic' + 9:r+20.
11. a;'+2:r-120 and a;' — 2a;-80.
12. 4 (x'-x-hl) and 3 (a;'+a;' + l).
13. x'-xy-ny' and a;'+5a:?/+6^'.
94. The G.C.D. of two polynomials cannot be generally
fonnd, however, thus by inspection. Hence, for more com-
plex examples it is necessary to adopt another method — the .
same given in Arithmetic for two numbers.
GREATEST COMMOIS^ DIViSCR. 71
95. Let there be given then two algebraic quantities of
which it is required to find the g.c.d.
EuLE. — Arrange the quantities according to poivers of
some common letter, and divide the one of higher dimen-
sions dy the other ; or, if the highest exponent happen to he
the same in each, taJce either of them for dividend. Take
now, as in Arithmetic, the remainder after this division for
divisor, and the preceding divisor for dividend, and so on until
there is no remainder ; then the last divisor tvill ie the g.c.d.
of the tiuo given quantities,
Ex. Find the g.c.d. of 2;' - 7^ -f 10 and 4x'~25:z;'+20:r+25
a;'-7ii;+10)4a;'-25a;' + 20:2;-f25(4:?;+3
4a;^-28:^:'' + 40:y
3.^'- 20a; +25
x-h)x^'-nx^-\^(x-%
x^ — hx
-2^+10
— 2a;+10 Ans. ic— 5.
Examples — 16.
Find the g.c.d.
1. Of 32)'+ ^-2 and 3.T'+4a;--4.
2. Of 6:?;' + 7a;-3 and 12a;' + 16rr-3.
3. Of 9a;'-~2o and 9a;'' + 3:r-20.
4. Of 8a;' + 14.T-15 and 8a;' + 30a:' + 13a;- 30.
5. Of 4a;'+3a;-10 and 4a;' + 7a;' -3a; -15.
6. Of 2a;* +a;'- 20a;' -7a; + 24 and 2a;* + 3a;' -13a;' -7a; +15.
Rule for the g.c.d. of Polynomials.
72 ELEME^s'TARY ALGEBRA.
96. In order to prove the Eule above given, it will bt*
necessary to show first the truth of the following statement
If a quantity c he a common divisor of a and b, it Krill also
divide the sum or difference of any multiioles of a and b, as
ma ± nb.
For let c be contained p times in a and q times in I ; then
a—pc, h—qc, and ma ± nh — mi^c -ihnqG— {mp ± ncfjc ; hence
c is contained mp ± nq times in ma ± nh, and therefore c
divides am^ ± nh.
Thus, since 6 will divide 12 and 18 without remainder, it will also
divide any number such as 7x12 + 5x18, 11x12 — 3x18, 12 (or 1x12)
+ 7x18, 5x12—18, &c., /. 6., any number found by adding or subtract-
ing any multiples of 12 and 18.
97. To prove the Rule for finding the Greatest Common Di-
visor of tioo quantities.
First, let the two given quantities, denoted by a and 5,
have neither of them any monomial factor.
Let a be that which is not of lower dimensions than the
other; and suppose a divided by Z>, with quotient j!9 and re-
mainder c\ bhj c, with quotient q and remainder d, &c.
b) a {p 546) 672 (1
2)b_ 546
c)l){q 126) 546 (4
' qc_ 504
d)c{r 42)126(3
rd 126
Then, by (96), all the common divisors of a and Z>.are also
divisors of a—ph or c, and are therefore common divisors oT
J) and c; and conversely, all the common divisors oib and o
are also divisors oi p)b-{-c or a, and are therefore common di-
visors of a and h. Hence it is plain that b and c have pre-
cisely the same common divisors as a and b.
Troof of tlie Rule for the g.c.d. of Polynomials.
GREATEST COMMOI^ DIVISOR. ^73
111 like manner it may be shown that c and d have the
same common divisors as h and c, and therefore the same as
a and h.
And so we might proceed if there were more remainders,
the quantities* a, d, c, d, &c. getting lower and lower, yet still
being such that a and 1), h and c, c and d, &c. have the same
common divisors.
But, if d divides c without remainder, then d is itself the
greatest quantity that divides both c and d\ that is, d is the
greatest of the common divisors of c and d, and therefore is
the Greatest Common Divisor of a and h.
Thus, in the numerical example, the common divisors of 546 and 672
ax-e precisely the same as those of 126 and 546, and these again are the
same as those of 42 and 126 ; but 42 is the g.c.d. of 42 and 126, and is
therefore the g.c.d. of 126 and 546, and also of 546 and 672.
(See Venable's Arithmetic, Arts. 82, 83.)
98. If the original expressions contain a common fiictor F,
which is obvious on inspection, then this factor F will be a
factoi of the g.c.d. We strike it out from both the quanti-
ties and apply the rule to the resulting quantities. The
G.c.D. ihus found must be multiplied by F to get the g.c.d.
of the original quantities.
99. If either of the quantities contain a factor which is
obviously not a factor of the other, this must be struck out,
and the g.o.d. of the resulting quantities is the g.c.d. of the
original quantities.
So, whenever we take a Eemainder for a Divisor in apply-
ing the rule, we may strike out any simple factor it may
contain.
100. Again, if after having thus prepared the divisor, at
any step of the process we find that the first term of the divi-
dend is not exactly divisible by the first of the divisor, then,
other features of the process of finding the g.c.d.
4
74 ELEMEXTAET ALGEBRA.
in order to avoid fractions in the quotient, we may multiply
the whole dividend by such a simple factor as will make its
first term so divisible.
In general, we may divide the divisor hy any cxjjression
wliicli has no factor commo7ito the hoo quantities luhose G.C.D,
we are seeking ; or tue may multiply the dividend hj any ex-
pression luhich has no factor common to the divisor.
Ex. Find the g.c.d. of
^x'-^x^'-^l^x'-^x' + ^x and dx'-Qx'^-^x.
Here, striking out of the first the factor 2x (which is com-
mon to all its terms), and of the second the factor 3:r, we re-
duce the quantities to x^—4:X^-\-Gx^—4,x+l and ^^— 2^?;^+!;
but as 2x and 3x have themselves a common factor x, it is
plain that the original quantities have a common factor x,
which these latter quantities have not; hence the g.c.d. of
these, when found, must be multiplied by x to produce that
of the given quantities.
x''-2x' + l)x'-4:x'+6x'-4.x+l{l
-ix\-
- 4a;' + 8a;' -4a;
x'
-3a;+l
x"-
-2x+l)x*-
-3a;'
+ l(a;'-l-3a; + l
X*-
-2a;'
^-x'
%x'-
^3a;' + l
3a;'-
-4a;'+3a;
a;'— 2a; + l
a;'-3a;+l
In this example, the first remainder is reduced by dividing
it by —4a:; and, the g.c.d. of these two quantities being x^—
2a; 4-1, that of the two given q? antities will be x (.t' — 2a;+l)
or x^—2x^ + x.
GREATEST OOMMOK DIVISOll. 75
Ex. Find the g.c.d. of
Qx'y + ^xif — ^y^ and ^x^-^4cX^y—4:X7f,
Stripping them of their simple factors 2y and 4a; ( ind
noting that these contain the common factor 2), we } ave
^x^-{-2xy—y'^ and ^x^ + xy—y"^, and proceed with these qi en-
tities as follows :
2
2x' + xy'-y')^x^ + 4.xy—2y\^
6^jy3xy-3f
y)xy+y^
x + y)2x^ + xy—y^{2x—y
2x^ + 2xy
—xy—y"^
The G.C.D. then will be 2 (a; + y) ; it being plain that the
G.C.D. of 2{^x'-[-2xy'-y^) and 2x'-\-xy—y'' will be the same
as that of dx^-^^xy—y"^ and ^x^ + xy—y"^, because the 2 intro-
duced into the first is no factor of the second quantity.
Examples — 17.
Find the g.c.d.
1. Oilox''-x-Q and ^x'-Zx-^,
2. Of 6a;'-a;-2and21a;'-26a;' + 8a;.
3. Of 2a;' + 6a;' + 6a: 4-2 and 6a;'' + 6a;' -6a; -6.
4. Of 2/- 10?/' + 12?/ and 3?/*- 15/ + 24?/' -24.
5. Of x' - 6ax' + 12^'a;— 8^' and x' - 4a'.a;'.
6. Of 2x' + 10a;' + 14a; + 6 and x' + x'-}- 7x + 39.
7. Of 3a;^ + 3a;'-15a; + 9 and e3a;^ + 3a;^-21a;'-9aj.
76 ELEMEJS'TARY ALGEBRA.
9. Oi2a' + a'h-4.a'h''-Uh' and W ■\-a'h-2a''lf + ab\
10. OiM'-\-16a'h-'da'h''-lba'h'
and 10a'-^0a'h-lQa''¥-\-'d0al\
11. O^x'-^x'y + ^xy^-y' and x'-2x'y^^x''y''-'^xif + y\
12. Of a;'+6a;' + ll^'+42;-4and2;'+2a;'-5:?;'-12a;-4.
XIII. Least Common Multiple.
101. When one Q\ndin.i\ij contains another as a divisor with-
out remainder, it is said to be a multiple of it ; and a common
multiple of two or more quantities is one that contains each
of them without remainder.
Thus, Qx^y is a common multiple of 2a^^, Zxy, 6.2;^, &c., and any quan-
tity is a multiple of any of its divisors.
102. The Least Common Multiple of two or more alge-
braic expressions is a term not appropriate if we use it in the
arithmetical sense. We must understand it to mean the
quantity of lowest dimensions which is exactly divisible by
these expressions. As in Arithmetic, we will use the letters
L.C.M. for shortness.
103. To find the l.c.m. of simple expfressions or monomials :
Fi7id by Arithmetic the l.c.m. of the nwnerical coefficients ;
after this number put every letter luhich occurs in the expres-
sions, and give to each letter respectively the greatest exponent
lohich it has in the expressions.
Ex. Find the l.c.m. of IQa^'bc and 20aWd. Here the l.c.m.
of 16 and 20 is 80. The letters which occur in the expres-
sions are a, b, c, d; and their greatest exponents are 4, 3, 1,
and 1. The required l.c.m. is, therefore, 80a*b^cd.
Meaning of Least Common Multiple in Algebra. Rule for l.c.m. of Monomials,
LEAST COMMO]Sr MULTIPLE. 77
Ex. Find the l.c.m. of ^a^h'c'x'y^, l^a'hcxY and IMc'x'if,
Here the L. c. m. of the numerical coefficients is 48. The
letters which occur are a, h, c, x, y, and z] and their greatest
exponents are, respectively, 4, 3, 3, 5, 4, and 3. Thus we
obtain ^^a^lfc'x^y^z' as the required l. c. m.
104. We shall now show how to find the l.c.m. of two
compound expressions or polynomials.
Let a and h represent the two quantities, d their G. c. D. :
and let a~pd, h—qd, so that jt? and q will have no common
factor. Then the least quantity which contains p and q will
be pq, and therefore the least quantity which contains pd
and qd will \}q pqd, which is consequently the l.c.m. required
of a and d,
Bince pqd=- — --^=— — -, it appears that the l.c.m of
a and b may be found by dividing their product iy their
G.c.D. ; or, which is more simple in practice, by dividing
either of them by their g.c.d., a7id multiply i7ig the quotient by
the other.
For example: required the l.c.m. of 2;"^— 4x+3
and42;'-9:c'^-152;+18.
The G.c.D. is x—'d] see Art. 95. Divide x'—4zX-\-^ by
x—o\ the quotient is x—1. Therefore the l.c.m. is
{x—l){4:X^ — ^x'^—lhx-\-l'^)\ and this gives, by multiplying
out, 4.x'-l'dx'-Qx''-\-?,dx-lS,
It is, however, often convenient to have the l.c.m. ex-
pressed in factors, rather than multiplied out. "We know
that the g.c.d., which is ^—3, will divide the expression
-ix^'—^x^ — lbx + l^] by division we obtain the quotient.
Hence the l.c.m. is
{x-Z){x-l){4.x^-dx-Q).
105. The principle of the rule in Art. 103, with the aid of
L.C.M.. by Inspection.
78 ELEME]S"TAEY ALGEBRA.
tlie Chapter on Factors, will enable ns in many cases to de-
termine, by inspection, the L.C.M. of polynomial expressions;
as we liaye only to set doiun the factors luMcli compose them,
each affected with the highest exponent ivhich it has in the ex-
pressions ; and the product of these is the Jj.cm, required,
Ex. 1. Find the l.c.m. of ^hx, Qahxy, dacx.
Here the factors are 2bXf day, c ; and the l.c.m. is 6abcxy,
Ex. 2. Find the l.c.m. of 2a\a \-x), 4:ax {a—x), 6x''{a+x),
Here the l.c.m. of the simple factors is 12aV, and that of
the coniponnd factors is a^—x'^; therefore the l.c.m. required
is 12a'x'{a'-x'),
106. Every common multiple of two quantities A and B, is a
multiple of their l.c.m. For, let M denote the l.c.m. of A and
B, and ]N" some other multiple. Suppose that if possible when
!N" is divided by M there is a remainder K ; let ^^ denote the
quotient. Then,
Isr=M^ + E, and E=K-M^.
J^ow since A and B divide both M and IST, they divide, also^
IST—Mg or K Therefore R, which from the nature of di-
vision is of loioer dimensions than M, is a multiple of A and
B less than the l.c.m. This is absurd. Therefore there can
be no remaiiider E. That is, N is a multiple of M.
107. Hence to find the l.c.m. of several expressions, we may
find the l.c.m. of two of them ; then find the l.c.m. of this
first L.C.M. and the third expression, and so on.
Examples — 18.
Find the l.c.m.
1. Of ^a%c and Qali^c', of ^x^y and 12xy^ \ of axy
and a{xy—y'^) ; of ab + ad and ab—ad,
L.C.M. of more than two expressions.
( VNIVERSITY
^LE. 79
2. Of ^a% l^a% and 12a'Jf', of a% ba% lQa'h\ 10d'b%
bah\ and h" ) of ^x", 6ax, 8a^ 36x% dax", 60a%
and 24a'.
3. Of 2(a + Z>) and S{a'-b'); of 4(a'-a) and 6{a' + a)\
of 6(:^;' + :^•^), S{xy-y'), and 10(cz;'^-/).
4. Of 4(a'-aZ»^), 12(aZ)^ + Z^'), S{a'-a'b);
and of 6{x''y + xy^), 9{x^—xy''), 4z{y^+xy'').
5. Of a;'-3a;-4, :i;'-a;-I2.
6. Of i^H5.^•H7^^-2, ic^ + 6cc + 8.
7. Of 12a;^ + 5^-3, Qx'+x'-x,
8. Of a;*-' - 6a;' + 11:^-6,2;'- 92;' + 26a;-24.
9. Of x'-7x-6, a;''+8a;' + 17a; + 10.
10. Of x' + x' + 2x' + x + l, x'-l.
11. Of 4a^Z>'c, 6«^V, -^Sa'M.
12. Of 8(a'-^>'), 12(a + ^)', 20{a-'^)\
13. Of 4(a + ^), 6(a'-^'), 8{a' + ¥),
14. Of 15(a'Z>-a^'), 21(a^-a&'), 36{ab' + b').
15. Of i?;'-l, c?;^+l, :?;^-l.
IG. Of x'-l, x'+l, x' + l, x'-l.
17. Of x'-l, x' + l, x'-l, x' + l.
18. Of a;' + 3a; + 2, x' + 4:X-i-d, x' + 5x + 6.
80 ELEMENTARY ALGEBEA.
XIV. Feactions.
108. Algebraical fractions are for the most part precisely
similar, both in their nature and treatment, to common
arithmetical fractions. Hence, the student will find the
rules and demonstrations in the chapters on Fractions are
little more than a repetition of those with which he is
already familiar in Arithmetic.
109. The expression -7-, we have agreed shall denote that
a is divided by h. We now say -j- means that the unit is di-
vided into 1) equal parts, a of w^hich are taken : a is called
the numerator, and h the denominator, and the expression -j-
is a fraction. (We shall show, as in Arithmetic, that a frac-
tion does also express the quotient of the numerator divided
by the denominator.)
Every integral quantity may be considered as a fraction
whose denominator is 1. Thus,
a—-, o + c=—-—
110. llo multiply 2b fraction by an integer: Either multi-
ply the numerator, or divide the denominator hy the integer ;
and conversely, to divide a fraction by an integer : Either
divide the numerator, or multiply the deoiominator ly. the in-
teger.
Thus, -Xx=— ; for in each of the fractions ^, -^, the
00 ho
unit is divided into h equal parts, and x times as many of
them are taken in the latter as in the former ; hence the lat-
Algebraic Fractions. To multiply or divide a fraction by an integer.
FRACTIONS. 81
ter fraction is x times the former, that is, — =— -Xcc; and,
ax
by similar reasonino:, -^-^ccr=-
A^ain, —-^x—-i-\ for in each of the fractions -r-, t-» the
^ b bx' b bx^
same number of parts is taken, but each of the parts in the
latter is -th of each in the former, since the unit in the lat-
X
ter case is divided into x times as many parts as in the
former ; hence the latter fraction is -th part of the former,
that is, -—-—--^x\ and, similarly, -—Xx=-,
111. If any quantity be both multiplied and divided by the
same quantity, its value will, of course, remain unaltered.
Hence, if the numerator and denominator of a fraction be
both multiplied or divided by the same quantity, its value loill
remain unaltered,
^, a ax a^ ,, , a3 a ac ^
ilius, -T- =-7-=-T =<!^c., and -— - = -— — =&c.
b bx ab a be c c
This result is of great importance, and many of the op-
erations in Fractions depend on it.
112. To reduce an integer to a fraction with a given de-
nominator: Multiply it by the given de^iominator, and the
product will be the numerator of the required fraction.
The truth of this is evident from (110).
Thus, a expressed as a fraction with denominator a; is — ;
.,, , • . 7 . ctb—ac
or with denominator /9— c, is -^ .
b — c
113. Since a — —, and therefore a divided ]dy b =~-^b=--r
1 -^ 1 b
(109), it follows, as stated in (108), that a fraction represents
To reduce an integer to a fraction with a given denominator.
4«
82 ELEMENTARY ALGEBRA.
the quotient of tlie numerator by the denominator. In fact
we get -y-th of a units (or a-^h) by taking — th of each of
the a units, and this is the same as a such parts of one unit
which we haye expressed (108) by -^.
1 3
Thus, in Arithmetic, — of $3 is the same as — of $1.
III. The demonstrations giyen in the preceding Articles
are based on the assumption that eyery letter denotes some
2)ositive luliole numler. By the Rule of the Sigjis established
in Multiplication and Diyision, we haye the following :
c^' (^ CIO ^ ... ^ ^ ^ a —a
bmce — = — , by puttmg —1 tor c we haye — = — -,
Hence, we may change the signs of all the terms in loth the
numerator and denominator of a fraction without altering
its value,
^ x^—^ax—a^ . ., ^. , .,, a^-\-2ax—x^
Jix. — ^— is identical with ^ — .
dax—x X —3ax
^ ^ a ^ —a a
So also, -^=+-^-=-^5
—b ~ h ~ h'
In like manner, by assuming that -^-X^? is equal to -j-,
whateyer be the sign of c, Ave obtain such results as the fol-
lowing :
115. If the numerator of a fraction be of lower dimen-
sions than the denominator, the fraction may be considered
in the light of a proper fraction in Arithmetic : if the nu-
Changing the signs of tlie terms.
FRACTIOI^fo. 83
merator be of liiglier dwiensions than the denominator, it
may be considered in the light of an improper fraction,
which (Art. 112) in Algebra, as in Arithmetic, may be ex-
pressed as a mixed quantity by the rule :
Divide the nunierator hy the denominator as far-as the di-
vision is possible, and annex to the quotient a fraction having
the remainder for numerator and the divisor for the denom-
inator.
Thus,
24a „ , M
- — aJr^O-
= — — M. -r -vt/ 7.
x'-dx-V^ x'-dx + 4:' x'-3x-\-4:
This last step the student should particularly notice, as
an example of the use of brackets, namely :
+ (-x + 2) = -{x-2).
Examples — 19.
Express the following fractions as mixed quantities.
25x 36ac+4.c. Sa^' + Sb,
7 9 4flj
, nx'-hy ^ a;'4-3.'c+2 ^ 2x'-6x-l
6x x+3 x-3
^ x'-\-ax'-3a:'x-3a' ^ x'-2x''
x—2a ' ' x^ — x + 1'
To reduce improper fractions to whole or mixed quantities.
84: ELEMEKTAEY ALGEBRA.
x—1 x + 1
Multiply
■,•, 4a' ^, ,„ 8(0.' +b'') , „, ,,
11. g^ by 35. 12. A^_^by3(a-5).
Diyide
15. -^— by 2a;. 16. —r- by 3a— 2^.
18. ^^hjx^^x+1.
XV. Eeduction of Feactions.
116. The result contained in Art. 110 will now be applied
to the reduction of a fraction to its lowest terms, and the re-
duction of fractions to a common denominator.
117. Eule for reducing a fraction to its lowest terms:
Divide the numerator and denominator of the fraction hy
their Greatest Common Divisor,
For example : reduce ^7—^,7-^ to its lowest terms.
%i — ■ ■
To reduce a fraction to its lowest terms.
REDUCTIO^q^ OF FRACTIONS. 85
Dividing both numerator and denominator by AiCi^h'^, which
4i:ClC
is their g.c.d., we obtain for the required result -^j-. That
is, -^j-j is equal to KTrrfJl {\^^)y ^^^ i^ is expressed in a more
simple form.
Ao:ain : reduce -j—, — — ,\^ — — -r to its lowest terms.
Dividing both numerator and denominator by x—'d, which
x—1
is their g.c.d,, we obtain for the required result j-ti-q^ — F
^X ~\~ ijX — o.
118. In many examples we may apply the results ol the
chapter on Factors, and strike out the common factors
from the numerator and denominator without using the
rule for finding the g.c.d. ; or rather, we may ly mere in-
sjpedion find the g.c.d. (Art. 93), and strike it out from the
7mmerator and denominator.
Ex. 1.
o^x^y'^ _ a^x^y^ _^ axy
a^xy + axy"^ axy {a + y) a + y
^ a^-{-x^_{a-\-x) (a^—ax+x^) a^—ax^-x^
a —X {a+x) {a—x) a—x
Ex 3 t±^±lJ^^li^J^J^±l
a;' + 5:^ + 6 (2: + 3) {x±2)'~x-{-'-Z'
x'^x''-\-^x-^ _ {x-l) {x'-\-2x^b) _ x'' + '^x- \-5
^^' • x'-^x-\-'d ~~ {x-1) {x-d) ~ a;-3"~"*
-p, ^ {a—hy—c^_{a—h-\-c){a—h — c)_a — h-^c
a'—{h-\-cy~\a-^b + c){a—b — c)~a + b-\-c
119. Kule for reducing fractions to a common denominator :
Multiply the numerator of each fraction ly all the denom*
To reduce a fraction to its lowest terms by Inspection. To reduce fractions
to a Common Denominator.
86 ELEMENTARY ALGEBEA.
inators exceipt its own for the numerator corrcsjjonding to that
fraction ; and muUijply all the denominators together for a
common denominator.
The truth of this rule is eyident; since the numerator and
denon\inator of each fraction are hoth multiplied by the same
quantities (viz. the denominators of the other fractions), its
Taluo will not be altered, though all the fractions will now
appear with the same denominator.
For example : reduce -r ? -^ ? and -r , to a common denom-
mator.
a_adf c _chf e__ebd
iThdf 2~dbf'f~fkV
Thus, T-j-^, -7^, and -7^-7 are fractions of the same value,
odf dof fbd
respectiyely, as -tj-^? and — , and they haye the common
denominator hdf.
We often wish to reduce fractions to their lowest common
denominator, which the aboye process will not effect if there
ai-e any common factors in the denominators. It is there-
fore often conyenient, as in Arithmetic, to use another rule.
120. Eule for reducing fractions to the lowest common
denominator
Find the l.c.m. of the denominators. This ivill he the Lowest
Common Denominator. Then for the new numerators multi-
ply the numerator of each of the given fractions hy the quotient
ivhich is obtained by dividiyig the l.c.m. by its denominator.
For example: reduce — , — , — , to-their lowest common
yz zx xy
denominator. The l.c.m. of the denominators is xyz ; and,
To reduce fractions to their LoweBt. Common Denominator.
EEDUCTIO]^ OF FEACTIOKS. 87
a ax 1) hj c cz ^, , _ ^
— = — , _— -^ _— — . 'ii-^Q numerator and denom-
yz xyz zx xyz xy xyz
inator of each fraction has been multiplied by the same ,
quantity.
Examples — 20.
Eeduce the following fractions to their lowest terms.
l^a'lfx ^ a^ah a^ -\- ah
ISa'b'y T^* '^F^^b'
^ l Oa'x ^ 4:{a+I)y ^ a' + F
6a'x-15ay'' b(ce^}f)' d'-V'
x^-VZx ^l a;'^ + 10a;+21
22;' + ^^'— 15 x^^-{a-^l:))x-\-db
2.r' — 192; + 35* ' ' x^-\- {a^c)x-\-aG
x^-{a^h)x-\ -db 3^^3^_36
• x^-V{c-a)x-ac 4a;^+33i^;-27*
\x-\-hY-ia-VcY x'-^x^\^'
^^ x^—l^x-^'^X ^^ :z;' + 92; + 20
Id. —^ — — — -— . 16. -^
x^-\JoX'-V: ' x^ ^Ix^ ■\-\^x-\-'S
x' + aV + a* x'^-y
^' - « . 1«. ^2«y+l-
19.
X —a
x^—l x^-a"" a^ — V x^ — lfx d^ — al-Vax—lx
ax-^a' x'—aV d'-b'' x" -{•^hx + b'"' d'-\-ab-\-ax-\-hx
Eeduce the following fractions to their lowest common
denominator.
3 4 5 ^ 9. rr
20. ^, ^,^. 21.
^x' 6.^^' 12x'' x + V 4.'r4-4' :r'-l*
88 ELEMENTARY ALGEBRA.
fl^ X a^ ax
x—a' a—x :a
a l ah
24. —
1 a; 3 4
25.
26.
(x-iy {x-iy x+v {x+iy x'-i'
a a-\-x ax
x—a' x^-^-ax + a'"' x^—a^'
x^—ax-^a"' x''-\-ax + a'" ic^ + aV+a*'
XVI. Addition and Subtraction of Fractions.
121. To add or subtract fractions: Reduce tliem to a com^
men denomi7iator (if necessary), a7id add or subtract the
numerators for a neiu numerator and retain tlie common
de?iominator,
Ex.1. Add ^ and ^.
TT ,1 . a-\-c-\-a—c 2a
Here the sum is ^ =-7-.
Ex. 2. From take .
c c
^a-3h da-'4 :I?_ 4.a-3h-{3a-U) _ 4:a-3b-Sa + ^ b
c c ~~ c ~~ c
a + b
To add or subtract fractions.
ADDITIO:^" AISTD SUBTRACTIOiq" OF FRACTI0:N^S. 89
Ex. 3. Add -^ and "^
Here the common denominator will be the product of
a-^h and a — h, that is, a^^lf,
c __c(a—b)^ c _c{a + h)
Therefore, ^^ + -^='A^3^±^^
a + b a—b a -o"
_ca—cb-\-ca-\-cb__ 2ca
^ . _ a + b . , a — b
Ex. 4. From 7 take
a—b a+b'
The common denominator is a^—b^,
a + b_{a+b)\ a-b_{a-by
a-b~ a'-b''' a-^b" a'-b''
Therefore, ^"^^ a-^b_{a + by-{a-by
a—b a + b oi^—V
_ y + 2ab+F-{a'-2ab + b') _ iab
~ a'-b' ~a'-r
Ex. 5. Add -4^4-,+ ^ ^"^ , .
l-{-x + x l—x-\-x
{1 + x) {l-x^-x'')-\-{l-x) {l-^-x^-x"") __ 2
Ex.6. From . "^"^^ , take "^"^
l + x-\-x' 1 — x+x^'
{1 + x) (l-x-\-x')-{l-x) (1 + ^; + :^:') _ 2x'
{l + x + x'){l-x + x') '~l + x' + x*'
90 ELEMENTARY ALGEBRA.
122. AYe haye sometimes to reduce a mixed quantity to a
fraction ; this is a simple case of addition or subtraction of
fractions.
For example :
h a h ac I ac-\-h
c 1 c c c c
h __a h _ac h _ac—h
c~ 1 c~ c c~ c
Hence,
Multiply the entire part ly the denominator of the frac-
tion, add to or suMract from this the numerator of the frac-
tion, and place the result over the denominator,
-, ^ . 2«Z> a 2«Z> aia + i) 2ab a^ + 3ab
Ex. 1. a-\ 7=-T-H r-7=— r^ H 7= t— •
a + o 1 a+o a^-0 a-\-o a + o
x—2 x+3 x-2
Ex. 2. x + 3
x''-3x + 4. 1 x'-3x + 4:
_ {x + 3)(x'-3x + 4:) x-2
'~ x''-3x-\'4: x''-3x-\-4.
_ x^'-^x-\-12-{x-2) _ x^—bx + 12'-x-^ 2 _x^ -^x + l ^
Ex 3 1 ^'+^'-^ '_ ^^^ + ^' + ^' -^' _ ('^+^)'-^'
2ah ~~ 2ah ~' 2ab
_ {a + d + c)(a + h—c)
"~ 2ab
^ . ^, XT, 4. -, a^' + h^-c^ {a~d + c)(I)-a + c)
Ex.4. Snow that 1 ^r-^ ==^ -zr^ :
2ao 2ao
To reduce a mixed quantity to a fraction.
ADDITION AXD SUBTRACTIO:^" OF FRACTIOXS. 91
Ex. 5. Show that
a''^]f-c'\\_{a-^l) + c){a^-l)-c){a-\-c-l) {b + c-a
w
The attention of the student is again called to the fact
that the line which separates the numerator from the de-
nominator of a fraction is a vinculum or bracket. Hence,
he will apply the rules of brackets, Arts. 51 and 55.
123. Expressions may occur involving both addition and
subtraction.
Ex. Find the value of 2 + ^^^,'-^.
Ans 2(a^-~^^) + (^'4-y)-(a-Z>)(^-^) ^ci'-^'^al-W
Examples — 21.
Find the value of
a {a—l) a (a + I?) ^ Sa—4:b 2a—h—c 15a— 4c
Yb'^oTb) ' Yb'^3 (a-h) ' ~"2 3 '^~T2~"'
a
a b a b a—b ah
^' a-b ^' a + b'^ a-b' a-b a + b' a + b'^ a'-b'
a (ad—bc)x^ a^ + b"^ a—b 2x'^—2xij+i/^ x
c c{c+dx) ' c^—b^ a-\-b^ x^—xy ^—y'
1 1 ^ . ^ — ^ C'—a b—c
•• 2{a-x)'^2{a-\-x) '^ a'-^-x" ' ~W^~^^'^~bG~'
^^ 1 1 1 _1_ 1 3a
2{x-l) 2[x-\-l) x'' ' 2a + b'^2a~b ^a^-V
92 ELEMEKTARY ALGEBRA.
a {a!' — lf)x a(a^ — l'')x^ ^ \_ 1 x —\
x^—y x—y x + y x + y x ~y^ x -j-y
11 ^_ ^+^ 12. 2 ^'"" ^ ' • ^^^'
13. -^ 7 r. 14. ^ ; V-f j.
a a(a—x) y x+y xy—y^
^^ X x^ x?
15. -r, ri- + -
1-x {1-xy "^ (1-^)'*
16. ^-v+. ' '-^
8(1-^) ' 8(1+^) 4(1 + ^')'
Bemarlc. — In the preceding examples we have combined two or more
fractions in a single fraction. On the other hand, we may if we please
break up a single fraction into two or more fractions. For example :
8Z>c— 4ac + 5a5__35c 4<xc ^ab 3 4 5
abc ~abc abo abc a b c'
but the beginner must not confound -; — with -r .
b—c be
124. The addition and subtraction of fractions can often
be much simplified by observing closely the factors of the
denominators, and avoiding unnecessary multiplications in
reducing the fractions to a common denominator.
Ex. Find the value of
a l c ^
{a—d){a — c) {b~c){b—a) {c—a){c—'b)'
Here the beginner is liable to take the product of the de-
nominators for the common denominator, and thus to ren-
der the operations laborious.
How tlie addition and Bubtraction of fractious may often bo simplified.
ADDITIOIs^ AKD SUBTEACTIOiT OF FBACTIOXS. ^3
The second fraction contains tlie factor l—a in its denom-
inator, and this factor differs from the factor a — l), which
occurs in the denominator of the first fraction only, in the
sign of each term ; and by Art. 110 :
1)_ I
{b-c){b-a)~ {b-c){a-l))'
Also, in the denominator of the third fraction, by the
Eule of Signs we have,
(c — a){c—'b) — {a— c) {b—c).
Hence, the given expression may be written,
a b
+ -
{a-b){a-c) {b-c){a-by {a-c){b-c)'
And in this form we see at once that the L.C.M. of the de-
nominators is {a'-b){a—c){b—c).
By reducing the fractions to this lowest common denomi-
nator, we get
a{b—c) — b{a—c)-\-c{a—b) ab — ac'-ab + bc^-ac—bc_
{a—b){a—c){b—c) ~~ {a—b){a—c){b — c)
Examples — 22.
Find the value of
a
1.
{x—a){a—b) {x—b){b—ay
b'
3,
{x-a){a-b) ^ {x-b){b'-a)
1 1
{a-b){a-~c)'^ {b-a)(b-cy
^4 ELEMEIS-TARY ALGEBEA.
a
^' {a-~b){a-c) "^ {h-a)(})-G) '
6. . .^. . -f- ^
7.
a{a—b){a—c) h{b—a){h—c) abc
{a-b){a-c) "^ {b-a) {b-c) "^ (c-a) (c-b)'
XYII. Multiplication of Feactions.
125. To multiply fractions together;
Multiply the numerators together for a neiu numerator^
and the denominators for a neiu denominator.
Suppose that we have to multiply -r hy -r : let — =0?,
/>
-j—y\ :. a—bx, c—dy, and ac=bdxy\ hence (dividing
a
ac a c
each of these equals by bd), j-^^xy, but xy—~y,—^ and
ac aXc product of numerators , ,. , ,,
T-i=Ti — i^-^^—j — T-i-^ — 7 — y whence the truth of the
bd bXd product of denominators
rule is manifest.
Similarly we may proceed for any number of fractions.
a^b a-b 3_ 3(a+^)(o^~^) dja'-V)
^^' c + d^c-d^2~2{c-\-d){c-d)'~'2{c'-d')'
126. The Kule of Signs (Art. 61) gives the following re-
sults in the multiplication of fractions :
To multiply fractions together.
MULTIPLICATIOI!^ OF FRACTIO^^S. ^ft
a c _a — c — ac_ ac
a c ~ a c — ac ac
h^ d~ b ^ d~ bd ~ bd'
a c — a — c_ac
b d~ b d ~~bd'
127. We shall now give some examples. Before multiply
jng the factors of the new numerator together, and the
factors of the new denominator together, examine if any
factor occurs in both the numerator and denominator; in
which case it may be struclc out of both, and the result will
be more simple. Art. 116. (See method of cancelling in
Arithmetic — Vulgar Fractions.)
Ex. 1. Multiply a by — .
c
a a b ah
Hence a— and — are equivalent; so, for example,
. X 4:X T 1 /^ 0\ ^^ — 3
4— =r-^; and — (2x— 3) = — - — .
5 5 4^ ^ 4
Ex. 2. Multiply - by ^.
X X _xXx_x^ ^
y y'vy^'f'
thus (— ) =-a.
^yJ y
Application of the Rule of Signs. Cancellation,
96 eleme:n'taey algebka.
Ex.3. Multiply I? by |.
Sa Sc_3aXSc__2cXl2a_2c
4:b 9a~4:bx9a~dbxi2'a~3b'
Ex. 4. Multiply , , ,,, by -^—z.-^-
(a+^)'^ 3ab ~ b {a + b)Xda {a + b ~b{a-\-b)'
^ ^ a ex a c ac
Ex. 5. tX-t=-tX-j=j--,*
bx a b a bd
^ 5fl^.T xy-^-if^ _J)a {x-^ij) _ hax 4- 5^-?/
3c;z/ x'^~xy~'3c{x—y)~ 3c{x~yy
-^ 4fl^a; a^ — o;^ ^c + bx _ 4:r ((7- + x) _ 4rir.T + 4^^
3bt/ c^—x^ a^—ax~Sy{c—x)~oy{c—xy
Bemark. — The student slioulcl leave the denominators of fractions
■ with their factors unmultipUed^ as in Ex. 6 and 7, unless they combine
very simply. The convenience of this will be found in practice.
Ex. 8. Multiply 4 + -+1 by -^+--1.
^ '' b a '^ b a
b a ~ab ab ab~' ab ^
a b -, _^' ^' ab ^a^-^-b"^ —ab
b a ab ab ab~ ab
^_±^±ab a'-\-b'-a b_{a''-\-b'' -\-al)) (a' + b'-ab)
ab ab "~ a'^b^
__ {a' + by -a'b'_ a' + b'-^a''b'
"" a'b' - a'b' •
MULTIPLICATION OF FRACTIONS. 97
Or we may proceed thus:
fa h ^\ f a h ^\ fa hy" ^
therefore,
fa h ^ ^\ fa I , \ a\ ^ h^ , a" h\ ^
\ a J \h a J a la
The two results agree, for -p,-\ — ^ +1 =
2 7,2
Examples— 23.
Find the yalue of the following :
M ^hc ^ h^ c^
ob oa 00 ac ah
a^h Vc c'a . x-{-l x-h2 x—l
0» 2 X ^ X "^ • ^» T X 2 -1 X V ■ L-i \ i»
i?;i/ y z zx ^—1^—1 (a;4-2j
cc + a Va xy \ b / \ a/
'■ (»+^J (*-^)-
cc(6i^— cc) a(a + x)
2X-
a^ + 2a:?; + ^^ ^^ — 2ax + 2;^
^'— «/• ^' + ^' ^ + y
x'—(a-^h)x-}-ah x^—c^
J.l/. -~ '^ I r - X 2 72'
X —{a + c) x + ac x—b
5
98 elemejs^taky algebra.
X -\-y \x—y x-\-yJ
13. f« i_4_i)xfi — ¥_).
\oc ac ab a) \ a + o + cj
^^ fx"" a" X a . \ fx a\
13. (-T+— + l)x( .
\a X a X J \a X J
^ . f X a tf h \ ( X a y b\
14. ( + ^ )X — T + -)-
\a X b y J \a x by)
x^—%x-\-\ x''—4x_j-^ x^ — 6x+9
' x'-6x-{-6^x'-4.x+3^x'--3x-}-2'
XVIII. Division of Feactions.
128. Eule for diyiding one fraction by another :
Invert the divisor, and proceed as in Multiplication.
The following is the usual demonstration of the rule.
Suppose we haye to divide "t- by -7 ; put -j-=x, and -T=y'y
then, a = bx, and c=dy'y
and, ad=bdx, and bc—idy\
therefore.
ad bdx X
bc~~bdy~ y*
-n , X a c
y -^ b d^
., « a c ad a d
therefore, -r-f--7= r=irX— •
b d be b c
To divide one traction by another.
diyisio:n" of feactioxs. 99
129. The results giyen in Art. 125 give us the following in
connection with Division of Fractions :
o- ^ c ac ^ a c ac
b d M Id bd
, ac c a ^ ac c a
bd d b bd d b
• T . a c ac , ac c a
Also, since — 7- x — 7=^-75 we have :r-,-. -— — —.
b d bd bd d b
130. The student should, in the division of fractions, en-
deavor to simplify the operations as much as possible by
striking out factors which occur in both numerator and
denominator.
Ex. 1. Divide a by — .
__ a a b _a c __ac
Ex. 2. Divide ?f by I?.
3a_^9^_3^ 8£_2^
4.b'^SG~^b^'^a~W
Ex.3. Divide ^^, by ^'
{a-{-by ^ a'-b''
ab-y W ab-b' a'
{a+by ' a'-b'~{a + by^ b'
_ b (a-b) (a+b) {a'-b _ {a-by
"" b\d^by ~b{a'\-by
x^^xy ^ x'-if jc^^-xy ix-yy _ x
x-y (x-yy x-y x -y' x +/
100 ELEMENTARY ALGEBRA.
E]
5:amples — 24.
Divide
4.xyz' ^ 3xYz'
1 1 6{ab-I?') W
x'-y' ^ x-y a{a + bY ^^ a{a'-b'j*
, a'-4:x' , a'-^ax ^ 82;' ^ Ix"
a^4-4a:c "^ ax + 4a;^* * ^'— ^ x' + xy-{-y'^'
iz;^ + ^^ -^ x^—xy^y^'
o^-\-(a-k-c)x-^ac ^ x^—a^
x'+{h + c)x^c ^ ?^''
10. ?!i:^I&: by ^'-^^
a; 4-^ x—xy-\-y*
I. (i.|)(.-£)b,J^..
13. 5x^-4 by :r + -^. 14. a'-\hj a---.
15. — 5 by . 16. Sa-\ ^ by a; .
a X '' a x a x ^ x
x" 1 a: 1 1 _ a;\ ^ ^r' x ^ a
17. — , by-jH — . 18. -^+l + -,by 1+— .
y X ^ y^ y X a'^ ^ x^ ^ a ^ x
COMPLEX i^RACTIOXS. 101
XIX. Complex Fractions and other Eesults.
131. Hitherto we have supposed, in the chapters on Frac-
tions, that the letters represented whole numbers, but when
we come to interpret the multiplication of fractions we must
extend the meaning of the term, as we have done in Arith-
metic. Thus, to multiply "t- by -7, the fraction -j- is divided
into d equal parts, and c such parts are taken. Now if -7-
be divided into d equal parts, each of these parts is y-^; and
etc
if c such parts be taken, the result is j-y Then, too, to divide
a c
■j-^J -J "ineans to find a quantity such, that if it be multi-
C Cb
plied by -^ the product shall be — .
132. N^ow with our extended definitions we can easily
prove that all the rules and formulas given are true when
the letters denote any numbers ivliole or fractional. Take,
for example, the formula — =— , and suppose we wish to
show that this is true when
a——. 0—-, and c=-,
71 q s
^ ^ a _m ^ p __m q _mq
on q n p np
Also ac=^ — , and hc~ —,
ns qs
rpi ac _mr pr _vir qs mrqs wq
he ~ ns ' qs ~ ns pr ~ nsjjr ~ np '
Thus the formula is proved to be true.
-1
2-x
2
2-
2-x
4a; -
~ Ax '
~ 2
~ 8x "
102 ELEMEKTAKY ALGEBRA.
133. Complex fractional expressions may be simplified by
tbe aid of rules respecting fractions which have now been
given.
Ex. 1.
Hence obserye that, when a complex fraction is pnt into
the form of a 7; — -r — , the simple expression for it will be
traction ^ ^
found by taking the product of the upper and lower quanti-
ties, or extremes, for the numerator, and that of the two mid-
dle ones, or means, for the denominator ; and that any factor
may be struck out from one of the extremes, if it be struck
out also from one of the means.
Ex. 2. Ex. 3.
2x 20-ri;
2x
1 6x 6-ix 4 60 -3a:
^-3
~3aj-l~"3^-r x + l^~~dx+4.~4:{3x + 4:y
3 3
Ex. 4.
a+h a-h {a + hy+ia-hy 2a'+2h'
a-\-b a-b {a+by-{a-by Aab 2ab '
a-b a + b a'-b' a'-b^
Ex. 5.
n 1 .
0—a
Simplification of Complex Fractions.
COMPLEX FRACTIOJ^S.
1U3
3 — a~~d-~a 3—a~ d—a
3-a'
3 — a 4a 3— a 3-\-3a
a + --r- ^-r-] T— = -. .
, 3 + 3a 1 4 _ 4
• 4 ■~1^3+3a~3+3a'
Ex. 6. Find the value of ^ when x=
Here a—x=a~
and l—x—1)-
ab a^ + ah—ah a^
ab
"aVb
a-\-b a+b'
b^
= ab\-V—ab=^ r.
a-^b
Hence,
a—x a-hb _ a^
l^" b' ~J''
a + b
Simplify
1
1.
l-\-x
1 1 '
1 + X
1
X
Examples — 25.
':+.="
2.
l + iz; 1— a;
1— ^ l + a;
3x x—1
5.
2"*"" 3
-(.H-l)---2i
1-.
3.
1+-
a;-l +
x—6
X-2 +
x-6
104 ELEMENTARY ALGEBRA.
Find the yalue of
-, x—a x—b , a^
7. — ^ when x= ~.
o a a—h
TT when x=--}- -^.
a-\-b b{b + a)
8. -+T^
a o—a
9,
a'x-hh'ij . 2,^2
■ — ~ when a— — and h= — .
i?: + .^ 3 3
10. -— — — ^ ^— „ when t/ = — .
ic+y x—y X —y ^ 4
^^ x + 2a x—2a 4:ab . ^5
J-J-. ^77 +?rr"^ TU "2 when x= -.
2b — X 2b+x ^W—x^ a + b
^^ fx—aV x—2a + b , a + b
l/c. I — -1 ; — when 0:=— -— .
\x—bj x + a—2b 2
x+y—1 a-hl , ab + a
lo. ■ — — -— when xz= , ^ , and y=—, — -.
x-y + 1 ab-\-V ^ ab-hl
134. The following results should be noticed.
If —==—-. then
b d
^ , ci c b d ,.. abbe a b ,...
^--b=^-d' ^^ ¥=7«' 7X7=7X7' '' 7=7 (">'
a c a + b c-\-d ,....
CI' ^ ^ a—b c—d ,. .
a:^b b c^d d a:^b r
hence -7-X— ^-^-X— , or =
/; a an a
iXYOLUTio:Nr. 105
, a-^l h c-\-d d a + b c{-d, .^
and — T— X -j= — j-X 7' ^^^ 7= 7(^1):
a—o d c—d a—o c—d^ ^
and any of tliese last may be inverted by (i), or alternated
, ,... ,^ ct c a a^h a + b a—h .
by (n); tlms, -^^=y^^j, -=^-^, ^=^=5' &'^-
So that, If any tivo fractions are equal, ive may comhin^
hy addition or subtraction, in any ivay, the numerator and
denominator of the one, provided that ive do the same with the
other.
XX. Involution.
135. The jorocess of obtaining the poivers of quantities is
called Involution, A poiver has been defined to be the p)roduct
of two or more equal factors. All cases of Involution, then,
are merely examples of multiplication, where all the factors
are the same; and the rules given in the present chapter
follow immediately from the laws of Multiplication.
136. Any even p)Oiver of a negative quantity is positive.
Any odd power of a negative quantity is negative.
This is a simple consequence of the Eule of Signs.
Thus, —■aX—a—-\-a^', —aX—aX—a=-{-a''X-'a——a^'y
— aX—aX —aX —a=—a''X —a — -}-a*; and so on.
Let the student notice : 1. That any eve^i power of a
quantity is the same whether that quantity be negative or
positive. Thus ( + «)^ and { — ay are each=+^^; and
{ — {a + b)y and { + {a-\-b)y are each= + (a + ^)\ 2. ISTo even
power of any quantity can be negative, 3. Any odd power
of a quantity will have the same sign as the quantity itself.
137. The expo7ient of any poiver of a power is equal to the
product of the exponents of the tivo poivers.
Involution— Power— Signs of Powers.
5*
106 ELEME^^TAEY ALGEBRA.
Thus, tlie cube of a% that is, {ay=za'i foY,{ay = a'Xa'Xa*
Similarly, {ay = a''; {-ay=-a''; {-ay=z-a'', {a^Y
138. Eule for obtaining any power of a monomial ex-
pression :
Multiply the exponent of every factor in the expression hy
the index of the required power, and give the proper sign to the
result.
Thus, for example,
{a''l)y=a'h'] {-aWy=-a'b'; {abVy=a'bV';
{-a'bYy=-a''b'V'', {2ab'cy=2Vb'Y'^64:a'b''c'\
It is usual to raise the numerical coefficient at once to the
required power, instead of first writing it with an exponent.
Thus, ( - 2xy'z') ' = - 8xYz\
139. Eule for obtaining any power of a fraction : Eaise
both the numerator and deiiominator to that potver, and give
the proper sign to the result. This follows from Art. 122.
For example,
140. Some examples of Involution of binomial expressions
have already been given. Thus,
{a-\-by=a'' + 2ab + b\
(a-'by=a'-2ab+b\
By (137) we may shorten the operation, finding the 4th
power of a quantity by squaring its square ; and similarly,
to find the 6th, 8th, &c. powers, we may square the 3d, 4th,
&c. powers.
Rule for obtaining any power of a monomial exprebsion ; — of a fraction. In-
volution of binomial expressions.
IXYOLUTIOX. 107
So also to find the cube or 3d power, vre may take the
product of the quantity itself and its square ; to find the 5th,
we may take that of the square and cube, &c.
Thus we shall have,
(a-by={a'-2ab + b'){a-b)=a'-3a'b + 3ab'-b';
{a+by = {a' + 2ab+b'){a' + 2ab + ¥)=a' + ^a'b+ea'b*
+ 4.ab' + b';
{a-by={a'-2ab + b'){a''-2ab + b')=za*-'4.a'b + 6aV
^4.ab' + b';
{a+by={a + by{a-\-by=a' + 6a*b + 10a'b'-hl0a'b'-^6ab^
(a-by={a-by{a-by=a'-6a*b+10a'b'-10a'b'-\-5ab*
-b\
The student should remember the above results, though
the higher powers of binomial expressions are best obtained
by the Binomial Theorem, which we shall give subsequently.
It will be noticed in the above examples that any power
oi a—b can be immediately obtained from the same power
of a-\-b by changing the signs of the terms which involve
the odd powers of b.
141. The results of Art. 140, can readily be applied to tri-
nomial expressions,
Ex. 1. {a + b^cy=a'-Y2a{b^-c)^{b+cy
=a'' + 2ab + 2ac-{-b'-\-2bc-{-c\
Involution of trinomial expreseions.
108 ELEMEKTARY ALGEBEA,
Ex. 2. {a-\-l) + cy={a-\-{'b^-c)Y
Ex. 3. (a-^-c)^={^-(Z» + c)}^
==a'-3^'^-3aV + 3a5' + 6rtJc + 3r^c'~Z»'-3/;'c
-3Z^c^-6'^
Or thus :
(a-l-cy={{a-l)-cY^{a-l>y-^{a-'byc-{-^a-l)c^-c\
which, of course, when expanded, would give the same re-
sult as before.
Ex. 4. {^x-dy={2xy-4. . 3 . {2xy + 6 . 31 {2xy-^ . 3^(2a:) + 3*
Examples — 26.
1. Eind the yahies oi {2aiy, (-3a^^V)', (-^T.
Write down the expansions of
2. {x^%)\ 3. {x-'^y, 4. (^+3)^ 5. (l + 2a;)*.
6. (2m-l)'. 7. {3.r+l)\ 8. {9.x-ay. 9. (3.'?; + 2a)*.
10. (4a-3Z>)'. 11. {ax-yy. 12. (ax + rr^)'.
13, {%am-my, 14. (a-Z> + c)«. 15. (l-a:-f .t'*)'.
IXYOLUTIOX. 109
142. The square of imy 2)oIl/nG}mal expression may be ob-
tained by either of two rules. Take for example,
{a + b + c+dy.
We will find,
{a + b + c-hcl}''
= a' + Z>' + c' + cr + 2al) + 2ac + 2ad +2bc + 2bd+ 2cd.
We see from this — the square of any polynomial may be
found by setting down the square of each term., and then the
doiihle 2^rodiict3 of all the terms, talcen two and two,
iVgain, we may put the result in this form,
{a^l^c + dy
^a''-\-2a{l) + c + d)-Vl)'^2'b{c-Vd)+c'^2ci: vd^.
and this may be obtained by the following rule :
The square of any multinomial expression consist i df the
square of each term,, together ivith twice the product \ f cadh
term by the sitm of all the terms ivhich folloiv it.
Ex. 1. {l + 2x-\-^xy^l-\-2{2x + 'dx')+^x^-\-ix{^x'')-^.&J
= l-\-4cX-\-10x'-\-l2x' + ^x\
Ex. 2. {l-2xy=[{l-2xyY^{l'-^,x-\-12x''-^x'Y
=:l-12x-\-2ix''-Ux^
+ 144^^-1920;* f 64^"
= l-12.^' + 602;^-1602;^4-240.^'^-192a;* V^^x\
The square of polynomial expreseiona,— two rules.
110 elementary algebra.
Examples — 27.
Find
1. {a + h-^c-\-dy-{a-'b + c-d)\
2. {a + d-^c + dy+{a-b-\-c-dy.
3. {1 + x-hxy. 4. {l--x+xy. 5. (l+2;-ic^)\
6. {l-\-3x-h2xy. 7. {l-3x+dxy.
8. (2 + 3a;+4a;^)^+(2-3a; + 4:c^)^
9. (l_:c+a;^+a;y. 10. {l + 2x+3x'+^xy.
XXI. Eyolution.
143. Evolution is the inverse of Involution. Evolution is,
then, the method of finding the roots of quantities. It is usual
in this connection to use the word extract in the same sense as
find. Thus to extract the square root is to find the square
root,
144. It follows from (136) that—
1. Any even root of a positive quantity will have the douUe
sign db.
Thus the square root of a^ is ±a, the fourth root of a* is
2. Any odd root of a quantity has the same sign as the
quantity itself
Thus, for example, the cube root of a^ is a, and the cube
root of —a^ \^ —a,
3. There can he no even root of a negative quantity.
Hence the indicated even root of a negative quantity is
called an impossible quantity or imaginary quantity, ^ —a^,
^—a, "^ — 1, are imaginary quantities.
Evolutiou. Three Rules for the Signs of Roots. Imaginary quantities.
EVOLUTION-. Ill
145. Eule for finding any root of a monomial integral
expression. Extract the required root of the numerical coef-
ficient, divide the expo7ient of each literal factor hy the index
of the root, and give the proper sign to the result.
Since the cube power of a^ is a^, therefore the cube root of
a^ is a^, and so on.
Thus, for example, V {Ua^h')= V {^^a''h') = :^4.a'b\
V{-Sa'b'c'') = V{-2'a'bY') = -2aWc\
V{266xy) =V{4:'xy) = ±:4.xy\
146. To obtain any root of a fraction: Find the root of
the numerator and denominator, and give the proper sign to
the result.
For example, \/ {^)=\^ {-^)
2a
147. Suppose we require the cube root of a^. In this case the
exponent 2 of the quantity is not divisible by the index 3 of
the root ; then we cannot find the root of it, but can only
indicate that the root is to he extracted by writing it
thus, V^. Similarly, \f~^, v/^, V^, indicate roots which
we cannot extract. Such quantities are called surds, or
irrational quantities; the difference between surds and
imaginary quantities being that surds have real values,
though we cannot find them exactly, while there cannot be
a quantity, positive or negative, an even power of which
would produce a negative quantity.
Examples — 28.
1. Find the square roots of ^a^h'c\ 4t9xyz% 100a'b'Y\
Eule for finding any root of a monomial integral expression; of a fraction.
Surds, or Irrational quantities.
115> ELEMENTARY ALGEBIIA.
_ ^. , ., . . ^a'x\f 4.9xy mxY'
2. Find the square roots of ---4- , -777-^- , -ttt-ott-
27^;^'' ^125a^'^ ^ 343 *
3. Find v/-^^4^, V-^,, V^,T., V
*■ - v(w> i/(5^.> ♦/(^:->
148. To find the square root of a polynomial: We hnnw
that the square of a + ^ is a^-\-2ah-\-lf. Let us observe, then,
how from a'^ + 2ab-hh^ we may deduce its square root a + b.
This will lead us to a general method of finding the square
root of polynomial expressions.
a'+2ah + I)'{a+b
a^
2a + b)2ab + b'
2ab+b''
Arrange the terms according to the powers of one letter, a;
then the first term is a^, and its square root is a. Subtract
the square of a, that is, a^, from the whole expression, and
bring down the remainder 2ab + b^. Diyide 2ab by 2a and
the quotient is b, which is the other term of the root; lastly,
if we add this b to the 2a, multiply the 2a+b thus formed by
b, and subtract the product from 2ab + b'^, there is no re-
mainder.
Now we may follow this plan in any other case, and, if we
find no remainder, we may conclude that the root is exactly
obtained.
Ex.1. Ex.2.
9x' + exy+y\3x + y 16a'-56ab + 4:9b'{4.a-'7b
i)x+y) Qxy-Vy'' 8a-7Z>)-5G^/Z» + 49Z''
^xy-^y'' — 56«Z> + 49Z>'
To find the square root of a polynomial.
evolijtio:n'. 113
Ex. 3. ^a^-^ah-F^^a-l)
^4.ab + b'
-2b\
Here we find a remainder —25"^ ; we conclude, therefore, that 2a— b
is not tlie exact root of Aa^ —Aab—b'^ which is a surd, and can only be
written V^a'_4^ab-b-''
149. If the root consist of more than two terms, a similar
process will enable iis to fir 4 it, as in the following example,
where it will be seen that the divisor at any step is obtained
by doubliiig the quantity already found in the root, or
(which amounts to the same thing and is more convenient
in practice) by douhling the last term of the ])receding divisor,
and then annexing the neiu term of the root,
Ex. 162,-'-24^'+252;*-20a;'-fl0a;'-4::c+l(42.-'-32;^ + 2a;-l
—24:x'-{- Qx'
8x'-6x'' + 2x) 16x'-20x'-\-10x^
16x'-12x^ji^_i^
8x'-Gx'' + 4:X-l)- Sx'+ i5x''-4:X + l
- 8x '-\- 6x''-4:X-\-l
150. It has already been remarked that all even roots have
double signs. Thus, the square root of a'^-\-2ab + b^ may be
— {a+b), that is, —a—b, as well as a-\-b; and, in fact, the
first term in the root, which we found by taking the square
root of tt^ might have been —a as well as a, and b}^ using
this we should have obtained, also, ~~b.
So in (148) Ex. 1, the root may also be —3x—y; in Ex. in
(149), —4:X^ + dx'^~2x-{-l; and in all such cases, we should
get the two roots by giving a double sign to the first term in
the root.
When the root consists of more than two terms. Double Signs.
114 ELEMENTARY ALGEBRA.
151. As the 4tli poiver of a quantity is the square of its
square, so the 4tli root of a quantity is the square root of its
square root, and may therefore be found by the preceding
j*ule. Similarly, the 8th root may be found by extracting
the square root of the 4th root.
Thus, if it be required to find the 4th root of
a' + 4a'ic+ 6«^V + 4«a;' + x\
the square root will be found to be a^-\-^ax-\-x^, and the
square root of this to be a+ic, which is therefore the 4th root
of the given quantity.
Examples — 29.
Extract the squnre root of
1. a;'+2a;' + 3a;' + 2a; + l. 2. 1— 2^ + 5a;''-4a;'' + 4ct .
3. x'-V^x^ + '^bx^^-^^x+U. 4. x'^4.x'-V%x-\-4:.
5. l-^x^l()x^- 12x^ + 9a;*.
6. 4x'--4x''-7i«* + 4x' + 4.
7. x'-2ax'-\-6aV-4.a'x+4.a\
8. x'-2ax' + {a' + 2I}^)x''-'2ab'x + b\
9. ic*-12a;'+60^*-160:z:' + 240:r'-192a;+64.
10. x' + ^ax'-10aV-\-4:a'x + a\
11. l-2x-{-dx'-4.x' + 6x'--4:x' + 3x'-2x'' + x\
^x" X l^x^ 9f_ 6xy 16^
9p~7~l5y^^l6?"^ 57"^25;2^'
The fourth root, etc.
EVOLUTIO:^'. 115
Extract the 4tli root
13. Of l-4:X-{-6x''-4.x' + x* and of a'-8a'-f 24a'-32«4-16.
14. 0£16a'-96a'b + 2Wa'b'-216ah'-i-SU\
Find the 8th root
15. Of {x'-2x'y-{-3xY-2xy'-{-t/}\
152. The observation of the square roots of trhiomial ex-
pressions enables us to find the square root of complete (/. e.),
exact squares of these terms very easily, without going
through the entire process of Art. 148.
EuLE. — Arrange the terms according to the powers of some
one letter. Find separately the square roots of the extreme
terms, and take their sum or difference accordingly as the sig?t
of the middle term is -h or — .
Thus, a^ + 2ax -f 2;* is a complete square arranged according
to powers of a, and its square root is ^a^~^ ^x^, or a-\-x,
.*. a-{-x squared produces a^ + 2ax + x^. The square root of
a^—2ax + x^ is a—x, for the same reason.
Ex. 1. ^¥~+l-{-2a='^a' + 2a-hl=^^a^-{-^l=a-hl.
Ex. 2. ^¥~-{-9-6x= y/x''-6x + 9= v^?- >/d=x—S.
Ex. 3. v^4Ty^-4^= s/y^-4:y + A=: n/^- \^I=y-2.
Ex. 4. \/x^-px+^= v/i^_f^^^_|.
Ex. 5. \/x' + 3x+\=^x'^]/\=x-{-%.
To find the square root of complete trinomial squares.
116 ELEME^'TARY ALGEBRA.
Ex. 6. ^ ni'jf + '^iniix + n^ — "^ i)fx' + ^ if ~ mx + n.
Ex. 7. v'9^'7^to7/ + a~'== ^9xy- ^7'r=dxy-a,
Ex. 8. v/i,rZ^^4-^^^;-|-c-^ =: \/I^^" + V? = J-«Z^ 4- c.
Find the square roots of the following expressions :
Ex. 9. 16a^ + 40aZ>+25^^ 10. 49^*-84a'^^ + 36^>^
Ex. 11. 36:^;° + 12i^' + l. 12. 64a' + 48a/;c + 9Z^V.
-g .o 25^^^^«^4^^ 9 x^-242^^ + 16
^"^^ * 2ba' + 20ac-\-^o^ ' S^~12:z: + 9*
153. By observing the terms of a complete trinomial
square arranged according to one letter, we see that the mid-
dle term is twice the product of the square roots of the two
extreme terms. Hence, the quantity which must be added
to an expression of the form, x^ + 22jx, in order to form a
complete trinomial square, is the square of one-half of the
CO factor, or coef[icient, 2p of x\ that is, ]f. Observe that x
represents the square root of the first term.
Thus, m^x^ + ^mnx requires the square of the half of 2?^,
or 7f^ to complete it, giving m^x^ + 2in7ix + ^^^
dx^y^—Qaxy requires the square of the half of 2a, giving
9x'^y^ — 6axy + a^.
Complete the squares in each of the following cases :
(1.) x'-12x + -
(3.) x'+llx+-
(5.) x'- ix-h-
(7.) ■ x'-i- px-h-
7t
(9.) --fo+-
(2.) x=-|+-
(4.) x''— X +-
(G.) 36:^;H24:?;-|—
(8.) 16:?;'-56a; + —
(10.) 4«V + 4«^^+-
To complete the square of expressions of the form of a;2 + 2/xc.
EVOLUTIOIS^ 117
154. The method of finding the square root of numbers
is derived from the methods of Arts. 148 and 149. (See
V^enable's Arithmetic — Square Eoot.)
The square root of 100 is 10 ; the square root of 10000 is
100 ; the square root of 1000000 is 1000, and so on. Hence,
it follows that the square root of any number between 1 and
100, lies between 1 and 10, that is, the square root of any
number haying one or hvo figures is a number of one figure ;
ISO, also, the square root of any number between 100 and
1000, that is, having three or four figures, lies between 10
and 100, that is, is a number of hvo figures, and so on.
Hence, if we set a dot over every other figure of any given
square number, leginning ivitli the units figure, the number
of dots will exactly indicate the number of figures in its
square root. Thus, for example, the square roots of 256 and
4096 consist of two figures each, and the square roots of 16384
and 6il524, of three figures each.
155. Find the square root of 3249. 9^00
Set the dots accordinsr to the rule. The ^^^ ZZ^TT^
. , . , . X . T ^ 100 + 7 749
root must consist oi two figures. Let ^aq
a-^-h denote the root, where a is the value
of the figure in the tens place, and h of the figure in the
units place. Then a must be the greatest multiple of ten,
whose square is less than 3200, that is, a must be the square
root of the greatest exact square contained in 3200. Now,
as 25 is the greatest square in 32, 2500 must be the greatest
in 3200 ; hence, a is 50. Subtract a^ — that is, the square of
50 — from the given number, and the remainder is 749. Di-
vide the remainder by 2a — that is, by K'O — and the quotient
is 7, wiiicli is the value of h. Then {2a-{-b)b — that is,
(100 + 7)7, or 107x7 = 749—18 the number to be subtracted ;
and as there is no remainder, we conclude that 50 + 7, or 57,
is the required square root. If the number be such that its
root consists of three places of figures, let a represent the
118 ELEME:^?TARY ALGEBRA.
value of the liuDdreds figure, and h of the tens fignie; then
having obtained a and h as before, let the hundreds and tena
together be a new yalne of a, and then as before find a ne«»'
yalue of h for the units.
Example. 186624 (400 + 30 + 2
160000
800 + 30 =830) 26624
24900
800 + 60 + 2=862) 1724
1724
Here the number of dots is three, and therefore the num-
ber of figures in the root will be three. ISTow the greatest
square-number contained in 18, the first period (as it is
called), is 16, and the number evidently lies between 160000
and 250000, that is, between the squares of 400 and 500.
We take therefore 400 for the first term in the root,
and proceeding just as before, we obtain the whole root,
400+30 + 2=432.
186624(432
16 The ciphers are usually omitted in practice, and it
83)366 will be seen that we need only, at any step, take down
__ the next period, instead of the whole remainder.
862)1724
1724
156. Kule for finding the square root of any given number :
Set a dot over every other figure, beginning Ex. 1.
with that in the units' place, and thus divide 3249(57
the whole number into periods. Find the 25
greatest number whose square is contained in i07) 749
the first period ; this is the first figure in the 749
root; subtract its square from the first period,
and to the remainder bring down the next period. Divide this
quantity, omitting the last figure, by twice the part of the root
Rule for finding the square root of any given number.
EVOLUTION. 119
already found, and. annex the residt to the root and also to the
divisor ; then midtiply the divisor as it noio stands by the
part of the root last obtained for the subtrahend. If there be
more periods to be brought doivn, the operatio7i must be re-
peated.
Ex. 2. In Ex. 2, notice (i) that the second remainder, 49,
77841 ( 279 i^ greater than the divisor 47 ; this may sometimes
4 happen, but no difficulty can arise from it, as it
47)378 would be found that, if instead of 7 we took 8
^^^ for the second figure, the subtrahend would be
549)4941 384, which is too large. And (ii) that the last
figure, 7, of the first divisor, being doubled in order
Ex. 3. to make the second divisor, and thus becoming 14,
10291264 (3208 ^^^^^es 1 to be added to the preceding figure, 4,
9 which now becomes 5. In fact, the first divisor
62)129 is 400-i-70, which, when its second term is doubled,
124 becomes 400-f 140, or 540.
6408) 51264 In Ex. 3, we have an instance of a cipher oc-
^^^^^ curring in the root.
157. If the root have any number of decimal places, it is
plain (by the rule for the multiplication of decimals) that
the square will have twice as many, and therefore the number
of decimal places in the root will be half that number.
Hence, if the given square number be a decimal, and one of
an even number of places, we set, as before, the dot over the
units' figure, and then over every other figure on both sides of
it. The number of dots on the Uft of the decimal point tvill
indicate the number of integers in the root, and the number of
dots to the right, the number of decimal places in the root.
For example :
The square root of 32.49, is one-tenth of the square
root of 100x32.49; that is, of 3249. So, also, the square
root of '003249 is one thousandth of the square root of
1000000 X -003249, that is, of 3249.
If the number have decimal places, how do we proceed?
120 ELEMENTAEY ALGEBPtA.
Tlnis 10.201264 would be dotted 10.291264 the dot being
first placed on the units-figure 0; and the root will have one
integral and three decimal places, that is, would be (Ex. 3
above) 3.208.
If, however, the given number be a decimal of an odd
number of places, or if in any case of finding the square root
there be a remainder, then there is no exact square root ; but
we may approximate to it as far as we please, by dotting, as
before (rememhering ahvays to set the dot first over the U7iits
figure), and then annexing ciphers (which by the nature of
decimals will not alter the value of the number itself), and
taking them down as they are wanted until we have got as
many decimal places in the root as we desire.
Ex. Eind the square root of 2 and of 259.351, to three
decimal places.
2 (1.414 &c.
1
24)100
96
259.3510 (16.104 &c,
1
26)159
156
281)400
281
321)335
321
2824)11900
11296
32204)141000
128816
Examples — 30.
Eind the square roots
1. Of 177241, 120409, 4816.36, 543169, 1094116, 18671041.
2. Of 4334724, 437.6464, 1022121, 408.8484, 16803.9369.
3. Extract to five figures the square roots of 2.5, 2000, .3,
.03, 111, .00111, .004, .005.
evolutions". 121
158. To find the cithe root of a polynomial expression:
AYe know that the cube root of a^ -\-da'h-\-'dah'' -\-lf, is a + ^;
and we shall be led to a general rule for the extraction of the
cube root of any polynomial by observing the manner in
which a^-h may be derived from a^ + ^a^h + dalf + ^^
Arrange the terms according to the a^ + Za^h-\-Zab'^ + 1}^ {a-\-b
dimensions of one letter, <?-; then tlie first a^
term is a^, and its cube root is a, which 3^2 \ 3^2^ ^ g^^a ^ ^s
is tlie first term of the required root, ^d^b-\-^aJ? -^-h^
Subtract its cube, that is, a\ from tlie ^
whole expression, and bring down the remainder, 3a^& + 3a5^ + 6".
Divide ^a^b by 3a^, and the quotient is Z>, which is the other term of the
required root ; then form ( 3<x'"* + ^ab + b"^) 6, (i. e.) '6a^b + ^ab"^ + ^^, and sub-
tract it from the remainder, and the whole cube of a + 6 has been sub-
tracted. This finishes the operation in the present case.
If any quantity be left, proceed with a + b a^ a new a-^ its cube, that
is, c^' + 3a'^5 + 3<3!&"-^ + 6^, has already been subtracted from the proposed
expression, so we should divide the remainder hj Z{a + bY for a new
term in the root; and so on.
That the rule may be thus extended will be obvious from
comparing the form of the cubes 0^ a-\-h-\-c, a + 5-f c + ^, &c.,
with that oi a + b, from which the rule was deduced.
For,
{a + b + cY={a + bf + ^{a + bYc + ^(a^b)c'' + c\
==d' + {W + ^ah + b'')b+\^{a + bf + ^{a^-b)c-^c']c,
Similarly,
{a-^b + c-vdy=d'-v{Za''-^%ab + b'')b+[^{a + bY + ^{a + b)c-^c''\c
■¥\^{a + b + cf-\-Z{a + b + c)d+d'^d;
and so on.
Pursuing the same course as above in any other case, if
there be no remainder, we conclude that we have obtained
the exact cube root.
^x^ + 12x''y + Q,xy'^+y^{2x + y Here the quantity corresponding to
^x^ the trial-divisor ^a? is 3 ( 2x f— 12a;\ that
X^x^) 12x'^y + 6xy^+y'^ to Sd'b is 12^'^y, that to Sab"" is 6xy'', and
12x^y + 6xy'^-]-y^ that to b^ is y^ ; so that the w^hole sub-
■ trahend is 12x^y + Qxy^ + y^.
To find the cube root of a polynomial expression.
6
ELEMENTARY ALGEBRA.
By attending, however, to the following hint, the subtrahend mav be
more easily constructed.
da -f- b 8a'
{Sa'}-b)b
dai + dab + W
Sa'b + Sab'' + b^
Wb + Sab'' + W
Set down first 8a, some little way to the left of the first remainder,
and then, multiplying this by a, obtain Sa'^ as before ; by means of this
trial-divisor find Z>, and annex it to the 8a, so making 8a + i ; multiply
this by Z), and set the product {Sa-\-b)b or Sab-\-P under the 3a'^, and
add them up, making Sa' + 'Sab + b'' ; then, multiplying this by 5, we
have da''b-{-Sab''-{-b'\ the quantity required.
The value of the above method, in saving labor, will be more fully
seen when the root has more than two terms, or, if numerical, more
than two figures..
Ex. 8^H '^2x'y + 6xy' -hy^2x + y
6x + y 12^;'
-\-fjxy + y''
12X'''}' Qxy^y''
nx'y + Qxy'' + y-'
12x''y + Gxy' + y^
Examples — 31.
Find the cube roots
1. Ofx' + Cx'y + 12xtf + Stj\ 2.0ia'-9a'+27a-27.
3. Of a;^ + lXV+48a3+64. 4. Of Sa'-36a'b + 64.ab'-27b\
5. Ofa'+24:a'b + ld2aI)' + 612h\
6. Of Sx' - Ux\j + 2^4.xtf - 343^^
7. Of 771' -12m'nx + 4.SmnV - 64:7i'x\
8. Of aV-loa'bx' + 75abV -126Fx\
9. Oi a' + (^a'-\-lDa'-{-20a' + 15a' + 6a + l.
10. Ofa;"-12:?;^ + 54:r*-112a;^ + 108.'r''-48:r + 8
11. Of a'- da'b + 6a'b' - 7a'b' + ^a'b' - Sab' + b\
12. Ofa'-b' + c'-3{a'b-a'c-ab''-ac'-b'c + bc')-(jabc.
EYOLUTIOJS^ 123
159. The method of finding the cube root of an algebraic
expression suggests a method for the extraction of the cub^
root of any number.
The cube root of 1000 is 10 ; the 'cube root of 1000000 is
100, and so on ; -hence, it follows that the cube root of a
number less than 1000 must consist of only one figure; the
cube root of a number between 1000 and 1000000, of two
places of figures, and so on.
If, then, a point be placed over every third figure in any
number, beginning with the figure in the units^ place, the
number of points will show the number of figures in the
cube root. Thus, for example, the cube root of 405224
consists of two figures, and the cube root of 12812904
consists of three figures.
Suppose the cube root of 274625 required.
180 + 5 10800 274625(60+5
925 216000
11725 58625
58625
Point the number according to the rule ; thus it appears
that the root must consist of two places of figures. Let
a-^-h denote the root, where a is the value of the figure in
the tens' place, and h of that in the units' place. Then a
must be the greatest multiple of ten which has its cube
less than 274000 ; this is found to be 60. Place the cube
of 60, that is 216000, in the third column under the given
number and subtract. Place three times 60, that is 180,
in the first column, and three times the square of 60, that
is 10800, in the second column. Divide the remainder in
the third column by the number in the second column,
that is, divide 58625 by 10800 ; we thus obtain 5, which
is the value of h. Add 5 to the first column, and multiply
To find the cube root of any number.
124
ELEME^^TARY ALGEBRA.
the sum thus formed by 5^ that is, multiply 185 by 5 ; we
thus obtain 925, which we place in the second column and
add to the number already there. Thus we obtain 11725 ;
multiply this by. 5, place the product in the third column,
and subtract. The remainder is zero, and therefore 65 is
the required cube root.
The ciphers may be omitted for brevity, and the process
will stand thus :
185
108
925
11725
274625(65
216
58625
58625
It will be seen by the following example, where the root
has more than two figures, how the numerical process cor-
responds to the algebraical. The ciphers are omitted, ex-
cept that in the numbers corresponding to 3a^, 3a' ^ &c., it
is better to express two at the end: thus 'a is really 4000,
and therefore oa"^ is 48000000 ; but, as in the first remainder,
we only need the figures of the first and second periods, cor-
responding to 43 in the root, we may treat the a as 40, and
thus So" will be 4800, and 3a will be 120, so that 3a + h will
become 123.
Ex.
80677568161 (4;
321
64
3^4-^ = 123 3a'^=4800
16677
a' = 43
{3a-i-b)b= 369
a"=432
3a'-i-3ab+F = ^169
15507
3a' + d=1292 3a'^ = 554700
1170568
{3a'-\-b)b= 2584
da" + 3a'b-{-I)'=657284.
1114568
12961 55987200
56000161
12961
56000161
56000161
EYOLUTIOK. . 125
XoTE. — Our trial-divisors may frequently give figures too large for
the next figure of the root. In such case try the next less figure, and
if necessary, the next less, until we get the right one.
100. If the root have any number of decimal places, it is
plain by the rule for the multiplication of decimals, that the
cube will have thrice as many ; and therefore the number of
decimal places in every cube decimal will be necessarily a
midti]jle of three, and the number of decimal places in the
root will be a third of that number. Hence, if the given
cube number be a decimal, and consequently have its num-
ber of decimal places a multiple of three, by setting as be-
fore the dot upon the units-figure, and then over every third
figure on loth sides of it, the number of dots to the left will
still indicate the number of integral figures in the root, and
the number of dots to the right the number of decimal
places.
If the given number be not a perfect cube, we may dot as
before (always setting the dot first upon the units figure), and
annex ciphers as in the case of the square root, so as to ap-
proximate to the cube root required, to as many decimal
places as we please.
Example. Extract the cube root of 14102.327296.
641
1200
14102.327296(24.16
»\
256'
8
721'
2/
1456
6102
16,
5824
7236
172800
278327
721'
173521
173521
104806296
1
104806296
1742430
4341
6
1746
771
6
126 ELEMEKTARY ALGEBKA.
Note. — A careful examination of the two columns of figures on the
left will disclose a much abbreviated process of finding the divisors. In
the left-hand column, adding to 64, 721, etc., twice the units-figure
gives the same result as multiplying the root already found by 3. In
the second column, adding the three numbers enclosed by a brace {i.e.^
the last true divisor, the number above it, and the square of the last
root figure), and annexing two ciphers, gives the next trial-divisor.
The examples and explanations above furnish us the follow-
ing rule, given also in the Arithmetic :
I. Place a dot over the U7iits-figu're of the nuwher, and over
every third figure to the left, and also to the right tuhen the
number contains decimals {altvays tahing care in this latter
case to mahe the number of decimal figures a multiple of 3).
II. Find the greatest cube in the nuniber which forms the
first period on the left, and place its root after the manner of
a quotient in division. This root is the first figure of the re-
quired root. Subtract its cube from the first period, and to
the remainder bring doivn the figures of the second period for
a First Dividend.
III. Multiply the square of this first figure by 3, annex
two ciphers, and find hoio often this Trial-Divisor ^6' con-
tained in the first dividend ; place the quotient as the second
(trial) figure of the root. Then to three tiines the first figure
of the root annex this second figure, and multiply^ the result
by the second figure ; add the product to the Trial-Divisor,
and call the sum the First Divisor.
IV. Multip)ly the First Divisor by the second figure of the
root ; if the product be greater than the First Dividend, use a
lower figure for the second figure of the root, and thus repeat
the process III. until the product be less than the First Divi-
dend ; subtract this product from this dividend, arid to the
remainder bring doiun the figures of the third period for a
Second Dividend.
Rule for extracting the cube root of any number.
EYOLUTIOISr. l:i?
V. Mulfiphf the square of the tioo figures of the root hy 3;
annex tc^o cipher ^^, and proceed as in III. and lY. Proceed in
this manner until all the 'periods have heen drought dotv7i.
iNOTii. — In extracting either the square or cube root of any number,
w2ieo a certain number of figures in the root have been obtained by the
iommoii rcle, tiiat number ma?/ be nearly doubled by dicidoii only.
1. In the e.xtraciion of the square root, when n + 1 figures are found
in the root, n more may be found by merely dividing the last remain-
der by the trial-divisor. For, let N be tlie number whose square root *
is to be found, conoi&ting of 2/i + 1 figures.
Let a= the part ai/eady found (consisting of n + 1 figures, and n ci-
phers after them, that i3, altogether of 2/2. + ! figures).
Let x-= required rema.uh\g part of tlie root, consisting of n figures.
So that
V j.Y~—a + x\
Then i7 ^ vj'-' + 2ax + x" ;
J^'—a"^ x^
?.'.i 2a
Now J^^—a^ is the remainder, after n + 1 figures of the root are found,
x^
and 2a the trial-divisor; if, then, we can show that — is ^proper frac-
fja
tion^ it will follow that the integer obtained by dividing N—a^ by 2a
will be X., the remaining part of the root.
But as X contains n figures, it must be <10'^, which has n + 1 figures,
and x^ <W-'^ '^ and since a contains 2?i + 1 figures, it cannot be <W'^
(which is the smallest number of 2n + l figures).
rj^ 10-" 1
Hence, ,r-< ^ ^^, <^, and is therefore a proper fraction. That is, if
2a 2.10-'^ 2 '
N—d^
the quotient of —z be taken for the n remaining figures of the root,
the siim is less than 1.
2. In the extraction of the cube root, when n + 2 figures are found
in tlie root, n more may be found by dividing the last remainder by the
trial-divisor.
For let N=^ the number ;
a= the part of root found (consisting of n + 2 figures followed by
n ciphers, that is, of 2n + 2 figures altogether) ;
a'= the required part of root (consisting of n figures).
. Then N=a^ + Za^x + ^ax^ + x\ and -~--^—=.v, + — + -^ ; and here
da^ a ?ia^
128 ELE3IE:N^TAIiy ALGEBEA.
2;<10«; and a^ since it contains 2n + 2 figures, cannot be <10-"+^-,
— + ^r-T, <1. That is, if tlie quotient of -tt-^t- t)e taken for the n re«
a Sa^ Sa^
maining figures of the root, the error is less than 1. Now iV^—f^^-*— re-
mainder after n-\-2 figures of root are found, and da^ is the Trial- Di-
visor for the next figure. Hence the rule as above.
* Examples — 32.
Find the cube roots of
1. 9261, 12167, 15625, 32768, 103.823, 110592, 262144,
884.736.
2. 1481544, 1601.613, 1953125, 1259712, 2.803221, 7077888.
3. Extract to 4 figures the cube roots of 2.5, .2, .01, 4.
XXII. Simple Equations.
161. The statement of the equality of two algebraical
quantities which differ only in form, is called an " Identity, ^^
An Identity is true for any value whatever of the letters
which enter it.
Thus, 2.'c + 5a;=7a;; 2{a-{-x)='Za + 2x',
{x-\-ay=x^-\-2ax + a'^] {x + a) (a;— a) = 0;^ — a', are Identities.
Up to this point we have been using Identities — especially
to express general facts — by means of letters. Our formulas
heretofore given are Identities.
16.2. An equation, however, is the statement of the equality
of two cliff event algebraical expressions; in which case the
equality does not exist for all values, but only for some par-
ticular values, of one or more of the letters contained in it.
Thus the equation x—b=i, will be found true only when
An Identity. An Equation.
SIMPLE EQUATIONS. l^/Q
we give x the value 9; and x'^=z3x—2, true only when we
give X the value 1 or 2.
In equations, the question always is, what value of the
letter or letters not already known will verify or satisfy, (i. e.)
make true, the expressed equality. The finding of such value
or values is called solving. the equatio7i.
163. The Iwo expressions connected by the symbol = are
called sides of the equation, or memlers of the equation. • The
expression to the left of the sign of equality, is called the
first side; and the expression to the right is called the
second side.
164. Those quantities to which particular values are to be
given in order to satisfy the equation, are called the unhnoiun
quantities. The last letters of the alphabet, x, y, z, &c., are
usually employed to denote these quantities.
165. An equation is said to be satisfied by any value of the
unknown quantity which makes the values of the two sides
of the equation the same, {i. e.) which makes the Equation an
Identity.
This includes the case where all the terms of an equation
lie on one side and on the other, as in x^—^x + '^ — Q, which
is satisfied by 1 or 2, either of which being put for x makes
the first side also 0.
Those values of the unknown quantities by which the
equation is satisfied, are called the roots of the equation.
Thus, '7 is the root of x—d=z4:; 1 and 2 are the roots
ofx'-dx + 'Z^O.
168. An equation of one unknown quantity, when cleared
of surds and fractions, is said to be of as many dimensions as
there are units in the index of the highest power of the un-
known quantity. 1^0
Thus, cc— 5=4 is an equation of one dimension, or, of
Solving the Equation. Sides or Members of the Equation. Unknown Qua?:
titles. Satisfying an Equation.
130 ELEMEKTARY ALGEBRA.
the first degree, or a simple equation; x^ = 'dx—^, is of
tiuo dimensions, or, of the second degree, or a quad-
ratic equation; x^ — h^=.^oi^ is of three dimensions, or of
the tliird degree, or a cuhic equation; x^ — ^x^=.V6, is of
four dimensions, or the fourth degree, or a hiquadratic
equation, &c., &c.
167. In the present chapter we shall show how to solye
simple equations. We have first to indicate some operations
which we may perform on any equation without destroying
the equality which it expresses.
168. If every term of each side of an equation he multiplied
ly the same quantity, the ttvo sides ivill still he equal.
For, if equals he multiplied by the same quantity, the
results are equal.
This principle is chiefly used for clearing an equation of
fractions, if they stand in the way of solving it.
Thus, taking the equation ^x—^ — -^, multiplying every
o
term by 3, the denominator of the fractional term, we have
21:?; — 18=3X-^, or 21:^^—18 = 5^, in which no fraction ap-
o
pears.
An equation of several fractional terms may be cleared of
fractions by multiplying every term by any common multiple
of all the denominators. If the L.o.M.of the denominators be
employed, the equation will be expressed in its simplest terms.
Take, for example, — + — + — = 9.
o 4 D
Multiply every term by 3 X 4 X 6, or, 72 ; thus,
72a; 72a; 72a; '^^
that is, 24a; + 18a:+12a:=648, cleared effractions.
RootR of an Equation. Difi'firont kinds of Equations.
SIMPLE EQUATION'S. 131
Instead of multiplying every term by 72, we may multiply
eyery term by 12, the l.c.m. of 3, 4, and 6. We would
\2x VHx 1.2x
thus liaye — — H — -- + —- = 108; that is, 4tx + dx+2x=108,
O ~c K)
expressed in simple terms.
-^ ^T ,. ,. 6x + 4: 7x + 5 28 x—1 „^
hx. Clear the equation — ttt"— ~^~~o~"^ ^^ ™^'"
/C 10 o /^
tions.
The L.C.M. of the denominators is 10. Multiply by 10.
Thus, 6{6x + 4:)-{7x + 5) = b6-6{x-l);
that is, 26x-{-20^'7x-6=z66-6x + 6.
The beginner should write the operations out in full, as
above, using brackets, in order that he may attend to the
. . . 7.T + 5 , x—1
signs 01 such expressions as — r— , and — .
10 Zi
169. Any term may he transferred from one side of a7i
equation to the otlter side ivitliout destroying the equality, j^ro-
vided 2ve change the sign of the term.
This transference is called transposing.
Suppose, for example, x—a — h—y.
Add a to each side (which of course will not destroy the
equality) ;
then, x—a-\-a—'b—y-^a\
that is, x='b—y-\-a.
NoAV subtract h from each side ; thus,
x—'b—lj^a—y—'b\
that is, x—t^za—y.
Here we see that —a has been removed from one side of
the equation, and appears as +a on the other side; and -\-'b
has been removed from one side and appears as — Z> on the
other side.
Transposition of Terms.
132 eleme:n^tary algebra.
170. If the sign of every term of an equation he changed the
equality still holds.
This follows from Art. 169, by transposing every term.
Thus, suppose, for example, that x—a=t)—y.
By transposition, y—h—a—x'^
that is, a—x=:y—ib.
And this result is what we shall obtain if we change the sign
of every term in the original equation.
It is also clear that if the same quantity occur with the
same sign on both sides of the equation, it may he erased
from hoth sides. For, by the erasure w^e either subtract
equals from equals or add equals to equals, according as the
sign of the term erased is + or —.
171. Every term of each side of an equation may he divided
hy the same quantity luithout destroying the equality expressed
hy it.
For, if equals be divided by the same qua;ntity, the quotients
are equal.
Thus, the equation 12a; + 5.^=1 3 6, or 17a? = 13 6, gives
17^_136 _1^_Q
17 — 1 7 ' ^^ '^ — 17 —
.. .n -, ct^ ^ 'b
Ai^o,\iax=o. — = — , or x=:—,
• a a a
Again: if 24:?;+18:?; + 12:^: = 648 be divided by 6, we get
4a;+32: + 2a-108, or 92;=108.
9ix 108 108 ,^
Hence, ~a—~K~^ ^"^ x——=V^, ^
then
Again: M ax-\-hx—cx^^d^(dY (a-\-h--c)x—d*y
{a-\'h—c)x d _ ^
a-\-h—c a + h — c^ a-\-h-
Chaflge of the signs of an equation. Division of the terms of an equation.
SIMPLE EQUATIOXS. 133
172. The operations indicated in the preceding articles
may be performed upon equations of any degree, and con-
taining any number of unknowns ; for they depend on
principles true of every equation. Of course all these opera-
tions can be performed on Identities, or identical equations,
as they are sometimes called.
173. To solve a simple equation of one unknown quantity :
Eule. Clear the equation of fractions, if necessary. Collect
all the terms involving the unknown quantity on one side of
the equation and the known quantities on the other side, tra^is-
posing them, when necessary, with change of sign. Add to-
gether the terms of each side, and divide both sides dy the coef-
ficient or sum of the coefficients of the unknoiun quantity ; and
thus the root required will he found.
Note I. — Erase terms by Art. 170, or simpUfy the equation by di-
vision, Art. 171, at any stage of the process.
Note II. — It is usual to collect all the unknown quantities on the
first side, and the known on the second side of the equation.
174. We shall now give some examples.
Ex. 1. Solve 7:?; + 25 = 35 + 5.T.
Here there are no fractions; by transposing we have
7:2;-5:z:=35-25;
that is, 2<c=10;
dividing by 2, x=—- = o.
We may verify this result by putting 5 for x in the' original
equation ; then each side is equal to 60.
Ex. 2. 4.x + b = 10x-U.
Here Wx-4.x=b-\-l(j; ,\ Qx=2l, ^ndi x^zz^ z=z^=^,
Ex. 3. b{x-\-l)-2 = ^{x-b).
Tlule for solving a Simple Equation. Note I. Note II.
134 ELEMENTARY ALGEBRA.
Here^ remoying the brackets, 6x+6—2 = dx—16;
.-. 5x—3x=:—16—6-i-2, or 2^=— 18, and /. ^=— 9.
Ex. 4. Solye4:{3x--2)-2{4.x-3)-3{4:-x)=0.
Performing the multiplications indicated,
12x-8-{Sx-6)-{12-3x)=0.
Eemoving the brackets,
12a;-,8— 82: + 6— 12 + 3^=0;
collecting the terms, 7a;— 14=0;
transposing, 7^== 14 ;
14
dividing by 7, x=—=2.
The student will find it a useful exercise to yerify the cor-
rectness of his solutions. Thus, in the aboye example, if we
put 2 for X in the original equation, we shall obtain 16 — 10—6,
that is 0, as it should be.
Ex. 5. dx-^2x—a=3x-{-2c.
Transposing, I)X + 2x—dx=a + 2c;
or, dx—x=a + 2c;
collecting the coefficients, (^—1) x=a + 2c;
a -{-2c
dividing by Z> — 1,
b-1
Examples — 33.
1. 4.x-2 = 3x-{-3. 2. 3^ + 7=:9a;-5. 3. 4.x-\-9=zSx-3,
4. 3-{-2x=7—6x. 5. x—7 + 16x. 6. 7nx-{-a = nx-{-d.
7. 3{:?:-2)+4=4(3-.t). 8. 5-3 (4-^-) +4 (3-22:)r=0. -J
9. 13:i;-21 (a;-3) = 10-21 (3-a;).
10. 6{a + x)-2x = 3{a-5x),
SIMPLE EQUATIOXS. ' 135
11. 3{x-3)-2{x-2)+x-l:=x-{-3 + 2{x + 2) + 3{x+l).
12. 2.i;-l-2 (3:?;"- 2) +3 (4a;-3)-4 (5:?;-4):=0.
13. {2-{-x){a-3) = -4.-2ax.
14. {7)1 +n) {m—x)=7n {?i—x).
15. 5.'?;-[8a:-3{16-6a;-(4-5^)}]=:6.
175. The following examples will illustrate the solution of
equations containing fractions.
Ex.1. If |-|:- 1=^-3, find:..
Multiplying by 2 X 3, or 6,
3:?;-10a;-8 =8x-lS;
transposing, 3.^—10:^—82:=: 8 —18;
combining, — 15a;=— 10;
dividing by —15, x= — — =— .
— J o
Ex. 2. ix-^x+lx=ll+^x.
Here we first clear the equation of fractions, by multiply
ing every term by 24, the l.c.m. of the denominators, and
(observing that in the first fraction ^^ = 12, in the second ^^
=8, and so in the others) thus we get 12:?;— 8x2:^:+6x3a;
= 264 + 3a:, or
12x-iex-\-lSx=264:-{-dx;
.-. 12.i;-16r^ + 18.T-3.T=264; .-. lla;=:264, and ^^.-^^V^^^-
Ex. 3. If x + —^:=:12-—,:p^, find X.
To clear of fractions, multiply by 2x3, or 6, and we have
62^ + 3 (32:-5)=.72-2 (2x-4) ;
or 62;+(9a;-15) = 72-(42:-8);
.-. C):j^ + 9.7:-15 = 72-4.t4-8.
136 ELEMEKTARY ALGEBRA.
Transposing, 6x-{-9x + 4:X=l!2-{-8-{-16;
combining, 192;= 95;
95
dividing, 0:=— =5.
Ex. 4. Solve h6x+3)-^{16-5x)=31!-4.x.
This is the same as
6x^-3 16-5x ^^ ,
—- ^ — =37-4:X.
Multiplying by 21, 7 {6x + 3)-'d (16-52;) = 21 (37-42;)
that is, 352; + 21-48 + 152;=r777-842;;
transposing, 352^ + 152; + 842;= 777 -21 + 48;
that is, 1342; = 804;
therefore, ^= ttt: = 6.
134
Ex.5. i{x-\-l)+i{x-^2)=16-i{x + 3).
Multiplying by 12, we have
6(2; + l)+4(2; + 2) = 192-3(2;+3),
or 62;+ 6 + 42; + 8 =192 — 32;— 9;
.•.62;+42; + 32;=192-9-6-8; .•.132;=169, and x=\%^z=:13.
Examples— 34.
1. ^2;+-j2;=2;— 7. 2. ix—^x=lx—l.
o
2r (T 2
5. y+-^=a;-4 6. i(9-3a;)=|-TV(7a.- 18).
7. x+\{U-x)=^{%l-x). 8; 2a;-^=|(3-2a;)+ia;.
9. ^(2a:+7)-yT(9.'P-8)=i(a;-ll).
SIMPLE equatio:n"S. 137
,^ x-a 2x—3b a—x ^ ^^ 6x—7 2x+'7 ..
-, a;-2 «— 3 „ ,„ x+3 x+4: x+5 ,„
13. ^_i_ -^+^=0. 13. _ + _+_=:16.
14. !^=7 + .-^-i^i. 15. ^i-^i.£^.
3a;-5 5a;-3 „„ „ ,„ 7:r-4 „„ 4-7a; 7
12'
IC. -3 ^+2|=0. 17. ^^- + 3| + -^=a;-^.
,„ 3— a; 3— a; 4—2; 5— a; 3
18.— + — + —+— + ^=0.
,„ 5a;-3 9-a; 5x 19,
XXIII. - Simple Equations CoNTrNUED.
176. We shall now give some examples which are a little
more difficult than those in the preceding chapter. In many
of these examples the common multiple of all the denomina-
tors is too large to be conveniently employed. In such a case
we may see whether two or three of the denominators have a
simple common multiple, and get rid of these fractions first,
observing to collect the terms and simplify as much as pos-
sible after each step.
^ ^ 2.^ + 3 :i;— 12 32:4-1 ^, . 4:X + 3
^-^- ^- ^T 3-+-^=^*+-]^-
Here the l.c.m. of all the denominators would be 132 ; but
as 12 will include three of them, multiplying by it (having
iirst changed 5^ to ^3^), we get
^^?-^tl)-4(a;-12) + 3(3a; + l) = 64 + 4a;+3;
.\\^{2x-^d)-4cX+4:8 + 9x + 3 = 64:+4.X'\-3;
When the Common Multiple of all the denominators is too large for coa
venience, what course may you pursue?
138 ELEMENTARY ALGEBRA.
hence, collecting terms and simplifying, we liaye
m2x+d)-4.x-{-9x-4:X=64.-\-3-4:8-3,
ovii{2x + 3)+x=:16;
.-. 12 (2:2;+3) +11^=176, or 24^; + 112;= 176 -36;
/. o6x=:^14:0, and 2;=V¥=^-
177. It will often happen that the icnlcnoion quantity is
found in the denominator of one or more of the fractions.
x^ooi 24352
Ex.2. Solye —+—=_-] _.
X X x X 11
Since the first four fractions haye a common denominator,
by addition, _=,___;
8 6 2
transposmg, ^~"^""l7'
2 2
combining, '^=17'
.-. a; =17.
Multiplying by 3^;, 9 — 2 = 5 + :?; ; :. x=2.
If any of the denominators which contain the unknown
quantity consists of two or more terms, it will generally be
adyisable to follow the method of Art. 176, and clear the
equation of the simplest denominators first, leaying tho
others to be dealt with afterward, when, by transposing,
collecting the terms, &c., the equation has been reduced to
feiver terms. Or, if all the denominators consist of two or
more terms, then they may be cleared off singly, one hy cne,
till all haye disappeared.
How may you proceed when the unknown quantity is found in the denora
Inator? When any of the denominators coneists of more than two terms?
Ex. 4. Solve
SIMPLE EQUATIONS. 139
6x-\-13 3.T+5 2x
15 5:^;— 25 5
Multiply by 15 to clear away the simjyie denominators first,
and we have
., . ^^ 15(3:^4-5)
y, and transposing, 13= \ ^^ ?
erasing, ana transposing, 16= k _<^-
or, dividing numerator and denominator of the fraction by 5,
^^^ 3(3a; + 5) ^
x—b
Multiplying by a;— 5, 13:?;— 65 = 9^; + 15 ;
transposing, 13^;— 9^=65 + 15 ;
combining, 4^ = 80 ;
80
dividing, x=——20,
Ex.5. Solve— g ___^_-_.
To remove first the denominators 18 and 9, multiply the
whole by 18, and we have
10^_,17_-j^_^.10:.-8;
erasing, and transposing,
216r.+36.
, . . ^^ 2162: + 36
combining, 2o=-— r— ;
Wx — o
multiplying by 1Lt-8, 25 (ll:r— 8)=216:?: + 36;
or, 275:^:-200=:216a; + 36;
transposing, 275a;— 2l6x==200 + 36;
combining, 5 9a; rr: 23 6 ;
dividing, 0;:=— ^=4.
140 ELEMENTARY ALGEBRA.
-^ ^ ^ , 2x-hS ^x + 6 3x + d
JbiX. 6. 'bolve — r^^z 7 + ?; T.
x+1 4.X + 4: Sx + l
Here it is convenient to multiply by ^x-\-4:, that is bv
4(0^+1);
. /^ ox . . . 4 (i?; 4- 1)3(^+1)
thus 4(2ic + 3) = 4a;+5+-^-— -^^--^--^;
12 (a^ + l)' ^
therefore, 8a; + 12— 4^— 5 =
that is^ 42^+7=
12 {x-\-iy
dx-\-l
Multiplying by 3a; + l, (3:2: + l) (42^+7) = 12 (a:+l)';
that is, 12:c'+25^ + 7=12a;'+24x+12.
Here l^x^ is found on both sides of the equation. Ee-
move it by subtraction, and the equation becomes a sim2)le
equation; that is,
25^ + 7=24:^;-fl2;
or, 252J— 24^=12-7;
Ex. 7. Solve
x—1 x—2 x—4: x—5
x—2 x—3 x—5 x—6'
Eeducing the terms on the first side to a common denom-
. ^ ^ x^~Ax + 3-(x'-4:X + 4.) 1
mator, we get ^, — , or — -, -^, — -^,
" {x—2) [x—d) {x^2) (x—d)
Eeducing the terms of the second side to a common de-
nominator, we have for this side
.^'-rl0:c + 24-(.'r'-10:r+25) 1
(x-b)(x-Q>) ' (x-b){x-Qy
Thus the proposed equation becomes
1 1^ ^
"" {x-2) {x-^)" (x-5) (.T-6) '
SIMPLE EQUATIONS. 141
1 1
Changing the Signs, l^_^)j^_^^-^',j(^^Ze)^
clearing of fractions, (x— 5) (x—Q) = {x—2) 0^—3) ;
that is, x^--llx + 30=x''—6x + 6;
whence, — lice + 5a; =6— 30;
that is, — 6.T = — 24 ;
therefore, a: =4.
A5x-,76 1.2 .dx-.6
Ex. 8. Solve .5^;+-
.6 ~ .2 .9
To insure accuracy, it is advisable to express all the deci-
mals as common fractions ; thus
6x 10/45^ __75^\_10 l^_10/3^_j6\
Simplifying, | +|(|-|) -=6- (-|-|-) ;
,, , . a; 3cc 5 ^ a; 2
that IS, _ + _^_:.6--+-.
Multiplying by 12, 6x+9x-16=:'72-4:X + S ;
transposing, 19a;=72+8 + 15 = 95;
therefore, x=—=5.
178. Complex fractions in an equation should first be re-
duced to simple ones, by the known rules.
T, ^ 25-4:?; 16.^ + 44 . 23
^"^•'- -^+-3-^+?=^+^T
Hei e, first simplifying the complex fractions, we get
' 75-x 80:r + 21 _ _23_^
3(a;+l)"^5(32: + 2)~ ^^+1'
Complex fractions in an equation.
142 ELEMEKTARY ALGEBRxV.
,^. , . ^ ,. 375-5.-?; 240:^+63 ^. , 345
then, multiplying by lo, —— + -———— = 7o +
x + 1 ^ 32^+2 ' x + 1'
whence, multiplying hj x+1,
o^. r. . 2402;'+303.T+63 ,,^ ^. . o...
875— 5:^;^ ^-— -^ =75iz; + 75+345;
OX -J- /i
or, simplifying,
J— — r = 752;+5aj + 75+345— 375=80:^+45:
3a;+2
/. 240^;' + 303:^ + 63 = 240;^'+ 295a; + 90, and 8a;:=27, or x=3%.
179. We will now solve three more equations, in whi^'^
letters are used to represent known quantities.
Ex. 10. Solye - + ^=^.
a
Multiplying by ^Z>, Ix-^-ax—dbc^
that is, {a-^'b)x=dbc\
dbc
,\ x=-—-.
a-\-o
Ex. 11. Solve a— ^ — yd —X,
a
Multiplying by ab, a^ {x—d)-\- If (x—l) = abx ;
that is, a'^x—a^-\-l'^x—lf=:abx]
transposing and collecting, a^x-\-lfx—abx—a^ + V ;
that is, {d^—ah-\-¥^ x^cc" -\-W \
dividing by c^-\-db-\- Z>^ x = ,_ ^ ;
.•. x=a+b,
-n -.n CI 1 ^— ^ (2x~ay
Ex. 12. Solve 7=77^ r(?.
x—o {2x—o)
Clearing of fractions,
(x-a) {2x-hy = {x-b) {2x-ay;
that is, (x-a) {^x''-ihx-^b')=z{x-b) (4a;'-4«a; + a'),
SIMPLE EQUATIONS. J 43
multiplying, we obtain
4.x'-4.x\a^h)^x{4.a'b-\-l)'')-ah''
= 4a;' - 4a;' {a -\-i)-\-x {4.ah + a') - a'Z> ;
whence, lfx—ah'^=a^x^a^h\
/. {a'—V^) x—a^l—aV—ab (a—V) ;
__ah{a—l)_ ah
Examples — 35.
12 _1__29 _iL__££_ 3 l^B _ 216
'^"^12a;~24* ' a;-2~a;-3* ' 3a;-4~5a;-6'
2^T3~4^^' ^' 2 ~ 3 "^ 4 6~^'^ '
1 3
--a;— 3 -a;— 10 . ^_
2 ,4 , 4:—x 10— a;
5 2 4 6
5a;~8 ^ 3a;-8
. a;-2 , 1 2a;-l ^. ^ ^ a;^+3 ^
9. _+_ = ^____. 10. a;+l-^^=2.
a;-l_7a;-21 7a;-4 7a;-26
* a;-2~7a;-2G' * a;-l ~" x-3 '
X dx 71_ 3a; + l . . 2a;-6 _ 2a;— 5
7 ~ 2 "^ 7 ~ 2 "^ ■^^* 3a;-8~3a;-7'
15. a;--3~(3-a;)(a; + l)=:a;(a;-3)+8.
IG. 3-a;-2(a;-l) (a; + 2)^(aj-3) (5-2a;).
17. I+^_i+^=,7^^. 18. (a; + 7)(a:+l) = (a-4-3r.
144 ELEMEKTARY ALGEBRA.
19. ^{2x-10)-^{3x-4.0) = 1d-^{d^-9:).
2x+l ^+12 ~
21. ^i^{2x-3)-i{dx-2)=i{^x-3)-d^^^.
22. 5(^__9)+_T(^_5)^9(^_7) + l|.
23. ,-V(2^-l)-3J^(3^-2)=^V(^-12)-^VG^+12).
24. ^(^x+20)-^\{3x + 4.)=^^-^{3x + l)-M^9-Sx).
^^ dx-1 4x-2 1 ,,^ 2 1 6
20. rr T — n 7^ = 7r- ^6. 5 +
27.
2:^-1 3:z;-2 6* ' 22;-3^a;-2 3:z; + 2'
x—4: x—6 x—7 x—8
x—b x—^ x—d> x—9
28. JL, + ^=^ + ^'
x—2 x—l x—1 x—6
29. ^-l±^.- ~=:1. 30. ,6x-2=--.26x+,2x-l.
3 — X 2 — x l-\-x
31. .5:z; + .6a3-.8=:.75.'?; + .25.
.135aj-.225 .36 Mx-.IS
32. .15a; + -
.6 ~.2 .9 •
fl^— a: ^ J+a: ^, x^ — a"" a-x 2x a
33. a~^f =x, 34. —j tt—t •
h a ox X
35. x{x—a)-{-x{x — 'b)=2{x—a){x—h).
36. ix-a){.-l) = {.-a-iy. 37. ^-^j=^.
28 -1 ^=-^^.
* x—a x—h x^ — ab
39.
40.
SIMPLE EQUATIOJS'S. 145
111
x—a x—a-\-c x—b — o x—V
m X —a — h mx —a — c
nx—c~d nx—h—cV
41. (ci-l) {x-c)-{b-c) {x-a)-{c-a) {x~b)=:0.
x—a x + a 2ax
42.
a — b a + b a^ — b'''
,_ x—a x—a—X x—b x—b—1 /rj t^ ^
43. r -= J — ~ r~^. (See Ex. 6,
x—a—1 x—a— 2 x—b—1 x—b— 2 ^
Art. 177.)
XXIV. Peoblems solved by Simple Equations.
180. We shall now see tlie practical application of the above
methods in the solution of many arithmetical problems. In
these problems certain quantities are given, and another
quantity, Avhich has certain given relations to these, has to
be found : the quantity which has to be found is called the
unhioivn quantity. The relations between the given quanti-
ties and this unknoiun are expressed in the enunciation of the
problem in ordinary language, -and these are to be translated
into algebraical expressions, to be used in the solution of the
problem. The method of solving the problem may be given
in general terms, as folloAvs :
Put X to represent the unhnoion quantity. Set doivn, in
algebraical language, the statements made in the problem, and
the relations betiueen the unhnoiun quantity and given quanti-
ties derived from these statements, using x tvhenever the un-
hnoion quayitity occurs. We shall thus arrive at an equation
from luhich the value ofx may be found.
Solution of problems by means of Simple Equations. State in general terms
the steps to be taken.
7
146 ele:>ii:xtaky algekha.
Ex. 1. What number is that, to which if 8 be added, one-
fourth of the sum is equal to 29 ?
Let X represent the number required.
Adding 8 to it, we have x + 8, and one-fourth of this is \ {x 4- 8) ;
Ave have, therefore, the equation i(a; + 8)=29; whence it; = 108.
Ex. 2. What number is that, the double of which exceeds
its half by 6 ?
Let a'= the number ; then the double of x is 2.r, and the half of x is
\X', hence, 2x—\x=^%\ whence a;=4.
Ex. 3. The ages of 3 children together amount to 24
years, and they were born two years apart : what is the age
of each ?
Here we have
Known quantities.
1. The sum of the ages of all
three, 24 years.
2. The difference between the
ages of any two of them.
Unknown and required.
1. Age of youngest.
2. Age of next.
3. Age of the oldest.
But, in reality, we have only one unknown quantity to find^ because,
when we know the age of one of the children, the ages of the two
others immediately follow. So that we say,
let x be the age of the youngest ;
then X + 2= next ;
and « + 4= oldest.
Thus far we have expressed, algebraically, 07ie of the two known
coftditions of the problem. There still remains to notice, that the sum
of the ages is 24 years. Now this sum is ^x + 6, adding together x^ a;+2,
and x-\-^;
:. 3.-?? + 6=24, an equation from which to find x.
Transposing, 3a=24— 6, or 18;
dividing, ^="5" » ^^ ^'
o
.'. the age of the youngest is 6 years,
next . . 8 .
oldest . . 10
SIMPLE EQUAT]0]S'S. 147
Ex. 4. A cask, wliicli held 270 gallons, was filled with a
mixture of brandy, wine, and water. There were 30 gallons
of wine in it more than of brandy, and 30 of water more
than there were of wine and brandy together. How manv
were there of each ?
Let ^=: number of gals, of brandy ;
.*. x + dO= wine;
and2aj + 30= wine and brandy together;
... 2a,' + 30 + 30 or 2x + 60=gals. of water ;
but the whole number of gallons was 270 ;
.-. a^ + (aj + 30) + (2,2; + 60)=270;
whence x= 45, the number of gals, of brandy,
25 + 30= 75, wine,
22^ + 60=150, water.
Ex. 5. A sum of £50 is to be divided among A, B,.and C,
so that A may have 13 guineas more than B, and C £5 more
than A : determine their shares.
Tjet x=B^s share in sliilUngs :
.'. « + 273=^'s, and .(;r + 273) + 100 or ^ + 373=6''s ;
.-., since £50=1000s., {x + 27S) + x + {x + d7S)=Sx + 64.Q=1000;
.-. 3^r=354, and 0^=118, :c + 273=391, aj + 373=491,
and the shares are 391s., 118s., 491s., or £19 lis., £5 18s., £24 lis., re-
spectively.
Ex. 6. A, B, C divide among themselves 620 cartridges, A
taking 4 to ^'s 3, and 6 to (7's 5 : how many did each take ?
Let .Tr=^'s share ; then |a;=jB's, ■|.c=(7's ;
.-. aj + |aj+ 5,^=620; whence ir=240, |.t'=180, |-a^=200.
We might have avoided fractions by assuming 12x for J.'s share, whea
we should have had 9aj=^'s, and 10.2^=: C^ ;
.'. 12a^ + 9^ + 10i=620 ; whence a'=20 ;
and the shares are 240, 180, 200, as before.
148 ELEMEXTARY xVLGEBRA.
Ex. 7. A line is 2 feet 4 inches long ; it is required to di-
vide it into two parts, such that one part may be three-
fourths of the other part.
Let X denote the number of inches in the larger part ; then — will
denote the number of inches in the other part.
The number of inches in' the whole line is 28 ; therefore
. + 1-28;
Whence, ^ + 3a?=112 ;
that is, 7^=112 ;
and, x=10.
Thus one part is 16 inches long, and the other part 12 inches long.
Ex. 8. A grocer has some tea worth half-a-dollar a lb., and
some worth 87^ cents a lb. ; how many lbs. must he take
of each sort to produce 100 lbs. of a mixture worth 62^
cents a lb. ?
Let x= the number of pounds of the first sort; then 100— a? will de
note the number of lbs. of the second sort. The value of the x lbs. is
ix dollars, and the value of the (100— 3j) lbs. is KlOO— .r) dollars; and
the whole value is to be | X 100 dollars.
Therefore, § X 100= Jc + 1(100 -x) ;
multiplying by 8, 500=4a; + 700- 7^ ;
whence, 3x=200 ;
Thus there must be 66| lbs. of the first sort, and 33^ lbs. of the sec-
ond sort.
Ex. 9. A person had $5000, part of which he lent at 4
per cent., and the rest at 5 per cent. ; the whole annual in-
terest received was $220 : how much was lent at 4 per cent. ?
Let x= the number of dollars lent at 4 per cent. ;
then 5000— a;= the number of dollars lent at 5 per cent. ;
SIMPLE EQUATIOXS. 149
and T7m~ ^^^^ aimual interest from the former;
5(5000 x)
and ' — the annual interest from tli£ latter;
4x 5(5000 -.'?) ^^^
therefore, ___ + 1-__J=220;
whence, 4x + 5(5000-.t)=22000 ;
that is, 47?-f 25000-52=22000 ;
/. -a;=-8000, or 2=3000.
Thus $3000 was lent at 4 per cent.
Ex. 10. Divide 42 into 4 parts which shall be 4 consecu-
tive numbers.
Let X be one part ;
then ic-{-l^ ^+2, x-{-S, are tlie other parts ;
and X + (2+l)+(2+2)+(2+3)=42, by the question ;
combining, 42j-|-6=42,
or, 4^=36 ;
.-. 2=9, and 2+1=10, 2 + 2=11, 2+8=12 ;
.*. 9, 10, 11, 12, are the required parts.
181. The great difficulty which the beginner finds m solv-
ing these problems is in translating the statements of the
enunciation into algebraical language. In this, practice alone
can give readiness and accuracy. The teacher will find it
advantageous to train the student orally in such transla-
tions, by means of examples like those given in Art. 40.
182. The student should always read carefully and con-
sider well the meaning of the question proposed ; and in or-
der to avoid error, he should observe that x represents an
unJcno2vn number of dollars, poimds, feet, miles, liours, or in
general, an iinhnoivn nwnher of tilings or units, and both
the land and denomination of the units of x should be dis-
tinctly noticed in the statement.
What caution is given to the student in Art. 181?
150 ELEMEJ^TARY ALGEBRA.
183. Many of the problems given below may be solved
readily by Arithmetic, but the student will soon perceive
the superiority of the method of solution by Algebra, in
power and generality and easy application.
Examples — 36.
1. What number is that which exceeds its sixth part
by 10?
2. What number is that, to which if 7 be added, twice
the sum will be equal to 32 ?
3. Find a number, such that its half, third, and fourth
parts shall be together greater than its fifth part by 106.
4. A bookseller sold 10 books at a certain price, and after-
ward 15 more at the same rate, and at the latter time re-
ceived $8.75 more than at the former : what was the price
per book ?
5. What two numbers are those, whose sum is 48 and dif-
ference 22 ?
6. At an election where 979 votes were given, the success-
ful candidate had a majority of 47 : what were the num-
bers for each ?
7. A spent 62J- cents in oranges, and says, that 3 of them
cost as much under 25 cts., as 9 of them cost over 25 cts. -■
how many did he buy ?
8. The sum of the ages of two brothers is 49, and one of
them is 13 years older than the other : find their ages.
9. Find a number such that if increased by 10 it will be-
come five times as great as the third part of the original
number.
10. Divide 150 into two parts, so that one of them shall be
two-thirds of the other.
SIMPLE EQUATIONS. 151
11. A child is born in November, and on the tenth day of
December he is as many days old as the month was on the
day of his birth : when was he born ?
12. There is a number such that, if 8 be added to its
double, the sum will be five times its half. Find it.
13. Divide 87 into three parts, such that the first may ex-
ceed the second by 7, and the third by 17.
14. Find a number such that, if 10 be taken from its
double, and 20 from the double of the remainder, there may
be 40 left.
15. A market-woman being asked how many eggs she had,
replied : If I had as many more, half as many more, and one
egg and a half, I should have 104 eggs : how many had she ?
16. A is twice as old as B ; twenty-two years ago he was
three times as old. Required ^'s present age.
17. Divide $64 among three persons, so that the first may
have three times as much as the second, and the third, one-
third as much as the first and second together.
18. A workman is engaged for 28 days at 62^ cts. a day,
but is to pay 25 cts. a day, instead of receiving anything, on
all days upon which he is idle. He receives altogether
$13.12.j : for how many idle days did he pay ?
19. A person buys 4 horses, for the second of which he
gives £12 more than for the first, for the third £6 more than
for the second, and for the fourth £2 more than for the
third. The sum paid for all was £230. How much did
each cost?
20. A person bought 20 yards of cloth for 10 guineas, for
part of which he gave lis. 6d. a yard, and for the rest 75. 6d.
a yard : how many yards of each did he buy ?
152 ELEMEXTARY ALGEBllA.
21. Two coaches start at the same time from A and B, a
distance of 200 miles, travellirng one at 9^ miles an honr,
the other at 9^: where will they meet, and in what time
from starting?
22. A father has six sons, each of whom is four years older
than his next younger brother ; and the eldest is three times
as old as the youngest : find their respective ages.
23. ^ 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,
24. Two persons, A and B, are travelling together; A has
£100 and B has £48: they are met by robbers, who take
twice as much from A as from B, and leave to A three times
as much as to i? : how much was taken from each ?
25. Find two consecutive numbers, such that one-half and
one-fifth of the first, taken together, shall be equal to one-
third and one-fourth of the second, taken together.
26. A cistern is filled in 20 minutes by 3 pipes, the first of
w^hich conveys 10 gallons more, and the second 5 gallons less,
than the third, per minute. The cistern holds 820 gallons.
How much flows through each pipe in a minute ?
27. A garrison of 1000 men was victualled for 30 days ;
dfter 10 days it was re-enforced, and then the provisions were
exhausted in 5 days: find the number of men in the re-
enforcement.
28. In a certain weight of gunpowder, the saltpetre com-
posed 6 lbs. more than a half of the weight, the sulphur 5 lbs.
less than one-third, and the charcoal 3 lbs. less than one-
fourth : how many pounds were there of each of the three
ingredients ?
29. A general, after having lost a battle, found that he
had left, fit for action, 3600 men more than half of his army;
600 men more than one-eighth of his army were wounded ;
SIMPLE EQUATIONS. 153
and the remainder, forming one-fiftli of liis army, were slain,
taken prisoners, or missing: what was the number of the
army ?
30. A tradesman starts v/ith a certain sum of money : at
the end of the first year he had doubled his original stock,
all but £100; also, at the end of the second year he ha\
doubled the stock at the beginning of that year, all but
£100; also in like manner at the end of the third year; -and
at the end of the third year he found himself three times as
rich as at first : what Avas his original stock ?
XXV. Problems — continued.
184. We shall now^ give some examples rather more diffi-
cult than the examples of the preceding chapter.
Ex. 1. Find a number, such that if | of it be subtracted
from 20, and y^ of the remainder from | of the original
number, 12 times the second remainder shall be half the
original number.
Let X = the number ;
.-. 20— |.T=lst remainder, and l-a'—-f^{20—^x) = 2d remainder;
.*. 12[J:r-y^Y(20— |.?0]=4.r, by the question; whence ^ = 24.
Ex. 2. A certain number consists of two digits whose
difference is 3 ; and, if the digits be inverted, the number so
formed will be ^ of ^^^^ former : find the original number.
Let a;=lesser digit, and .*. ^+3==ihe greater: then, since the value of
a number of two digits = ten times the lirst digit+the second digit,
(thus 67=10x0+7), the number in question=10 ( a'+a ) H-^ ; similarly,
the number formed by the same digits inverted=10.2'+(.'c+3); hence,
by question, 10c+(a?+3)=f[10(;c+3)+.i'], whence ii=3, ic-\-'S=6, and
the number required is 63.
Ex. 3. A can do a piece of work in 10 days; but, after he
has been upon it 4 days, B is sent to help him, and they
154: ELEMENT AH Y ALGEBRA.
finish it together in 2 days. In what time would B have
done the whole ?
Let a'^u umber of days B would have taken, and IT denote the work :
W W
.-. — , — , are the portions of the work which A, B would do in one
10 X
4Tr
day ; hence in 4 days, A does — -r-, and in 2 days, A and B together
^" 10+^= ••• TO+lO +-^-^^' whence .==0.
It is plain that, in the above, we might have omitted W altogether,
or taken unity to represent the work, as follows :
J., 5 do — , — of the work, respectively, in one day, and therefore,
4 2 2
reasonino^ lust as before, -7:+-^H — = the whole w^ork = 1.
[In all such questions, the student should notice that, if a person
does — ths of any work in one day, he will do — th of it in — th of a day,
and therefore the whole work in — days.
Thus, if he does -| in one day, he will do ^ in -J- of a day, and there-
fore the whole, or -^ , in 3^=2^ days.]
Ex. 4. A alone can perform a piece of work in 9 days, and
B alone can perform it in 12 days : in what time will they
perform it if they work together ?
Let X denote the required number of days. In one day A can perform
1 X
— th of the work ; therefore, in x days he can perform ^ths of the work.
In one day B can perform r^ of the work ; therefore, in x days he can
perform T^ths of the work. And since in x days A and B together
perfoi-m the whole work, the sum of the fractions of the work must be
equal to unity ; that is,
9^12
Multiplying by 36. 4^'+3:i;=36,
that is, 7a?=36 ;
36
therefore, if=— =5-JJ-
SIMPLE EQUATIOIs^S. 155
Ex. 5. A cistern could be filled with water by means of
one pipe alone in 6 hours, and by means of another pipe alone
in 8 hours ; and it could be emptied by a tap in 12 hours, if
the two pipes were closed : in what time will the cistern be
filled if the pipes and the tap are all open ?
Let X denote the required number of hours. In one hour the first
1 X
pipe fills — th of the cistern; therefore, in x hours it fills Trths of the
6 6
cistern. In one hour the second i3ipe fills — th of the cistern ; therefore,
o
X 1
in X hours it fills ^ths of the cistern. In one hour the tap empties — rth
X
of the cistern ; therefore, in x hours it empties T^ths of the cistern. And
since in x hours the tohole cistern is filled, we haye
XX ^ _-,
Multiplying by 24, 4x + dx—2x=24:,
that is, 5x=24 ;
therefore, a*==^==4|-.
185. It is sometimes convenient to take x to represent not
the quantity which is actually demanded in the question,
but some other unknown quantity on which if depends.
This we will illustrate by some examples. But experience is
the only guide to the best selection of the unknown quantity.
Ex. 6. A colonel, on attempting to draw up his regiment
in the form of a solid square, finds that he has 31 men over,
and that he would require 24 men more in his regiment in
order to increase the side of the square by one man : how
many men were there in the regiment ?
Let X denote the number of men in the side of the first square ; then
the number of men in the square is .^''^, and the number of men in the
regiment is x"^ + 31. If there were x-hl men in a side of the square, the
number of men in the square would be (x + lf ; thus the number of
men in the regiment is (:r' + l)'^— 24.
156 ELEMEXTAEY ALGEBRA.
Therefore,
(^+1)^-
-24=.
i'' + 31;
that is,
x' + 2x + l-
-24-
,v'' + Sl.
From these
two
equal expressions we can remove x"^,
which
occurs
in both ; thus,
2^ + 1
-24=
=31;
therefore.
2.r=31-l
+ 24=
=54;
or,
a:
54_
=27.
Hence the number of men in the regiment is (27f + 31; that is,
729 + 31; that is, 760.
Ex. 7. A starts from a certain place and travels at the :ate
of 21 miles in 5 hours ; B starts from the same place 8 hoars
after A, and travels in the same direction at the rate of 15
miles in 3 hours : how far will A travel before he is overtaken
by 5?
Let a^=the number of hours which A travels before he is overtaken ;
then ic— 8=the number of hours B travels.
21
Now since A travels 21 miles in 5 hours, (i. e.) ~ of a mile in one
o
hour, therefore,
~— = the number of miles which A travels in x hours ;
5
15
and similarly, —(^—8)= number of miles which B travels In x horn's
o
Therefor^' ^ (aj-8)=^ ;
o O
25(«-8)=21a?;
25iz;-21^=200 ;
a— 50.
21 V 21
Therefore, -—=— X 50=210 ; so that A travelled 210 miles before
5 5
lie was overtaken.
186. The principles of proportion, as taught in Arithmetic,
are often used to form the equations.
Ex. 8. It is required to divide the number 60 into three
• parts, such that they may be to each other in the proportion
of the numbers 3, 4, and 5.
SIMPLE KOUATIOXS. 15?
Let the number x denote the first part. Then, since
1st part : 2d part : : 8 : 4, therefore — - is the second part;
o
and since 1st part : 3d part : : 3 : 5, tlierefore — is the third part
o
Therefore, the sum of the parts
x + -x-\--x—m',
3a? + 4aj + 5aj=180; ,
12.r=180 ;
a:'=15, the first part.
4 5
Hence the 2d part is — Xl5, or 20, and the 3d part is ^Xl5, or 25.
o o
The preceding mode of solution, and many similar solu-
tions involving proportions, may be shortened after the
following manner :
Let 3.^ denote the first part ; then the second part must be 4a;, and
the third part must be 5.r.
Therefore, ^x + ^x + ^i =80 ;
12, =60; a =5.
. . 3x5=15, the first part; 4x5=20, the second part; 5X5=25, the
third part.
Ex. 9. There are two bars of metal, the first containing 14
oz. of silver and 6 of tin, the second containing 8 of silver
and 12 of tin : how much must be taken from each to form
a bar of 20 oz., containing equal weights of silver and tin ?
Let a:=number of oz., to be taken from first bar, '^0 — x from second;
now ^0" of ^^^ fi^st bar, and therefore of every oz. of it, is silver ;
and, similarly, -^ of every oz. of the second bar is silver ;
Hud there are to be, altogether, 10 oz. of silver in the compound ;
.'. ii^' + ^o (3O-.70=lO, whence .r=r)f, and 20-.?=13i.
Ex. 10. Find the time between two and three o'clock, when
the minute-hand of a watch is exactly over the hour-hand.
Find, also, the time between 2 and 3 o'clock, when the hands
are exactly opposite each other.
158 ELEMEXTAilY ALGEBEA.
1st Ciise. — Let x denote the required number of minutes after 2 o'clock.
In X minutes the minute-hand moves over x divisions of the watch-face ;
and as the long hand moves 12 times as fiist as the short hand, the latter
"will move over — divisions in x minutes. At 2 o'clock the short hand
is 10 divisions in advance of the long hand ; so that in the x minutes
the long hand must pass over 10 more divisions than the short hand.
Therefore, ^=T^ + 1^ 5
12^=^ + 120;
110^=120;
aj=-— 7- =:10yt minutes.
Or more briefly, thus : The minute-hand in every minute gains —
of one minute-division on the hour-hand. Hence, in x minutes, it gains
-r^r- divisions. Therefore, -3-^=10. :. x=———10\f.
1/^ \Z 11
. 2d Case. — Here the minute-hand must not only overtake the hour-
hand, but advance so as to leave it 30 minute-divisions behind. Then
let (C= required number of minutes after 2 o'clock ; then the gain of the
llx
minute-hand, or — ^ =10 -\- 30.
llx
:. -—-=40, lla?=480 ; x=A:^-^j minutes. Hence the hands are in the
required position at 43x\ minutes past 2 o'clock.
Ex. 11. A hare takes four leaps to a greyhound's three,
but two of the greyhound's leaps are equivalent to three of
the hare's ; the hare has a start of 50 leaps : how many leaps
must the greyhound take to catch the hare ?
Suppose that ^x denote the number of leaps taken by the greyhound ;
then Ax will denote the number of leaps taken by the hare in the same
time. Let a denote the number of inches in one leap of the hare ; then
3rx denotes the number of inches in three leaps of the hare, and there-
fore also the number of inches in two leaps of the greyhound : therefore
3<2
-— denotes the number of inches in one leap of the greyhound. Then
^x leaps of the greyhound will contain ^xX~ inches. And 50-l-4iB
leaps of the hare will contain (50-f-4.T) a inches; therefore,
^xa _. . .
— -=(oO -I- 4a;) a.
SIMPLE EQUATIONS. 159
Dividing by a, i7=50 4- 4c ;
therefore, 9.^=100 + 8^;
or, . a.'=100.
Thus the greyhound must take 300 leaps.
Here an auxiliary symbol a has been introduced to enable us to form
the equation more easily. Being in every term, it is removed by di-
vision Avhen the equation is formed.
Ex. 12. If tlie specific gravity of pure milk be 1*03, and
a certain mixture of milk and water be found (by means of
an instrument for the purpose) to be of specific gravity
1*02625, how much water has been added ?
[Definition.— ^By the specific gramty of a substance is meant the
number of times which its weight is of an equal bulk of water. Thus
the specific gramty of silver is 10"5, or 10 j, which means that any
quantity of silver is lOJ times the weight of the same hulk of water.
The specific gravity of milk being 1.03, signifies that milk is lyl-g-
times as heavy as water ; and so on.]
Let 1 quart of water be added to x quarts of pure milk to form the
mixture; then,
Since the w^ eight of x quarts of pure milk
=1 .03 times the weight of x quarts of water,
— 1.03X X X weight of 1 quart of water;
.-. whole weight of water and milk
=(1 + 1.03.r)X weight of 1 quart of water.
But there are 1+x quarts of the mixture whose specific gravity is
1.02625; .*. the Avhole weight of this
=1.02625(1 +cr)X weight of 1 quart of water;
.-. l + 1.03.T=1.02625(l + ir).
Therefore, (1. 03 -1.02625>r=l. 02625-1 ;
that is, .00375.t;=.02625 ;
_. 02625
•*• "^"".00375
Hence, 1 quart of water has been added to 7 quarts of milk ; (i. e.)
o7i€-eigWi of the mixture is water.
160 ELEMENTARY ALGEBRA.
Examples — 37.
1. Out of a cask of wine of which a fifteenth part had
leaked away, 12 gallons were drawn, and then it was two-
thirds full : how much did it hold ?
2. In a garrison of 2744 men there are two cavalry sol-
diers to twenty-five infantry, and half as many artillery as
cavalry : find the number of each.
3. The first digit of a certain number exceeds the second
by 4, and when the number is divided by the sum of the
digits, the quotient is 7 : find it.
4. The length of a floor exceeds the breadth by 4 feet; if
each had been increased by a foot, there would have been 27
more square feet in it : find its original dimensions.
5. In a mixture of copper, lead, and tin, the copper was
5 lbs. less than half the whole quantity, and the lead and
tin each 5 lbs. more than a third of the remainder : find the
respective quantities.
6. A horse was sold at a loss, for $210; but if it had
been sold for $262.50, the gain would have been three-
fourths of the former loss : find its real value.
7. A can do a piece of work in 10 days, which B can do
in eight ; after A has been at work upon it 3 days, B comes
to help him : in what time will they finish it ?
8. There is a number of two digits whose difference is 2,
and, if it be diminished by half as much again as the sum
of the digits, it will give a number expressed by the digits
inverted : find it.
9. A number of troops being formed into a solid square,
it was found that there were 60 over; but when formed into
a column with five men more in front than before, and three
less in depth, there was just one man wanting to com-
plete it : what was the number of troops ?
SIMPLE EQUATIONS. 161
10. A person has travelled altogether 3036 miles, of which
he has gone seven miles by water to four on foot, and five by
Avater to two on horseback: how many did he travel each
way ?
11. A mass of copper and tin weighs 80 lbs., and for every
7 lbs. of copper there are 3 lbs. of tin : how much copper
must be added to the mass, that there may be 4 lbs. of tin
for every 11 lbs. of copper?
12. A does I of a piece of work in 10 days, when B comes
to help him, and they take three days more to finish it : in
what time would they have done the whole, each separately,
or both together ?
13. A and B were employed together for 50 days, each at
$1.20 a day; during which time A, by spending 12 cents a
day less than B, had saved three times as much as B, and 2^
days' pay besides : what did each spend per day ?
14. There are two silver cups, and one cover for both ; the
first weighs 12 oz., and with the cover weighs twice as much
as the other cup without it ; but the second with the cover
weighs a third as much again as the first without it : find
the weight of the cover.
15. Find a number of three digits, each greater by unity
than that which follows it, so that its excess above one-
fourth of the number formed by inverting the digits shall be
36 times the sum of the digits.
16. If 19 lbs. of gold weigh 18 lbs. in water, and 10 lbs.
of silver weigh 9 lbs. in water, find the quantity of gold and
silver respectively in a mass of gold and silver weighing 106
lbs. in air and 99 lbs. in water.
VI, A and B can reap a field together in 12 hours, A and
(7 in 16 hours, and A by himself in 20 hours : in what time
could, 1st, B and C together, and, 2d, A, B, and G to-
gether, reajf it?
162 ELEMEXTARY ALGEBRA.
18. Find two numbers whose difference is 4, and the dif-
ference of their squares 112.
19. Divide the number 88 into four parts, such that the
first increased by 2, the second diminished by 3, the third
multiplied by 4, and the fourth divided by 5, may all be
equal.
20. Three persons whose powers for w^ork are as the num-
bers 3, 4, 6, can together complete a piece of work in 60
days : in what time could each alone complete the work ?
21. A and B are at present of the same age; if ^'s age be
increased by 36 years, and ^^s by 52 years, their ages will be
as 3 to 4 : w^hat is the present age of each ?
22. Divide 100 into two parts, such that the square of their
difference may exceed the square of twice the less part by
2000.
23. A cistern has two supply-pipes which will singly fill
it in 4|- hours, and 6 hours, respectively ; and it has also a
leak, by which it would be emptied in 5 hours: in how many
hours will it be filled when all are working together ?
24. A market-woman bought a certain number of eggs at
the rate of 5 for 2 cents ; she sold half of them at 2 for a
.cent, and half of them at 3 for a cent, and gained 4 cents by
so doing : what was the number of eggs ?
25. A and B shoot by turns at a target ; A puts 7 bullets
out of 12 into the bull's-eye, and B puts in 9 out of 12 ; be-
tween them they put in 32 bullets : how many shots did
each fire ?
26. Two casks, A and B, contain mixtures of wine and
water ; in A the quantity of wine is to the quantity of water
as 4 to 3 ; in B the like proportion is that of 2 to 3. If A
contains 84 gallons, what must B contain, so that when the
two are put together the new mixture may be half wine aud
half water? "^
SIMPLE EQUATION'S. ' 163
27. How many minutes does it want to 4 o'clock, if three-
quarters of an hour ago it was twice as many minutes past
two o'clock?
28. What is the time after 6 o'clock at which the hands of
a watch are, 1st, directly o23posite, and, 2d, at right angles
to each other ?
29. It is between 11 and 12 o'clock, and it is observed that
the number of minute-spaces between the hands is two-
thirds of what it was ten minutes previously : find the time.
30. The national debt of a country was increased by one-
fourth in a time of war. During a long peace which fol-
lowed, $125,000,000 was paid off, and at the end of that
time the rate of interest was reduced from 4|- to 4 per cent.
It was then found that tfle amount of annual interest was
the same as before the war : what was the amount of debt
before the war ?
31. Find three numbers, the sum of which is 70, and such
that the second divided by the first gives 2 for quotient, and
1 for remainder ; and the third divided by the second gives
3 for quotient, and 3 for remainder.
32. Shells are thrown from two mortars in a besieged city ;
the first mortar has thrown 36 shells before the second com-
mences its fire, and it sends 8 shells for every 7 sent by the
second ; but the second expends as much powder in 3 dis-
charges as the first does in 4 : how many shells must the sec-
ond mortar throw in order to expend the same amount of
powder as the first ?
187. We shall now give a few problems in which the
known quantities are represented by the first letters of the
alphabet, instead of numbers.
Examples — 38.
1. Find a number, such that being divided successively by
m and n, the sum of the quotients shall be equal to a.
T64 ELEMENTARY ALGEBRA.
2. Divide a number a into two such part3 tV^ai the quo-
tient of the one divided by m, and the other divided by n,
may be equal to h.
3. Divide a number a into two parts proportional to the
numbers m and n,
4. Divide a number a into three parts, such that the first
may be to the second as m is to n^ and* the second to the
third asj9 is to q.
5. Two numbers, a and I, being given, what number must
be added to each one of them in order that the ratio of the
'}7l
two sums may be equal to — ?
6. 'Three fountains will fill a ce]i;ain reservoir, when each
one runs alone, in the times a, t, and c, respectively. In
what time will they fill it, all running together ?
7. Two couriers, whose distance apart > when they set out
was d miles, travel toward each other, the one moving at the
rate of h miles an hour, and the other at the rate of c miles
an hour. In what time after starting will they meet ?
8. A can do a piece of work in h days, and B can do the
same work in c days. In what time can they together do the
work?
XXYI. SiMUIiTANEOUS EQUATIONS OF THE FlRST DEGREE.
188. If one equation contain tivo unknown quantities,
there are an infinite number of pairs of values of these by
which it may be satisfied.
Thus in .t=10 — 2?/, if we give any value to y, we shall get
a corresponding value for x, by which pair of values the
equation will of course be satisfied. If, for example, we take
When one equation contains two unknown quantities, what values may they
have?
SIMULTANEOUS EQUATIONS. 165
t/—l, we shall get :?;r=:10 — 2 = 8; if y^2, x=z6; if ^=3,
x=4cf &c.
One equation then, containing ttvo unknown quantities
(or, as it is expressed, "between two unknown quantities"),
admits of an infinite number of solutions ; but if we have
as many different equations as there are quantities, the num-
ber of solutions will be limited ; for it will be seen that they
can always be reduced to a single equation containing a sin-
gle unknown quantity.
Thus, while each of the equations, 2:= 10— 2^, 4:X=:32 — 6y,
separately considered, is satisfied by an infinite number of
pairs of values of x and y, we shall find there is only one
value of X, and one value of y, which will satisfy both equa-
tions; for, multiplying the first equation by 3,
dx=30 — 6y; now take this from
the second equation 4:^=32 — 6^, and we get x=i2.
Thus 2; =2 is the only value of x common to both equa-
tions. Put this value of ;c in either of the two given equa-
tions — for example, in the first; and we obtain,
2=^10-2^;
/. 2y=S; .-. y=^.
Thus, x=2, y=^4:, are the only pair of values which satisfy
both equations.
189. Equations of this kind, which are to be satisfied by
the same pair or pairs of values of x and y, are called simul-
taneous equations. In the present chapter we treat of si-
multaneous equations of the first degree, (i. e.) where each
unknown quantity occurs only in the first power, and the
product of the unknown quantities does not occur. If
there be three unknown quantities there must be three equa-
tions, and so on.
Simultaneous EquaUons.
166 ELEMEl^TARY ALGEBRA.
190. These simultaneous equations must all express dijf ev-
ent relations between the unknown quantities.
Thus, if we had the equation x=10—2y, it would be of
no use to join with it the equation 2iz;=20— 4y (which is the
double of the former), or any other deriyed like this from
the former.
191. There are generally given three methods for solving
simultaneous equations of two unknowns; but the object
•aimed at is the same in each, viz., to combine the two equa-
tions in such a manner as to' expel, or eliminate, one un-
known from the result, and so get an equation of 07ie un-
known only.
192. First Method. — Multiply fhe equations by the least
numbers wliicli will maJce the coefficients of 07ie of the tin
Icnoivn quantities the same in both resulting equations ; then
adding or subtracting the two equatio7is thus obtained, accord
mg as the equal terms have different or the same signs, these
terms ivill destroy each other, and, the elimination will be ef-
fected.
Ex.1. ^x-vZy^ 4 1(1)
3a:-2^=-7) (2)
Multiply (1) by 3, 6a; +9^/= 12.
Multiply (2) by 2, 62;-4y--14.
Subtracting, 13?/=26 and /. ^=2.
Then put this value of y in either (1) or (2) ; for example,
in (1). We have thus:
2a; + 6=4; .-. 2a;==4-6=:-2; .-. a:=-l.
Ex. 2. 8a;+7?/=100 ) (1)
12a;- 5y= 88 ) (2)
What is said of the relations expressed by simultaneous equations ? Of the ohject
aimed at by each of the three methods of solving them ? First method ?
SIMULTANEOUS EQUATI02s^S. 167
We might multiply (1) by 12 and (2) by 8, giving thus :
96x-4:0y= 7 04
Subtracting, 124^= 496 .-.«/ = 4.
• But the process is more simple if we multiply equation (1)
by 3 and equation (2) by 2. Thus :
24:X-\-21y==300
24.x-10tj=176
3 1^=124; .•.^=4; and, substituting
m (1), 8a;+28 = 100; /. Sx=z'72, and x=9.
We see here the advantage of multiplying by the leasi
numbers which will make the coefficients the same, though
we may multiply by any nmnbers which will effect the same
object.
It is sometimes possible to multiply one of the given equa-
tions by some number which will make the coefficient of x
or y in it the same as in the other equation. The process is
in this case much shortened.
Ex.3. 4.x -^y^U) (1)
4y+a;=16j (2)
Here multiplying (2) by 4, 16^ + 42^=64;
but y+jtx=U; (1)
/. subtracting, 16y =30, and ,\y=2-^
and (2) .r=:16-4y = 16-8=8.
Ex. 4. 4:x- y= 7) (1)
dx + 4:y=29 ) (2)
Here 3.T + 4y=29,
and, multiplying (1) by 4, 16a;— 4^=28;
.*. adding, 19a; —57, and ,\x=d;
and (1) ^=4a;~7n= 12-7=5..
168 ELEMEKTARY ALGEBRA.
193. Second Method. — Express one of the unhnoiun quan-
fities 171 terms of the other by means of one of the equations^
and put this expression for it in the other equation.
Thus, taking the example 1 in the preceding article,
2.T + 3?/=: 4)(1) •
^x-^^-l) (2)
4—3^
From {l),x^=z — ^r-^; substituting this expression in (2), we
obtain
whence 3(4— 3?/)— 4^= — 14;
that is, 12-9«/-4^=-14; /. -133/^-26; .-. iy=2.
4-3y 4-6
,,x- ^ _ ^ _ i.
Ex. 5. 7^' + i(2y + 4) = 16 ) or reducing, 35:^ + 2;/= 76 ) (1)
3y-i(a: + 2)= 8) 12^- a;=:34j (2)
Here from (2), i?;=:12^-34, and from (1), 35(12i/-34)+2«/
= 76; whence y=^, and /. ir=2.
194. Third Method. — Express the same unlcnoivn quantity
in terms of the other in both equatio7is, and put these expres-
sions equal
Thus, taking again Ex. 1, 2x + 3y= 4 ) (1)
3x-2y=-7) (2),
(1) gives y=—^; (2) gives y=-^-;
^, ^ 3a; + 7 4.—2X
therefore, — - — = — - — .
Second method ? Third method ?
SIMULTAXEOUS EQUATIOKS. 1G9
Clearing of fractions,
9x + 21=zS-4.x. .\13x=z-r6, x=-^.
Ex.6. 5.7;-i(5y + 2) = 32) or reducing, 20.-?;— 5^=130 ) (1)
3?/+t(^ + 2)= 9) 9ij+ x= 25) (2)
Here in (1), y=^{20x-ld0), in (2), y=^{2o-x) ;
.^|(20.^•-130)3^i(25-^), whence a;=7, y=^2.
The first meiliod is to be preferred generally; but the second
may be used with advantage whenever either x or y has a co-
efficient unity in one of the equations.
Note. — It may be well to give, here, an abbreviation of the first
method, which saves much trouble when the coefficients are large.
Two examples will serve to illustrate it.
Ex. 54^-121^=15 ) (1)
^ . y to find X and y.
36a?-88?/=-12;)
36a.'- 77^= 21 ;f
(2)
Subtracting, 1 8a?— 44y= - 6 ;
multiplying by 2,
from 2d equation,
subtracting, 11^=33 ;
And I8.r=44?/-G=132-6=126; .'. a'=7.
Ex. 101aj-242/=G3 ) d)
103..-28^=29f^?^^^^"^^^-
Subtracting, 2x- 4?/= -34;
multiplying by 6, 12^'— 24?/= - 204 ; )
but 101a;- 24y= 63; )
subtracting, 89a; ==267 ;
267 ^
And 4^=2aj + 34=40 ; .-. y=10.
Which method is preferable ?
8
170 £Lemp:ktar.y algebra.
195. Ex. 7. Solve —+-=:=8l (1)
X y ^
^^-1? = 3! (2)
X y J
If we cleared these equations of fractions they would con-
tain X y, and could not then be solved by the methods of this
chapter. But if we do not clear them of fractions, they may
be solved readily by the methods given. Thus,
multiplying (1) by 3,
'-V?-*=24;
X y
multiplying (2) by 2,
X y '
adding,
-=.30; .-.90=
X '
.-. (1) gives
o
therefore, — = 8 — 4 =
=4; .-.8=4./; .
=2.
Ex. 8. Solve a''x-\-Vy:=^c'' (1), ax-^hy^c (2).
Multiplying (2) by Z>, and subtracting it from (1),
(^x^-h^y—&
abx-\-h''y — 'bG
G^x—abx—c^ — 'bc\
that is, a(a—h)x—c{C'—y)\
' ~a(a-by
substituting this value of x in (2),
acic—V) 7
~f iT + oy=c\
in
therefore,
SIMULTANEOUS
EQUATIOJfS.
hy-~
c{c-
-b)
c(a-
-c)
' («-
h)-
a—
-b
:. ij~
c{a—c)
Or this value of y might be found in the same way as that
of X was found.
Examples — 39.
Find the values of x and y in the following pairs of
equations :
1. 3a:-4y=2, Ix-^y^^t,
2. 7^-5?/=24, 4.x-3y=ll.
3. 3^^+2^=32, 20x-3y=:l,
4. llx-7y=37, 8^+%=41.
5. '7x-i-5y=^60, 13:r-lly=:10.
6. 6:^-7^=:. 42, 7x-6y = 75.
9. V+ 2 =^' 2+V=°-
4~3~' 3 + 6~
„ l-3a; 3?/— 1 „ dx + y
11. -nj^ + -Y-=2' ^^ + .V=9-
13. 3(3a; + 3j/) = 3(2a;-32/)+10, 4a;-3y=4(6y-2a;)4-3.
172 ELEMENTARY ALGEBRA.
13. x{ij^7)=y{x + l)l 14. i.^+i7/ = 13)
2a;4-20 = 3^+l ) :^x + i2j=D)
15. 1.^+^^=43)
16. 3.^+%=2.4, ,21x-My=z.03.
17. ,dx + A26tj=x-6, 3x-.6y^2S—,26i/.
18. ,0Sx-.21tj=.S3, .12x-}-.'7y=3M.
x y X y
20. — +-|-=2, Ix—ay—^.
21. ir + ^=:<2+Z>, 'bx-\-ay—2ab,
22. - + 4=1' -^ + -^=1.
ah a
23. {a-]-c)x—'by—'bc, x-\-y—a-\-h,
a ha
25. x-}-y=c, ax—hy—c{a—h),
26. « (^ + ^)+Z>(^— ^) = 1, a {x~y)+h {x-{-y) = l.
27. ?i:i^+l(z:^=o, ?±i/:i^+?i=lr^=o.
196. Simultaneous equations of three unknown quantities
are solved by eliminating one of the unknown quantities by
means of any pair of the equations, and then the same un-
known by means of another pair; we shall then hare two
equations involving only two unknown quantities, which we
may solve by the preceding rules. The remaining unknown
is found by substituting the values obtained for the other
two in any of the given equations.
Simultaneous equations of three or more unknown quantities.
SIMli/rAXEOUS EQUATIONS. 173
Similarly for simultaneous equatiojis of more than three
unknown quantities.
Ex.1.
x-2y+ 3^=2] (1)
2^-3^+ z=l I (2)
dx- y\ ?.z-^\ (3)
From (1)
2.^--4y+ 6^=4
(^)
"^x-Zy^ z=l
•. - y+ 5^=3 (4)
Again, from
(1)
dx—6y+ 9z=6
(3)
3x- y+ 2z=d
\ -6y+ 7;2=-3 (5)
but from (4)
-6y + 25z=15
—18^=— 18, and z=l
hence (4),
y= 6z-3=2
and (1),
x=2+2y-3z=:2 + 4:-3=S,
Ex.2.
1+^-^=1 (1)
t+± + l=2^ (2)
X y z ^ ^
I-l+i=14 (3)
X y z ^ '
Multiplying (1) by 2, | + A^| = 2
5 4
- + — ■.
X y z
^ + -^=36 (4)
Multiply (1) by 3, |+|_|=3
Adding (3) to this, ^ + -+-=24:
174
ELEMENT AKY
ALGEBKA.
Add
(3) to this,
1
X
y "
= 14
10
X
_2_
y"
-.11
(i
NoTv
' multiply (5) by 4,
40
X
8_
'y~
:68
Add
(4) to this.
X y
:26
47_
X ~
--M-
,-.47=!
__47_1
•''-94-2'
Substitute the value of x in (5) ; thus,
20--=17; /. -=20-17^:3; /. v=-|-.
y y ' ^ z
Substitute the values of x and y in (1) ; thus,
Ex.3. Solve -+^=1 (1)
From (1) -+-|-=1
subtract (2) — I — — 1;
a c
to the result,
add (3)
y
z
=
b
c
z
c
=1
^=1 y-^
Sr^iULTAXEOUS EQUATIONS. IvO
Substitute this value in (1) ; thus,
Substitute the same yalue in (3) ; thus,
1 z z 1 __c
These values of z and x might have been written down at
once from the symmetry of the equations, since it is obvious
that the values of x and z will be of the same form as that
of y, only interchanging a and c, respectively, with h.
Examples — 40.
1. 2a;-h3v + 4^=20l 2. 5^ + 3^^:65^
3a:+4y + 5;^=26 I 2y- z=ll
3:?; + 5^ + 6^=31 J 3^;+ 4;2=57^
3. dx^-2y- ^=201 4. x^y-^z=b^
'2x-^dy-\-Qz='10 \ x-\-y=z-l
X-
y-\-Qz—Al\ x-3=y + zj
5. x + 2y=7 ^ 6. xy=x + y
y-{-2z=2 I xz=2{x + z)
dx+2y=z—i J y^=^(y+^).
7. 2(2;-^) = 3^-2 1 8. ix + iy=12-iz
x + l = ^y+z) I iy+iz= S+ix
2.r+3^=4(l-^)J ix + iz=10
9. y + iz=:ix + 5
i(^-l)-i(y-^)-TV(^ + 3)
t.-i(2t/-5)=:lj-^^.
176 ELEMEXTARY ALGEBRA.
10. = -, - + - = 3|, - + - = -.
X y h y z ^ X y z
11. y-\-z=a, z-\-x=h, x-\-y=c,
12. x-\-y-\-z—a-^h-\-Cy x + a—y^h—z-\-c.
13. y^z—x=a, z^x—y—l^ x + y—z—c.
^ , X y z ^ X y z ^ x y z .
a o € a c a c
KoTE. — In Ex. 6, divide the equations by xy, xz,, and yz, respectively,
and they will then be of the form of those given in Ex. 10.
XXVII. Problems solved by Simultaneous Equations
OF THE First Degree.
197. Prob. 1. There is a certain fraction which becomes
equal to -J- when both numerator and denominator are dimin-
ished by 1 ; but, if 2 be taken from the numerator and added
to the denominator, the fraction becomes equal to ^ : find it.
Let X denote the numerator, and y the denominator of the required
fraction ; then the conditions of the problem give,
f-i~Y' y + 2~Y'
Clear the equations of fractions, — transpose, and reduce. We obtani
thus,
2x-y=l (1)
dx-y=S (2)
Subtracting(l)from(2) we getir=7; .'. (1) gives 14— 2/= 1; .*. 2^=1«>;
therefore, the required fraction is J^.
Prob. 2. There is a certain number composed of two fig-
ures or digits, which is equal to four times the sum of its
digits ; and if the digits exchange places, the number thus
formed is less by 12 than twice the former number: what is
the number?
SIMULTANEOUS EQUATION'S. 177
Let X be the digit in the tens^ place,
y ". . . . ^m^ts^ ;
then lO-r+y is the number (just as 23 = 10x2-f3), /. by the condi-
tions of the question,
10.zj + 2/=4(.i' 1-2/),
that is, =4:X-\Ay\
transposing, 10x — 4:X=A:y—y\
uniting, ^x=^y\
or, 2x=y (1).
Again, if the digits be reversed, 10y-{-x will be the number ; /. by
the question,
102/ + ^=2(103:-l-^)-12;
that is, =20.2^+2^-12;
transposing, l^x—Sy=VZ\
or, [••• 2/=2^', (1)], mc-\Qx=12', (2)
uniting, 3.r=12;
.'. ir=4; and ^=2.2?= 8.
. . the number requu'ed is 48.
pROB. 3. A railway train after travelling an hour is de-
tained 24 minutes, after which it proceeds at six-fifths of its
former rate, and arrives 15 minutes late. If the detention
had taken place 5 miles further on, the train would have ar-
rived 2 minutes later than it did. Find the original rate of
the train, and the distance travelled.
Let 5.^' denote the number of miles per hour at which the train origi-
nally travelled, and let y denote the number of miles in the whole dis-
tance travelled. Then y—t)X will denote the number of miles w^hicli
remain to be travelled after the detention. At the original rate of the
train this distance would be travelled in -~ — hours : at the increased
rate it will be travelled in ^ — hours. Since the train is detained 24
K)X
minutes, and yet is only 15 minutes late at its arrival, it follows
8*
1T8 ELEMKJSTAKY ALGEBiiA.
that the remainder of the journey is performed in 9 minutes less than
it would have been if the rate had not been increased ; and 9 minutes
9
is 7^ of an hour ; therefore,
00
y—6x_y-6x 9^
'~Q^~~5x~ 60 ^ ^*
If the detention bad taken place 5 miles further on, there would
have been y—6x—6 miles left to be travelled. Thus we shall find
that
y—6x-5 _y-5x-5 7
Qx ~ 6x 60 ^ ^'
Subtracting (2) from (1),
Qx~5x 60'
therefore, 50=60-2aj;
whence, 2a?=10; and x=5.
Substitute this value of a? in (1), and it will be found by solving the
equation that y=4!7^.
Prob. 4. A and B can together do a piece of work in a
days; A and can together do it in b days; B and C can
together perform it in o days: find the number of days in
which each alone could perform the work.
Let X denote the number of days in which A alone could perform
it, y the number of days in which B alone could perforai it, z the num-
ber of days in which C alone could perform it.
Then we have,
X z b ^ ^' *
1.1=1 (3).
y z c
Bubtractmg (2) fi-om (1) we obtain,
1_1=1_1 (4)
y z a b ^ '
SIMULTAIS^EOUS EOTATIOXS. 170
1 1 _ 1 ^
y z c '
2_1 1 . 1
y a b c
bc + db—ac
~~ aba
2abc
bc + ab—ac'
2abG
■■—r- T-, and x=
ab + ac—bc'
2abc
ac-\-bc—ab'
adding (3),
Therefore.
these latter vahies bemg written out at once hj the symmetry of the
equations.
Or we might have solved the problem thus :
Let ic=the number of units of work performed hj A in one day;
Let y= "
" " performed by B in one day ;
Let z=z
" " performed by C in one day.
hen,
<^+y=^ (1);
X + 2=ry (3);
y+z=\ (8).
These give by eliminating, as before,
bc-{-ab—ac
7/ = — -— .
^ 2abc '
Therefore J5's time of performing the whole work, or — =7 7 •
^ ° y bc + ab—ac
as before.
198. A problem may often be solved, as readily by a single
equation and one unknown quantity, as by simultaneous
equations with two or more unknown quantities. The ad-
vantage to a beginner in taking several letters to denote the
unknowns is that, though he has more equations and longer
work, he can more easily follow the steps by which the equa-
tions are formed.
Thus Ex. 19, Chap. XXV., may be solved by four simul-
taneous equations, involving four unknown quantities.
180 ELEMEJs^TARY ALGEBRA.
Examples — 41.
1. What fraction is that, to the numerator of which if 7 be
added, its yahie will be | ; but if 7 be taken from the denom-
inator its value will be | ?
2. There is a number of two digits wliich, when divided
by their sum, gives the quotient 4; but if the digits change
places, and the number thus formed be increased by 12, and
then divided by their sum, the quotient is 8: find the
number.
3. A rectangular bowling-green having been measured, it
was observed that, if it were 5 feet broader and 4 feet longer,
it would contain 116 feet more; but if it were 4 feet broader
and 5 feet longer it would contain 113 feet more: find its
present area.
4. A person rows on a uniformly flowing stream a distance
of 20 miles and back again, in 10 hours ; and he finds that
he can row 2 miles against the stream in the same time that
he rows 3 miles with it. Find the time of rowing down and
the time of rowing up.
5. A and B can do a piece of work together in 12 days,
which B, working for 15 days, and C for 30 days, would to-
gether complete. In 10 days, working all three together,
they would finish the work : in what time could they sepa-
rately do it ?
6. Some smugglers found a cave which would just exactly
hold the cargo of their boat, viz., 13 bales of silk and 33
casks of rum. "While unloading, a revenue cutter came in
sight, and they were obliged to sail away, having landed only
9 casks and 5 bales, and filled one-third of the cave. How
many bales separately, or how many casks, would it hold ?
SIML'LTAXKOrS IX^UATLO^^S. 181
7. Seven years ago, A was three times as old as B was;
and seven years hence, A will be twice as old as B will be :
find their present ages.
8. A certain fishing-rod consists of two parts : the length
of the npper part is to the length of the lower as 5 to 7 ; and
9 times the npper part, together Avith 13 times the lower part,
exceed 11 times the whole rod by 36 inches : find the lengths
of the two parts.
9. If the nnmerator of a certain fraction be increased by 1,
and the denominator be diminished by 1, the yalne will be 1 ;
if the nnmerator be increased by the denominator, and the
denominator diminished by the nnmerator, the valne will be
4: find the fraction.
10. A nnmber of posts are placed at equal distances in a
straight line. If to twice the nnmber of them we add the
number of feet between two consecutive posts, the sum is 68.
If from four times the number of feet between two consecu-
tive posts we subtract half the number of posts, the remainder
is 68. Find the distance between the extreme posts.
11. On the addition of 9 to a certain number of two digits,
its digits change places; and the sum of the first number
and the number thus formed is 33 : find the digits.
12. A and B ran a race which lasted five minutes. B had
a start of 20 yards ; but A ran 3 yards while B was running
2, and won by 30 yards: find the length of the course, and
the speed of each.
13. A person has two casks, with a certain quantity of
wine in each. He draws out of the first into the second as
much as there was in the second to begin with; then he
draws out of the second into the first, as much as was left in
the first; and then again out of the first into the second, as
much as was left in the second. There are then exactly 8
gallons in each cask. How much was there in each at first?
182 ELEMEIN^TAIIY ALGEBRA.
14. The year of our Lord in which the ^ cltangc of style^
from the Julian to the Gregorian Calendar was made in
England, may be thus expressed : The first digit being 1 for
thousands, the second is the sum of the third and fourth,
the third is the tJiird part of the sum of all four, and the
fourth is the fourth part of the sum of the first two. De-
termine the year.
15. A and B can together perform a certain work in 30
days : at the end of 18 days, howeyer, B is called ofi", and A
finishes it alone in 20 more days. Find the time in which
each could perform the work alone.
16. A cistern holding 1200 gallons is filled by three pipes,
A, B, C, together, in 24 minutes. The pipe A requires 30
minutes more than G to fill the cistern; and 10 gallons less
run through C per minute than through A and B together :
find the time in which each pipe alone would fill the cistern.
17. Find two numbers, the sum of which is equal to 3
times their difference, and their product equal to 4 times
their difference. (See ISTote at the end of "Examples — 40.")
18. The sum of two numbers is 13, and the difference of
their squares is 39. What are these numbers?
Note. — Here divide the second equafion by tlie first.
19. ^ and B are two towns, situated 24 miles apart, on the
same bank of a river : a gentleman goes fi^om A to B m^
hours by rowing the first half of the distance, and walking
the second half. In returning, he walks the first half at
three-fourths his former rate; but the stream being with
him in returning, he rows at double his rate in going ; and
he accomplishes the whole distance in 6 hours : find his rates
of walking and rowing.
20. Two trains, 92 feet long and 84 feet long, respectively,
are moving with uniform velocities on parallel rails : when
SIMULTAXEOUS EQUATIOKS. 183
tliey move in opposite directions, they are observed to pass
each other in one second and a half; but when they move in
the same direction, the faster train is observed to pass the
other in six seconds : find the rate at which each train moves.
21. A raih^oad runs from A to C. A freight train starts
from A at 12 o'clock, and a passenger train at 1 o'clock.
After going two-thirds of the distance the freight train
breaks down, and can only travel at three-fourths of its
former rate. At 40 minutes past 2 o'clock a collision occurs,
10 miles from C, The rate of the passenger train is double
the diminished rate of the freight train. Find the distance
from A to C, and the rates of the trains.
22. A certain sum of money was divided between A, B,
and C, so that ^'s share exceeded four-sevenths of the shares
of B and (7 by $30 ; also 5's share exceeded three-eighths of
the shares of A and C by $30 ; and (7s share exceeded two-
ninths of the shares of A and B by $30 : find the share
of each person.
23. A, B, and C can together perform a piece of work in
30 days ; A and B can together perform it in 32 days, and
B and C can together perform it in 120 days : find the num-
ber of days in which each alone could perform the work.
24 Express the two numbers whose sum is a, and their
diflPerence h.
25. Find two numbers, such that the product of the first
increased by a, and the second increased by h, exceeds by c
the produce of the two numbers ; and the product of the first
increased by m, and the second increased by n, exceeds by h
the product of the two numbers.
26. The sum of two numbers is a, and the difference of
their squares is h : find the two numbers.
27. Find two numbers, whose sum is m times their differ-
ence, and their product is n times their difference.
184 ELEMEJS^TARY ALGEBJIA.
XXVIII. Indices.
199. The indices or exponents which we have used hith-
erto are positive whole numbers, whicli express briefly the
repetition of the same factor in any product. (See Art. 15.)
Under this definition we have proved (Art. 59),
Also (Art. 74), oT'-^cf—a'^'',
200. Hence it will follow that (a'")^=:«"^"=(<^")^
For {a;''y' = d!'\0r,ar\ &C. n factors =^^+»»+"^+&c. n terms ^^mn^
and {o}'Y=^d!'.d!'.a\kQ>. m factors ^z^'^+^+^+ac- ^ terms _^nm.
.-. since a'^''~-=^cr\ we have {(f''Y=or'''—(a!'Y \
that is, tlie n*^ power of the m*^ potver of 2^ — the m*^ potoer
of the Vl^ power of 2.\ and either of them is found by multi-
plying the two indices.
201. Hence, also, it will follow that Va"^=(V<^)^
For let Va"'=:^^'^ ; then rr = {pf'Y^ {x^'Y by (199) ;
hence a — x^, and .*. Va=2:, and {\/aY^^x'^\
but also, by our first assumption, V^"'^-^"';
hence, we have, Va"' = (V^)"' ;
that is, the n*^ root of the va^^ poiver of ^ — the m"' poiver of
the n*'' o^oot of a.
202. These results refer so far to positive integral indices.
But now suppose we write down a quantity with a positive
fraction for an index, and agree that the law of multiplica-
tion, a^XaP'=a'^^'', shall hold true for m a7id n fractions as
well as m and n j^ositive ivhole numbers. What would such
a fractional index denote ?
For example : required the meaning of aK
Indices. Positive fractional indices.
INDICES. 185
By supposition we are to liave a'^Xa^—ci'—a. Thus a^
must be such a ii umber that if it be multiplied by itself the
result is a ; and the square root of a is by definition such a
number; therefore a^ must be equivalent to the square root
of a, that is, a7^=\/a.
Again ; required the meaning of a^.
By supposition Ave are to have,
a^Xa^Xa^'--=a^-^^^"'^^=a'=a.
Hence, as before, a^ must be equivalent to the cube root
of fl^; that is, a'^ = ya.
Again ; required tlie meaning of a*.
By supposition, a^Xa^Xa'^X a*=a^ ;
therefore, a* = Va^.
To give the definition in general symbols :
1. Required the meaning of a" where n is any positive
whole numler.
By supposition,
JL - JL _L^ i-^J-_(. ... to n terms
a''Xa''Xa''X ... to n factors — a"" "" "" —a =a;
_i_
therefore a" must be equivalent to the n^^ root of a,
2_
that is, ar=Va.
m
2. Required the meaning of a" tuhere m a7id n are any pos-
itive ivhole numbers.
By supposition,
!!L i!i !!L ^',^.^..... ton terms
a'^Xce^Xa^'X ... to ^z factors = «j" " " ==«"*;
therefore a'' must be equivalent to the n^^ root of a^ ;
VI
that is, ^=Va"*.
ISG ELEMENTARY ALGEBRA.
m
Hence, a" means the 7^*^ root of the m^^ power of a ; that
is, in a fractional index the numerator denotes a power, and
the denominator a root.
203. Again ; if we wTite down a quantity with a negative
index, as ar^ (where p is either an integer or a fraction), and
agree that this symbol shall be treated by the same law of
multiplication as a positive index ; what would this symbol
denote ?
By this law of multiplication, a^-^^ Xor^—d^''^^—dJ^\
but we have also, oJ^^^-^oF— — — = — '—-=d!^\
oF oF ^
so that to multiply by a~^ is the same as to divide by oF ;
and therefore,
lXa-^=l-^«^ or or''——-.
Hence, any quantity with a negative index denotes the re-
ciprocal of the same quantity with the same positive index.
Thus a~^=~, a~^=r— , cri = -r= — — , or= >/a~^= V— ;
a a a^ Va a
^3 v/2 a
Hence, also, any power in the numerator of a quantity may
be removed into the denominator, and vice versd, by merely
changing the sign of its index.
Thus a ^trc ^— 1=^-^= — ^=&c.
c cr^c a
204. Lastly, if we write down a quantity with zero for an
index, as aF, and agree that this symbol shall be treated as
if the index were ai; actual number— what then would it
denote ?
Since, by this law, a°Xtt"'=flf'+"'=«^^ it follows that «° is
only equivalent to 1, whatever be the value of a.
Negative indices. Zero as an index.
INDICES. 18?
In actual practice, such a quantity as a^ would only occur in certain
cases, where we wish to keep in mind from what a certain number
may have arisen : thus (»" + 2d^-\-'da-\-&c.)-^d^=a-{-2 + 3a~^ + &c., where
the 2 has lost all sign of its having been originally a coefficient of some
power oia\ if, hoAvever, we write the quotient a + 2«" + 3«-^ + &c., we
preserve an indication of this, and have, as it were, a connecting link
between the positive and negative powers of a.
The quantity a* is still called a to tlw power of — , and similarly in the
case of a-^^^ a^ ; but the word power has here lost its original meaning,
and denotes merely a quantity with an index^ whatever that index may
be, subject, in all cases, to the Law, a"^. a^ = a^+^ .
Examples — 42.
Express, with fractional indices,
1. Vx' + Vx'-h{^xY+{Vxy; V{a'b')^V{a'b')-{-V{ab')
+ V{a'b').
2. aVb' + {s/ay + V{a'b)-i-V{a'b'); V{aW)+a{Vby
+V{a'b'')+V{a'b'y
Express, with negative indices, so as to remove all powers,
1st, into the numerators, and 2d, into the denominators.
1 2 3 4« 5Z> «^ U' 6a U 2F
^* '^'^b'^'7'^'b^/a'' J^^T^T'-^^'^-^'
3^V a'b' a ^ Zabc' %y c^ ?> sf a'^ ^V {a'bc'y d^^/W
Express, with signs of Evolution,
JL 3.7 X ^ 1. 3
5. «^ + 2r/ 3 4- 3^ 4 ^ 4^ 6 4. ^4 , _|. . _| —
b' 2c ^ db^
J)l c^ b^ c^
4^^ 6a^
Express, with positive indices, and with the sign of Evo-
lution,
188 ELEMENTARY ALGEBRA.
+a-H^ + b-^'
cr%^ 2a_ 3b-'c-' 1 ^ ^ ^ ^
^' C-' '^b-'c-'^ a-' '^a-'b-'c-'' ^'k'^a-i'^a-^^ ^~"'
205. From our definitions, {a^ y = a^ said (a^)'* = a's and
in general, (a «)"" = {a'"'') n ; then, also, [a'^)^ = a^^ -^ also, a^,
bn ,c^ , , , z=i[abc . . . )"^ since each, raised to the n^'^ power,
gives abc.
It follows, then, that ^vJiatever be the indices,
cd"^ 1
It will be observed, also, that since a fraction may take
different forms without changing its value, the form of a
fractional index may be changed without altering the valuo
of the quantity.
Thus (3^ 3 — ^(j • for either raised to the sixth power gives a* ;
and in general, a^=a^, for either raised to the np power
gives a"'^. Hence we can reduce the indices of two quantities
to a common denominator without changing the value of the
expressions.
20^. Hence, (1) to multiply together any powers of the same
quantity we must add the ijidices ; (2), to divide any one power
of a quantity by another power of the same quantity, subtract
the index of the divisor from that of the divideiid ; and, (3),
to obtain sluj poiuer of a poiver of a quantity, we must muIM-
ply together the two indices.
The signs of the indices must of course be carefully ob-
served.
Ex.1. Multiply a^'b^ c^hj a^'b^c^ .
Reduction of the indices of two quantities to a common denominator. Rule for
(1) multiplying together powers of the same quantity ; (2) for dividing one power of a
quantity by another power of the same quantity ; (3) for obtaining any power of a
power of a quantity.
IXDICES. 189
3 "^2 ""6' 4'^8~12' 3'^3~ *
therefore, c^ h^ c « x r^^ Z> -^ t^ ^ = ^^ b ^^ ^•
Ex.2. Divide x^ y^^ hj x^^ y'^ ,
4 2~4' 3 6~2'
tlierefore, ir* y-^ -^x' y^ =zx^ y~ .
Also,
a^Xa-^-:a^~'=^; «^--«-*=:a^+^^ri^; a-^-4-a-^=a-*+^=a-^'-<^ ;
Ex. 3. Multiply x-\-x^ -\- x~^ by cc^ + i?;"^ — x~^.
X -\-x'^
+ x-i
x^ ■\-x~^—x-'-
•
x^+x^
+ 1
x^
+ 1+x
4
- 1-x
-I.
-x-i
x^^lx-
i+l
-x^
Here in the first line, x^ Xx=x^^^^ — x^^\ x^ Xx^ = x^ \
x'^ X X '^=x'^ = l] and so on.
Ex. 4. Divide
x^ - a^ x'-Ux^ ^eJx~ 2a V' by x^ -4.ax^ +2 J •
x^ - 4:ax^ + 2a^ ) x^ - a^ x' - 4.ax^ + 6a^ x-2a''x^ (x-a^ x^
x^ — 4:ax^ + 2a^ x
-a^x' -h 4:J x-~2a''x^
-a^x' -h 4:a^ x-2a*x^
190 elemektaky algebka.
Examples — 48
Find the value of
1. 9-^. 2. 4~i 3. (100)-^. 4. (1000)^. 5. 81"^.
Simplify
6. (a=)-l 7. (a-^)-^ 8. ^/a-^ 9. V^"".
10. aixaixa~^'
Multiply
11. X"^ + ^4 ]3y ^f _ ^! ^
13. x-\-x^' + 2 by iz; + :2;* - 2.
14. .^*+:^' + 1 by ^'c-'-ii;-' + 1.
15. oT^^ar'-^l by a~i^l.
16. «3_2+^-^ by a^-^-t
17^ a + ^^Z** — ii;"^^/^ by a + a^^i+o;*^^.
Divide
18. :^;t-3^t byi^;*-^*. 19. a-5bya^-J^.
20. 64^-^ + 27y-^ by 4:?;-^ + Zy-^\
21. o;^ — x]p + i^^?/ - ^^ by o^ — ^-i.
22.. a^ ^-a^l^ ^ifih^ a^ ^a^lh j^l\^
23. fl^^ + ^>5_c§ +2^3^* by a^ + ^^ + c^.
24. ^^ - '^a^x^ + a' by x^ - %a^ x^ + «.
Find the square roots of the following expressions.
25. a:* -4 + 42;-*. 26. (a: + aj-^)2-4 (i?;~ar-^).
27. a^l-'' + 2a^-^ 4- 3 + ^ar^l + a-^Z>l
SURDS. 191
XXIX. Surds.
207. It was stated (Art. 147) that when any root of a quan-
tity cannot be exactly obtained it is expressed by the use of
the sign of Evolution, as \/5, V{3ab), \/(a' + c''); and such
quantities are called Iri^ational quantities, or Surds.
It is also stated in (147) that there cannot be any even
root of a negative quantity, but that such roots may be ex-
pressed in the form of surds, as \/ — 3, V— Z>^ V—{a^-\-lf)y
and are then called impossible, or imaginary, quantities.
208. Since every fractional index indicates by its denomi-
nator a root to be extracted, all quantities having such in-
dices are expressed as Surds.
When a negative quantity has the denominator of its in-
dex (reduced to its lowest terms) even, the expression will be
imaginary.
Thus, (—3)^, ( — 9)4, are imaginary quantities; but
(— 4)ti is not so, since it is the same as ( — 4)^, where the
root to be taken is odd,
209. The operations of the preceding chapter are opera-
tions on surds, but we may apply the rules which are de-
monstrated in that chapter by the use of fractional indices,
also to surds expressed by the sign of Evolution, or Radical
Sign,
2i0. In the case of a mmierical surd expressed with a
fractional index, should the numerator be any other than
unity, we may take at once the required power, and so have
unity only for the numerator, and simply a root to be ex-
tracted.
Thus, 2^ = {2')^ = 4* =V4; 3-^=:(3-^)* = (^)^ =V^S,
Surds. Surds expressed by the radical sign.
192 ELEMENTARY ALGEBRA.
211. Quantities are often expressed in the form of surds,
which are not really so — i.e., when we ca7i, if we please, ex-
tract the roots indicated.
Thus, Va, Vh {cc' + ah + h'')^ are actually surds, whose
roots we cannot obtain; but Va^, \/27, (4a^-|-4«Z> + ^^)J, are
only apparently so, and are respectively equivalent to a, 3,
2a-\-l).
Conversely, any rational quantity may be expressed in the
form of a surd, by raising it to the power indicated by the
root-index of the surd.
"For example, 3-= v/3'r:. v/9 ; 4=V4' = V64;
a^\/a'\ a + h=:l/{a-\-l)y.
212. In like manner a mixed surd, i. e. a product partly
rational and partly surd, or a surd with a rational coefficient,
may be expressed as an entire, i. e., complete surd, by raising
the rational factor to the power indicated by the root-index
of the surd, and placing beneath the sign of Evolution the
product of this power and the surd factor. An entire or
complete surd is one in which the whole expression is under
the sign of Evolution.
Thus,
2v/3=v/4Xv/3=v^l2; 3. 23=3^4^ V27x 1/4 rr:Vl08 ;
iCa /Ca
Conversely, a surd may often be reduced to a mixed form,
by separating the quantity beneath the sign of Evolution
into factors, of one of which the root required may be ob-
tained, and set outside the sign.
Thus, v/20=:n/(4x5) = 2v/5; V24=V(8x3)==2V3;
y/{^a'h)^^a>/{ZaV) ; \/{^a'h'c')=iahV(2ac').
Rational quantities in the form of a surd. Mixed surds and entire surds.
SUKDS. 193
213. A surd is reduced to its simplest form, when the
quantity beneath the root, or surd-factor, is made as small as
possible, but so as still to remain integral.
Hence, if the surd-factor be a fraction, its numerator and
denominator should both be multiplied by such a number as
will allow us to take the latter from under the root.
Thus,
/2 /2.3 1 ,^ 5 s /24 ^ ;/3 , f/3.5^ ,,^.
/3^ /3X2^ /_^^v^6
r ft r ftv2 y Ifi
8X2 ^ 16 4 '
y 2 _ ;/^X9_ yi8_ V18
^ 3-'^3x9~'^27~ 3 *
214. Surds which have not the same index may be trans-
formed into equivalent surds which have. (See Art. 204.)
For example, take \/5 and Vll ;
v/5 = 5^, Vll = (ll)3 ;
5^ =5^ = V5^= V125, (11)3 =111 =V(11)'=V121.
215. Similar surds are those which have, or may be made
to have, the same surd-factors.
• Thus 3 Va and Va, "^aVc and ?>Wc, are pairs of similar
surds ; bVa and '^VaJ^ are similar surds, because ^^/a" may be
written 2 V^^; and \/8, \/50, \/18 are also similar, because
they may be written 2v/2, 5\/2, 3\/2.
Examples — 44.
1. Express 4^, 9^, 3~s 2"^, (f)"^, (^)-^, with indices,
whose numerator is in each case unity.
Eeduction of surds lo their simplest form, —to equivalent surds having a given
index. Similar surds.
9
194 ELEME]S^TAIIY ALGEBRA.
2. Express 5, 2^, |«, ^a'^, 2(^^ + ^)^ ^^ surds, Avith indices
I and I".
3. Express d~% (3^)"^ a~\ ab~'c~\ with indices ^ and
Eeduce to complete or entire surds,
4. 5v/5, 2Vh |.3S fN/li i(|)-S 25(li)-^.
5. dV2, 8X2-S 4X2^ 3x3-^ Kf)"', id)"*.
6. 2v^^, 7a%/(22:), (^ + Z^) (^^-Z,^)-i.
^ a ^ 3x 3b ^ 2a 3 ^ 4a' ^ ^ '^ a-\-x
Eeduce to tlieir simplest form,
8. v'45,x/125, 3v/432, V135, 31/432, v/f, 2V|, 3Vi 4V3-|.
9. si, 32% 72*, (H)-^, (20i)-t, (30|)-^ fv/-^^, 5V4^V,
10. Show that x/12, Sv'TS, |V147, fV^. VyV and (144)-i
are similar surds.
216. To compare surds with one another in magnitude,
express tliem. as entire surds, and then reduce them, if neces-
sary, to a common siird-index, and simplify as in Art. 213.
Their relative values will then be apparent.
Thus, 3\/2 and 2\/3, expressed as entire surds, are \/18 and
\/12, and it is at once plain which is the greater.
To compare \/5 and VH :
v/5-:5^=5^r=V125;
V11 = 11^=::11*=n/121.
We see now that \/5 is greater than Vll.
217. To add or subtract similar surds, reduce them, luhen
necessary, to the same surdf actor, then add or subtract their
rational factors or coefficients, and affix to the result the com
mon surd-factor.
> compare surds with one another in magnitude. To add or suhtract similar surds.
SUEDS. • 195
Thus, v/8+v/50-v/18 = 2\/2 + 5v/2-3N/2:rr4v/2;
^aV{a'h')-\-bV{^a'h)-V{nba'h') ^A.a'Wl + WbVh-^
hcCWl^a^Wb.
2' Id 1' 7256 2' /12 1' /64xl2
2' /3 1' /256 2' /12 1' /6
27
__2_V12 1 4V12 _ 2V12
■~3 2 "^4 3 ~" 3 •
Dissimilar surds can be added or subtracted only by con-
liccting them with their proper signs.
218. To midtiply simple surds which have the same surd-
mdex, multiply separately the rational factors and the surd
factors, retaining the same surd-index for the product of the
latter.
Thus, 3v/2X\/3 = 3v/6; 4\/oX7v/6=:28n/30;
2V4X3V2=:6V"8 = 6X2 = 12;
2v/3x3n/10x4v/6 = 24v/180 = 144\/5.
219. To multiply simple surds which have not the same
surd-index, reduce them to the same surd-index and proceed
as hefore.
Thus,
4v/5x2V11 = 4V125x2V121:=8V(125x121) = 8V15125;
2^3 X 3 V2 =2V27 X 3 V4= 6V108.
Compound surd-expressions are multiplied according to
the method of compound rational expressions.
Ex. 1. (2±N/3)=^=:4dz4v/3 + 3 = 7±4y3.
Ex. 2. (2+ v/3) (2- v/3) = 4-3 = l.
Ex.3. (2+v/3)(3-\/2).= 6 + 3v/2-2v/2- v/6.
To multiply simple surds— two cases.
196 ELEMENTARY ALGEBIIA.
220. To divide one simple surd by another^ reduce hotli
surds to the same surd-index, ivlien necessary ; then divide the
coefficients and surd-factors separately, retaining the common
surd-index aver the quotient of the latter.
The result may be simplified by Art. 212.
— o f
125 X 121 X 121 4V1830125
"3" 121X121X121 3X11
Ex.3, (8^/2-12^/3 + 3^/6-4)-^2v/6
=:4v/|~6v/| + |--^=tv/3-3x/2+|-ix/6.
Ex. 4. (2v/3-6V2)-f:\/6=:2v/|-6V-^t6 = %/2-V864.
221. But, if the divisor be not monomial, the division is
not so easily performed. The form, however, in which com-
pound surds usually occur, is that of a l)inomial quadratic
Burd, i, e. a binomial, one or both of whose terms are surds,
in which the square root is to be taken, such as 3 + 2\/5,
2v/3— 3v/5, or, generally, s/a^s/h, where one or both terms
may be irrational; and it will be easy, in such a case, to
convert the operation of division into one of multiplication,
hy putting the dividend and divisor in the form of a fraction,
and multiplying loth numerator and denominator by that
quantity which is obtained by changing the sign between the
ttvo terms of the denominator. By this means the denomina-
tor will be made rational: thus, if it be originally of the form
\/a± \/Z>, it will become a rational quantity, a—b, when both
numerator and denominator are multiplied by Va^i y/b.
To divide one Bimple surd by another. To divide binomial quadratic surds.
SUKDS. 197
2+v/3 _ (2+v/3) (3-n/3) _ 6+3v/3-2v/ 3-3
• 3+v/3~(3+v/3)(3-v/3)~ 9-3
3+V/3
Ex.2.
6
2\/2+\/3 2v/2+v/3
2v/2~v/3~ 8-3 5
This process is called rationalizing the denominators of the
fractions, and the fractions thus modified are considered to
be reduced to their simplest form.
Examples — i.5,
1. Compare 6v/3 and 4v/7; 3V3 and 2V10; 2V15, 4V2,
and 3V5.
Simplify
2. 3v/2 + 4:n/8-v/32.
3.
2V4:+5V32-V10«
4. 2v/3 + 3v/(li)-v/(5i).
5.
1 1
V2 ~ V16'
Multiply
-1 -j
6. v/5+%/(li)--;^by v/3.
7.
'^^ Vl6+V3^y'^^-
8. l+%/3-v/2by v/6-v/2.
9.
v/3+v/2byj3H--i.
10. Divide *
2x/3 + 3v/2 + V30 by 3x/6, and 2^/3 + 3V2+V30 by 3^/2
11. Eationalize the denominators of
__1 4 3 8--5v^ 3 + \/5
2^/2-^/3' Vb-1' \/5 + v/2' 3-2v/2' 3-n/5'
12. Eationalize the denominator of —7 "-4: 77 7.
V{a-\-x) — y/{a—x)
222. The squar® root of a binomial, one of whose terms is
a quadratic surd and the other rational, may sometimes be
expressed by a binomial, one or both of whose terms are
Eationalizing the denominators of fractions. To express the square root of a
Dinomial, one of whose terms is a quadratic surd and the other rational.
198 ELEMEKTAKY ALGEBKA.
quadratic surds. (A quadratic surd is one whose indicated
root is the square root.)
Since {y/x:^Vyy=^x-±:^Vxy + y, therefore Vx^2Vxy-\-y
=z^xdci^y'^ hence if any proposed binomial surd can be put
under the form x±^Vxy + y, its root may be found by in-
spection to be \/x±^/y. To show how to proceed in any
proposed case, let us take the binomial 3 + 2\/2. To place
this under the form x + 2y/xy-\-y, we observe that 2\/2
=2v'2X\/l, and the sum of the two numbers under the
radical signs 2 + 1=3, the rational term of the binomial;
/. 34-2\/2=2 + 2\/2X\/l + l. Hence the square root of
3+2v/2='v/2 + 2V2X\/l + l=:\/2 + l.
Ex. 2. Kequired the square root of 7— 2\/10.
Here2yiO=2v/5X\/2.
Also, 5 + 2=7, the rational term. Hence,
7-2x/10=5~2v/5X\/2 + 2; .-. root required is Vo-V2.
Ex. 3. Eequired the square root of 11 — 6\/2.
Here 6x/2=2v/18=:2v'9X\/2, or 2v/6X\/3; of which the
former answers the condition that their sum 9+2=11, the
rational term; .-. ll-6y2=:9--2x/9x%/2 + 2.
A the required root is \/9 — \/2; that is, 3 — %/2.
These illustrations show the method to be, to put the term
loMcli contains the surd into factors, of the form ^VxXVy,
in such a manner that the sum, x-{-y, of the numbers under
the ttvo radicals may be equal to the rational term. Then the
y/x'±.\/y loill be the required square root,
Ex. 4. Eequired the square root of 7 + \/13.
Here N/13 = 2v'y = ^^^¥X^/i.
Also, 1^ +-|-=7, the rational term.
%/13+l
/. root required is x/V + v^i, or
n/2
SURDS. 199
Examples — 46.
Find the square roots^ of
1. 4+2x/3. 2. ll^-6^/2. 3. 8-2v/15. 4. 38-12v/10.
5. 41-24v/2. 6. 2-1— v/5. 7. 4|-fv'3.
2222^. It is often required to c/mr an equation of surds. An
equation may be cleared of a surd by transposing the terms
so that the surd shall form one side and the rational quanti-
ties the other side, and then raising loth sides to that poiver
which will rationalize the surd.
In the case of quadratic surds we square both sides. We
shall confine ourselves to clearing equations of quadratic
surds.
Thus if ^ya^-x—h — c, by transposition, '^« + :i'=:^ + c; and,
squaring both sides, a + x—{l)-\-cy. We thus have an equa-
tion without surds.
If the equation contain tivo surds connected by the signs
+ or — , then the same operations must be repeated for the
second surd.
Thus, if "^a + x ^s/x—l,
by transposition, '^ a-\-x=l)—\/x\
squaring, a-^-x^y^—^bs/x-^rX^
reducing and transposing, %b\fx—lf—a\
squaring, W'x^iV^ — aY,
an equation in which the surds do not appear.
Ex. 1. ^b-\-x-\- "^b—x—2\/x\ required^.
Transposing, ^b-\-x=-2Vx — ^b—x ;
squaring, b-\-x—A:X—4:^bx—x''-\-b—x]
To clear an equation of surds.
200 ELEMF.XTAllY ALGEBKA.
reduc'g and transp'g, 4:V5x—x^—2x;
%y/hx—x^= x;
squaring, 20x—^x^=x* ;
20x=bx'',
.'. 20= 5a;, and a;=4.
Examples— 47.
Find X in the following equations :
1. ^/(4a;)+v/(42;-7) = 7. 2. V{x-\-14.)+V{x-U)=lL
3. x/(a; + ll)4-\/(a;-9) = 10.
4. v/(9a;+4) + v/(9a;~l)=3.
5. \/(a;+4aZ>)=2a— \/:r.
6. v/(a;-a)+v/(2;-Z>)=x/(a~^>).
7. a+a;~N/(2a:?; + ir')=&. 8. a-{-x^-y/{a^ + 'bx + x^)=b.
XXX. Quadratic Equations.
223. Quadratic equations are those in which the square
of the unknown quantity is found, but no higher power of
it. Of these there are two species :
1. Pure Quadratics, in which the square only of the un-
known is found without the first power, as
a;'-9=0; ic'-a'=Z^^ &c.
2. Affected Quadratics, where the first power enters as
well as the square, as
a;'— 3a; + 2=:0; ax^-\-hx^c\ &c.
224. Quadratic equations are (Art. 1G6) also called Equa-
tions of tlie second degree. The two species also are distin-
Quadratic equations ; two species of, and two ways of designating them.
QUADEATIC EQUATI0:N'S. 201
guished as, 1st, Incomplete equations of the second degree ;
and 2d, Complete equations of the second degree. We shall,
however, generally use the notation of Art. 223.*
225, To solve a Pure Quadratic Equation :
Find tlie value of x^ ly the ride for solving simple equa-
tions ; then take the square root of both sides of this result,
we thus find the value of x, to which we must prefix the
douUe sign =b (144).
Such equations will therefore have two equal roots with
contrary signs.
Ex. 1. x^-^=^.
Here a:^=9, and x=^^.
If we had put db:?;=:±3, we should still have had only
these two different values of a;, viz., a; = +3, a;=— 3; since
—x—^'d gives x=—d, and —x=—^ gives x= +3. *
Ex. 2.' ^(3:?;= + 5)-i(a;'' + 21) = 39-5a;'.
Reducing, 121a;'^=:1089;
.*. x^=^, and ^=±3.
a;^ + 2 9
Ex. 3.
a;^-2~ 7'
To examples like this the principle of fractions, (Art. 134,
vi), may be applied with advantage when the unknown
quantity does not enter in both sides of it.
* The term affected was introduced by Vieta, about the year 1600.
It is used to distinguish equations which involve or are affected with
different powers of the unknown quantity from those which contain
one power only. (Lund.).
To Boh'e a pure quadratic.
202 ELEMEJS^TARY ALGEBRA.
{x'-i-2)-lx'-2)~9-7'
that is,
2^_16
T'~~2''
Ex. 4. - — =4.
ir"— 16, and x—±4c.
(134, Yi)
V4: + X' 5
X 3'
squaring,
4: + x' 25
x' -9'
again, (134, iy and i).
x' 9 ,9 , ,
-_-; r,x--; a:-±f.
Examples — 48.
1. ix'=U-3x\
2. x' + 5=^x'--16.
3. (a;+2)^=4^ + 5.
'■ it+i-.-»-
3 17
4.x' 6x'~3'
a; 7
dx' 15a;'' 4- 8
^'4 6 ~
2x'
, ^ x^ a;' -10 ^ 50+a?
^- ^- 5 15 =^ 26
. 3:2;*-27 90+4^;^ ^ Ar^^' + S 2.^^-5 7ii;'^-25
^' _2 , O + 2 . A ~7. lU.
11.
12.
a;''^+3 ^ aj'-h9 '10 15 ~ 20
10:r^ + 17 _ 12^N-2 _ 5a;'' -4
18 ■~lla;''-8~ 9~"'
140;" + 16 2a;'' + 8 ___ 2^
21 ~8a;''-ll~'3"'
QUADKATIC EQUATjg^is^S. 203
2 2
^16 + af-'^25^^_l_
226. An affected quadratic, or comiDlete equation of tlie
second degree^, may alivays he reduced to the form x^ + px + q
= 0, ivliere the coefficient of x^ is +1, and p and q represent
known numbers, luliole or fractional, positive or negative.
For, let all the terms be brought to one side, and, if neces-
sary, change the signs of all the terms, so that the coeflScient
of ^ may be a positive number ; then divide every term by
this coefficient, and the equation takes the assigned form,
X' -^px-{-q—^.
jN'ow in this equation we have x^ ^px——q', and adding
(kvf ^^ ^^^^ side, we get '^^ ^-px'\-\p' =\p^ —q\ by this step
the first side becomes a complete square (Art. 153); and
taking the square root of each side, prefixing, as before, the
double sign to that of the second side, we have
x-V\p^±^W^\
which expression gives us, according as we take the upper or
lower sign, two roots of the quadratic.
227. From the preceding we derive the following rule for
the solution of equations containing an afiected quadratic :
By reduction and transposition arrange the equation so
that the terms involving x^ and x ai^e alone on one side, and
the coefficient of x" is + 1 ; then add to each side the square of
half the coefficient of x, and talce the square root of each side,
prefixing the double sign to the second.
To what form may an affected quadratic always be reduced ? Rule for the solution
of equations containing an affected quadratic ?
204 ELEMENTARY ALGEBRA.
We thus obtain a simple equation from which x is readily
found.
Ex. 1. x'-6x==7.
Here x'-ex-}- 9=^7 + 9 = 16;
whence ^—3 = ±4,
and a;=3+4=7, ori^=3-4=-l;
so that 7 and —1 are the two roots of the equation.
Ex. 2. x''-i-Ux=95.
Here a;' + 14a; + 49=: 95 + 49 ==144;
whence a; + 7= ±12,
and ir= — 7±12=5, or —19.
Examples — 49.
Solve
1. x''-2x=8. 2. a:' + 10:?;=-9. 3. x^'-Ux^VZO.
4. a;''-12a;=-35. 5. a;' + 322;=320. 6. i^;' + 100a; =1100.
228. If the coefiicient of x be odd, its half will be a frac-
tion. Its square may be indicated on the first side bv using
brackets.
Ex.1. • x'-6x=z-6.
Here x''-5x+{iy= -6 + ^=1;
whence a;— 1=±|,
and a;=| + |=f ==3, or x=^—^=^=2,
Ex. 2. x'-'X=l.
Here x'-x+{iy=i+i=l;
whence cc— ^=±1,
and a;=i+l=l|; ora;=J— 1 = — ^.
quadratic equations s. 205
Examples — 50.
iSolye
1. ^' + 7a;=8. 2. ^j'^-IS^j^GS. 3. x' + 2dx=-100.
4. i?;' + 13ic=-12. 5. a;' + 190:= 20. 6. cc' + 1112:= 3400.
229. If the coefficient of a; be a fraction, its half will, of
course, be found by halving the numerator, if possible; if
not, by doubling the denominator.
Ex.1. Solve x' + y^x=19.
Here x' + \^x+{iy=19-i-^^=^^;
whence a;+|=±^,
and a;= — 1 + ^=3, or x=—^—^='-6^,
Ex.2. Solve x'+^x=U,
Here x'+h'-x+my=U-hm='^'^y
whence x+\^=±^^y
and x=-{i+U=n, or ^=_^— 1-|=-10.
♦
Examples— 51.
Solve
1. x'-ix=d4.. 2. x'-ix=27, 3. x'+^xz=86.
4. x'-^x=lU. 5. x'-{-j\x=U6, 6. ^'^ -f|:z;=147.
230. In the following examples the equations will first re-
quire reduction ; and since the rule requires that the coeffi-
cient of x^ shall be + 1, if it have any other coefficient we
must first divide each term of the equation by it ; and if its
coefficient be negative we must change the signs of all the '
terms.
206 ELEMENTARY ALGEBRA.
Ex. 1. Solve -3x' + 20x + 6 = 0.
Here 3:^;'- 20:^-5 = 0,
and x^—^^-x=^;
therefore, x'—^^-x+^^=^^;
/, o^^KlOiv'iis)^ the roots being here surd quantities.
Ex.2. Solve —. T\+-2 — T=-r-
2(a;-l) x^—1 4
Here we first clear of fractions by multiplying by ^x^^—l),
which is the least common multiple of the denominators.
Thus 2(a; + l)+12rr:cc'^-l.
By transposition, a:'^— 22;=15;
adding 1', x'-2x +1 = 16 + 1 = 16;
extracting the square root, a;—! = ±4;
therefore, a;=ldb4=5, or —3.
Ex.3. Solve _+^-^^-=_^_.
Multiplying by 570, which is the least common multiple
of 15 and 190, ^
10 + i?; ^ ^
whence 190(^x-50)^^^^_^^
10 + a; '
and 190(3^-50)=:(210~40^)(10 + ir);
that is, 570a;-9500=2100-190^-40a;';
therefore, 40a;'' + 7602:= 11600 ;
or a;' + 19a;=290;
addmg (^— J , a:'' + 19^+ (^yj =290+-^-=--^ ;
QUADRATIC EQUATIOJS^S, 20T
19 39
extracting the square root, x +—= dz— ;
whence a;=— — dz— =10, or —29.
^ , ^ , a;+3 i?:— 3 2:?;— 3
Ex.4. Solve -— ^4- ^ = r-
x+2 x—2 x—1
Clearing of fractions,
(x + d) {x-2) (^-l) + (^-3) {x + 2) {x-1)
= (2x-d) {x + 2) {x-2) ;
that is, x'-'7x + 6+x'-2x'-6x + Q=2x'~dx'Sx + 12;
or 2a;'-2a;'-122;4-12=2a;'-3:z;'^-8a;+12;
therefore, x^—4:X=0;
adding 2^ * x'-4.x + 2':=4:;
extracting the square root, a;— 2 = ±2,
whence a;=2±2 = 4 or 0.
Remark. — We have given the last three lines in order to complete
the solution of the equation in the same manner as in the former
examples ; but the results may be obtained more simply. For the
equation x^—Ax = ^ may be written (a?— 4)a?=0; and in this form it is
sufficiently obvious that we must have either oj— 4=0, or x—0, that is,
a^ = 4 or 0.
The student will observe that in this example 2a;^ is found on both
sides of the equation, after we have cleared of fractions ; accordingly,
it Ci,n be removed by subtraction, and so the equation remains a quad-
ratic equation.
Examples — 52.
6 1. ■ X
3. ^x''-^x=^^{llx + lS). 4. lla;^-9ri;-lli.
5. i{x'-3):=i{x-3), 6. 2r^Hl-:ll(.r+2).
S:08 ELEMEJS'TARY ALGEBRA.
X
1
^ a;+33 4 9a;-6 .. a;+3 4-3; _
4
^x-Q
a; '
~ 2 "
4
32
12 _JL__?L 12 ^ a;4-l_13
' 5-a;'^4-a;"a; + 2' a;+l ri: ~ 6 '
231. Sometimes, on completing the square, the second side
of the equation becomes 0. For example, take the equation
ic^ — 14:^j = — 49. This giyes
a;^-14a; + 49=0 ; .-. {x-iy=0'y .\ x=7.
In this case we say the quadratic equation has two equal
roots.
232. Solve x'-^x + lZ^O,
By transposition, x^ —Qx— — !^ ;
adding 3', a;'-6:z:+9 = -13 + 9 = -4.
If we try to extract the square root, we have
cr-3=:±Vir4.
In this case the quadratic equation has no real root, and
this is expressed by saying that the roots, are imaginary or
impossiUe,
233. An equation of the form ax^-{-'bx-\-c=0, or ax^ ^hx
= — c (where a, d, and c are any quantities whatever), may
be solved by what is called the Hindoo method,* as follows,
without diyiding by the coefficient of x^. Multiply every
term hy 4a, that is, 4 times the coefficient of x^ a7id add V,
that is, the square of the coefficient of x, to both sides : the first
side will be a " complete square." Thus,
4:aV + 4.adx + b'=I?'- 4.ac.
* This method is given in the Bija Ganita, a Hindoo treatise on
Algebra.
QUADRATIC EQUATIONS. 200
Extracting the square root,
.'.x=^{^d±>/¥^^^c). (1)
Ex. 1. If dx^ + 2:r=85, find x.
Multiplying by 4x3, or 12, 3 6a;' +240; =1020;
adding 2\ or 4, 36ic*+24a; + 4=1024;
extracting root, 6a; + 2=±32;
B:r=:±32-2=:30, or -34;
/. x=z5, or — 5f.
Ex. 2. If 6x' -dx^ 2i= 0, find x.
Transposing, 6x^ — 9x= — 2 J ;
multiplying by 4x5, or 20, 100:r'— 180^;= -45 ;
adding 9', or 81, 100cc'-180a:+81 = 81-45=36;
extracting root, lOx— 9 = dz6;
whence, 10a;= 9 ± 6 = 15, or 3 ;
15 3
.•.:.=^,or-;
= 4, or-.
The student will find it well to apply at once (by memory)
the formula (1) above obtained for x,
Ex. 3. (3:i;-2) {l-x)=4:, or dx'-5x-{-6 = 0.
Here a:=:i(5±^^25-72)=l(5±^^^^^), the roots being
impossible.
Or, since it appears that the equation ax^ + I)x-= — c is re-
ducible to the simple equation 2ax-\-b=:dt:^b''—4:ac, the
What two abbreviated forms of Bolution are suggested ?
14
^]0 ELEMENTAEY ALGEBRA.
student may readily acquire the habit of writing down the
simple equation from any proposed equation without the aid
of any intermediate steps.
Ex.4. Solve 3x'+6x=^2,
62; + 5 = ±V25 + 12X42;
that is, 6a; + 5 = ±23 ; ,\x= 3, or — 4|.
EXAMPLES-
_f;?
^ 2x 2x-6 _. _ 2x + 9 4.X-3 _ 3x-16
5a; 3x—2 _^ 4a;+7 5— a;_4a;
• ^T4~2a;-3~"' ' 19 "^3T^~"9''
x—1 ^-2_2^+13 ^+1 ^+^_2^+13
• x-\-l'^x + 2~ x + 1^' * a;-l'^a;-2~ ic+1 '
2a;-l 3^-l_5^--ll 14a:-9 a;^-3
a; + l "^ a; + 2 ~ a;-! * * ^~ 8a;-3 "ic + l*
9. a'a;^-2a'i?;+a'-l = 0. 10. 4a'a;=(a^-^^ + a;)^
^^ X a x h ^^1111
11- r+-=r+-. 12. r+i^^=-+:
a X h X ' X x + h a a+b'
234. If r, r' represent the two roots of x^+px+q=0,
then —p = r + r\ Sind q=rXr'.
YoTr=-lp-hViip'-q), r'^-ip-VUp'-q) ;
.-. r + r'=—2y, and rXr'==iy~(iy— g) = ^.
Hence, luhen any quadratic is reduced to the form
x^' + px + q^O, the coefficient of 2d term, ivith sig?i changed,
= su7n of roots, and the 3d term = product of roots.
To what are the sum of the roots and the product of the roots of a quadratic equa-
tion, respectively, equal ?
QUADEATIC EQUATIOXS. 211
Thus, in (Ex. 1, 227) the equation, when expressed in this
form, is x^ — 6x—7 = 0, and the roots are there found, 7 and
—1; and here -{■6 = '7-\-{ — l) =^ sum of roots, and —7 =
7X (— 1) = product of roots.
So, also, ax'^-\-hx + c=0, expressed in this form, becomes
x^-\ — x^ — = 0; .-. = sum of roots, — = product
a a a a "-
235. If T, r' he the roots of x^ ■\-px-\-q — ^, then
x^-\-px-\-q—(x—r^ (x—r').
For, (236) x^ -\rpx-\-q—x^ —(r ^r') x-^r r\
=x^ —rx--r'x+r r' = {x—r) {x—r').
So, also, if r, r' be the roots of ax'^+bx-{-c=0,
b c
that is, of x''-\ — X-] — =0,
a a
we have ax'' + bx + c~a (x^-i — x + —]—a{x—r) {x—r').
236. Hence we may form a quadratic equation with any
two given roots.
Thus with roots 2 and 3 w^e shall have
(^-2) {x-^)=x''-bx-{-^=0.
With roots —2 and J, we have {x + 2) (x—D—x' ■\-lx—i
=0; or clearing it of fractions, 4iz;^ + 7r?^— 2 = 0.
If one of the roots be 0, the corresponding factor will
be x—Oy or x.
Thus with roots. and 4, we have x{x~4,) — 0, or x^—A:X
= 0. (Compare Ex. 4, Art. 230.) In such a case, then, x will
occur in every terr)i of the equation, and may be struck out
of each; but we must notice, always, that when we thus
strike out x from every term of an equation, x—^ satisfies the
equation, and is therefore one of the roots.
A quadratic equation may be formed from any two given roots. a;=0, a root.
212 elementary algebra.
Examples — 54.
Form equations with the following roots :
1. 7 and -3. 2. f and -|. 3. -6 and -5.
4. 2| and 0. 5. 10 and -10.
6. a-f— and a . 7. — 1 + n/2 and — 1— \/2.
a a
m
XXXI. Equations which may be solyed like
Quadratics.
237. There are many equations which are not strictly
quadratics, but which may be solved by the method of com-
pleting the square. We will give some examples.
238. Ex. 1. Solve x'-W=^,
Adding (|)^ x'-W + {iy=^^-^=^i^;
extracting the square root, a;^— |==b|;
whence, a;'=|d=|=:8 or —1 ;
extracting the cube root, a; ==2 or —1.
This method applies, evidently, in all cases where the
lowest of the two exponents of the unknown quantity is one-
half of the highest exppnent.
Ex. 2. a; + 4a;2=21; required x,
a;+4a;^+4=r21+4=25;
iz;i + 2=:±5;
a;*=:±5-2=3 or -7;
:,x=^ or 49.
What other equations may be solved like quadratics ?
QUADRATIC EQUATIONS. 213
Ex.3, a;"' + ^""^=6; required ic.
X ^z= — - — =2 or —6;
.-. x=l or f
239. Equations may be proposed containing quadratic
iTirds, from which, by performing the operations of trans-
posing and squaring, once or oftener (Art. 223), we obtain
afiected quadratic equations.
Ex. 1. x+V6x-{-10 = S ; to find x.
By transposition, 'v/52;+10=:8— :?:;
squaring, 5a; + 10 =64— 16:?;+ a;';
x'-21x=-54c;
441 441 225
^«__21a;+— =— -54=— ;
21 ^15
21±15 ^_ ..
a;= — ^r — =18 or 3.
We have thus found two values of x ; but on trial we find
that 18 does not satisfy the equation if we suppose that
1/0:2; + 10 represents the positive square root. The value 18
satisfies the equation x— y 5.^ + 10=8.
Ex. 2. Solve 2x-V{x'-3x-3) = d,
Transposing, 2x—9 = V{x^—3x—3);
squaring, 4:?;^--36a; + 81=cc^— 3.^—3;
transposing, 3^;'— 33.^+84=0;
dividing by 3, ' x'- llo; + 28=0.
214 ELEMEKTARY ALGEBRA.
By solving this quadratic we shall obtain x^=^^l or 4. The
value 7 satisfies the original equation ; the value 4 belongs
strictly to the equation 2a; + \/(a;^ — 3a;— 3) = 9.
Ex. 3. Solve N/(aj + 4) + v'(2:?; + 6)3=N/{8^+9).
Squaring, aj4-4 + 2a; + 6 + 2v/(:2; + 4)v/(2a;4-6) = 8a; + 9 ;
transposing, 2\/(:?; + 4)\/(2a; + 6) = 52:— 1;
squaring, 4(2; + 4) (2a;+6) = 25^^— 10::i; + l;
that is, 8a;' + 56a; + 96 = 252;'— 10a; + l ;
transposing, 17:?;^— 66:?;— 95 = 0.
By solving this quadratic we shall obtain cc=:o, or — f^
The value 5 satisfies the original equation ; the value — ^
belongs strictly to the equation
v/(2^ + 6)-v/(:?; + 4)=:x/(8:2; + 9).
240. The student will see from the preceding examples
that in cases in which we have to square in order to reduce
an equation to the ordinary form, we cannot be certain, with-
out trial, that the values finally obtained for the unknown
quantity belong strictly to the original equation.
241. Solve x^ ■\-Zx-\-?>sf(x^ ^Zx-%):=^^.
Subtracting 2 from both sides,
a;'-f3:?;-2 + 3v/(^' + 3a;— 2)=4.
Thus on the left-hand side we have two expressions,
namely, \/(a^ -\-Zx—%), and x'^ + ^x—2, and the latter is the
square of the former ; we can now complete the square.
Adding (-|)^
i2;^ + 3^_2 + 3x/(^' + 3a;-2) + (|y=4 + |=^;
extracting the square root,
N/(a;'' + 3a;-2)+|=zb|;
therefore, v/(:r' + 3a;-2) = -f ±f =1 or -4.
QUADRATIC EQUATIONS. 215
First siii:)pos8 \/{x^ + 3x—2) = 1 ;
squaring both sides, 2;'^ 4-32'— 2 = 1.
This is an ordinary quadratic equation ; by solving it we
T. n T4- • ^ -3±^/21
shall obtain x= .
z
l^ext, suppose V{x^-{-3x—2) = —4z.
Squaring both sides, r?;^+3:?;— 2 = 16.
This is an ordinary quadratic equation ; by solving it we
shall obtain a; =3 or —6.
Thus on the whole we have four values for x, namely,
-3±n/21
3 or —6, or
2
o -4-/9-1
But we shall find, on trial, that only the values ■
2
will satisfy the given equation
x'+3x + 3V{x' + 3x-2) = 6,
but the values 3 or — 6 satisfy the equation
x'' + dX'-3V{x'-{-dx-2)z=6.
242. The method pursued in the example in the last article
applies whenever an expression may be formed which, con-
taining all the unknown terms outside of the surd, is the
same as the surd expressio?i, or is a rmilUple of it,
2IB. Equations of the third degree are sometimes proposed
in which it is intended to find one of the roots by inspection
or trial, and the two remaining roots by solving a quadratic
equation.
x-{-A: x—4: d + x 9—x
Ex. 1. Solve
X-\r4: 9 — x 9-hx'
Bring the fractions on each side of the equation to a com-
mon denominator. Thus:
What method of solution is explained iu Art. 243?
.216 ELEMEKTARY ALGEBRA.
16x d6x
that IS,
-16""81-cz;'
Here it is obyious that x=0 is a root (Art. 236). To find
the other roots we begin by dividing both sides of the equa-
tion by ^x. Thus :
4__ 9_^
ic^ - 16 ~ 81 -:?;'/
therefore, 4(81~:r^)=:9(a;'-16) ;
/. 132;'=324 + 144=468;
.-. x' = 36; .-. x=do6.
Thus there are three roots of the proposed equation,
namely, 0, 6, —6.
Ex. 2. Solve x'-7xa' + 6a'=0.
Here it is obvious that a;=fl^ is a root. We may write the
equation,
x^—a'' = '7a\x—a);
and to find the other roots we begin by dividing hj x—cu
Thus, x''-]-ax+a'=7a^.
By solving this quadratic we obtain x=2a, or —3a. Thus
there are three roots of the proposed equation, namely a, 2a
3a.
Examples — 55.
1. a;'-13.^'4-36=0. 2. x-5Vx-U=0.
3. x-^V{x+^)=::Z 4. x'+y/{x'' + 9)=21.
5. 2v/(a;''-2a:+l)+:?;'=23 + 2a;.
6. x'-2x'+x' = 36. 7. 9V{x'-9x-{-2S) + 9x=x'i-S(>,
QUADEATIC EQUATIONS. 217
9. x'-4.x''--2V{x'-4.x' + 4:)=31.
10. a; + 2v/(:?;' + 5:2; + 2) = 10.
11. ^x+^/{x''h'7x + 6) = 19,
12. v/(a; + 9) = 2v/.T-3. 13. 6 s/{l -x')+6x=7.
14. v/(3:c-3)+ N/(5a;~19) = v/(2aj + 8).
:?;+ \/(12<^''— ^) a + l
15
X— Vil^a^—x) a-
..> 1 1 1 1 ^
16. ^ + -H - + ^=0.
x + 7 ^— 1 cc + l x—1
x+V{2-x')^x- V{2-x-') ^ '
19
x-^a x—a b + x h—x
x—a x + a h—x b + x'
20. x' + 3ax''=4:a\
XXXII. Problems which lead to Quadratic Equa-
tions CONTAINING OnE UNKNOWN QUANTITY.
214. In the solution of problems depending on quadratic
and higher equations there may be two or more values of the-
root, and these values may be real quantities, or impossible.
In the former case, we must consider if any of the roots are
excluded by the nature of the question, which may altogether
reject fractional, or negative^ or surd answers; in the latter
case, we conclude that the solution of the proposed question
IS arithmetically impossible.
Prob. 1. Find two numbers, such that their sum is 13, and
their product is 42:
What is said of the different roots of quadratic equations ?
10
21H ELEMEJS^TARY ALGEBllA.
Let X be one of the numbers, tben Id—x will be the other ; and then,
x{lS-x)=i2;
x=7 or 6. .-. 13-a;=:6 or 7.
Thus the two numbers are 7 and 6. Here, although the quadratic
equation gives two values of i», yet there is really only one solution of
the problem.
Pkob. 2. What number, when added to 30, will be less
than its square by 12 ?
Let X = the number ; then
whence « = 7or— 6.
And here the latter root would be excluded if we required only posi-
tive numbers.
Pkob. 3. A person bought a number of oxen for $600 : if
he had bought 3 more for the same money, he would have
paid $10 less for each. How many did he buy ?
Let X be the number bought ; then the price actually given for each
was — , and therefore,
600 600 ^^
x + S~ X '
whence a^=12 or —15,
which latter root is rejected by the nature of the problem.
Prob. 4. There are four consecutive numbers, of which, if
the first two be taken for the digits of a number, that
number is the product of the other two.
Let X, x + 1, x + 2, x + S, be the four numbers required; then
10^ + (aj + 1) = the number whose digits are x, and x + 1.
Therefore, by the question, (x •^2){x + S) = IO.2; -\-{x + l);
or x'' + 5x + e=zllx + l;
whence ic = 5 or 1.
QUADKATIC EQUATIO^^S. 219
Hence the numbers required are 5, 6, 7, 8, or 1, 2, 3, 4, both of which
results satisfy the conditions of the problem.
Prob. 5. Find two numbers whose difference is 10 and
whose product is one-third of the square of their sum.
Let X = the smaller, and a; + 10 = the greater ; then,
whence 2J=— 5±5\/^,
which values are impossihle. And the solution of the question is
arithmetically impossible, as may easily be shown, since it calls for
two numbers whose product is equal to the sum of their squares.
245. The reason why results are sometimes obtained, as in
Prob. 3, which do not apply to the problem proposed, seems
to be that the algebraic language is more general than the
ordinary language in which the problem is stated ; and thus
the equation which expresses the conditions of the problem
will also apply to other conditions. It will be a profitable
exercise for the student, when it is possible, by suitable
changes in the statement of the problem, to form a new
problem, corresponding to the result which was inapplicable
to the original problem. Thus in Prob, 3 it will be found
that " 15 " oxen is the answer of the following problem : Find
the number of oxen bought for $600, when, if the person
had bought '^ fewer oxen, he would have paid $10 more per
head.
Examples — 56.
1. Find the three consecutive numbers whose sum is equal
to the product of the first two.
2. The sum of two numbers is 60, and the sum of their
squares is 1872 : find the numbers.
Why are some of the results ohtainefl inapplicable?
220 ELEMENTARY ALGEBIIA.
3. The difference of two numbers is 6, and their product
is 720 : find the numbers.
4. Find three numbers, such that the second shall be
two-thirds of the first, and the third one-half of the first,
and that the sum of the squares of the numbers shall be
549.
5. Find the number which added to its square root will
make 210.
6. There are two numbers, one of which is f of the
other, and the difference of their squares is 81 : find them.
7. A and B together can perform a piece of work in 14|
days, and A alone can perform it in 12 days less than B
alone : find the time in which ^. alone can perform it.
8. In a certain court there are two square grass-plots, a
side of one of which is 10 yards longer than a side of the
other, and the area of the latter is -^^j of that of the former :
what are the lengths of the sides ?
9. A detachment of troops was arranged in a column
with 5 more men in depth than in front; the arrangement
was changed so as to increase the front by 845 men ; this
left the column 5 men deep : find the number of men in the
detachment.
10. There is a rectangular field, whose length exceeds its
breadth by 16 yards, and it contains 960 square yards: find
its dimensions.
11. A person bought a certain number of oxen for $1200,
and after losing 3, sold the rest for $40 a head more than
they cost him, thus gaining $295 by the bargain: what
number did he buy ?
12. The fore-wheel of a carriage makes 6 revolutions more
than the hind-wheel in going 120 yards ; but if the circum-
ference of each were increased by 3 feet, the fore-whc^el would
SIMULTANEOUS EQUATIONS. 221
make only 4 revolutions more than the hind one in the same
space : what is the circumference of each ?
13. By selling a horse for £24, I lose as much per cent, as
it cost me : what was the prime cost of it ?
14. Find the price of eggs per dozen, when two less in 24
cents' worth raises the price 2 cents per dozen.
15. There are three equal vessels, A, B, and C ; the first
contains water, the second brandy, and the third brandy
and water. If the contents of B and C be put together, it
is found that the mixture is nine times as strong as if the
contents of A and (7 had been treated in like manner: find
the proportion of brandy to water in the vessel C,
XXXIII. Simultaneous Equations involving
QUADEATICS.
246. We shall now give some examples of simultaneous
equations which may be solved by means of quadratics.
There are three cases in which general rules can be given
for' the solution of these simultaneous equations of two un-
known quantities.
217. I. When one of the equations is of the first degree,
and the other is of the second degree :
Eule. — From tlie equation of the first degree find the ex-
pression for either of the unhnoivn quantities in terms of the
other, and substitute this expression in the equation of the
second degree.
This will give a quadratic equation from which the value'
of one unknown is found.
Example. Given, 3:r4-4?/=18 ] , ,, -, .
^ ^ ^ ^ r to hnd x and u,
hx^ — Zxy~ 2 ) -^
Solution of Simultaneous Equations involving quadratics. Case I.
222 ELEME2!^TAIIY ALGEBEA.
From the first equation, y= — ; substituting this
value in the second equation, .
whence, 20x''—54:X + 9x''=8;
that is, 29x''-6^x=8.
From this quadratic we shall find that
x=2,ov-±;
and tlien by substituting these in the first equation we find
that
2/=3, or --.
218. II. When the two equations are of the second de-
gree, and all those terms which contain x and y are homo-
geneous, with respect to these quantities :
EuLE. — Put J=YX in both equations; obtain by division
an equation in wJiich v is tJie only unknown; v being deter-
mined, X and J may then be found.
Example. Solve 2x''—xy=z6Q \
2xy—y^=4.8 )
Putting 't^ — vx and substituting for y,
x\2-v)=^Q, and ^^(2^;--^;')=48;
whence, by division,
2v-v\ 48 _ 6 ^
2~v ~56~ 7 '
f n
w^hence, v——,or,v=2. The latter value is inapplicable;
the first gives, ^=±7, ^— ± 6.
This method is also applicable to Case I.
Example. x^ + xy-\-y^=l
2x-\
Case n.
SIMULTANEOUS EQUATIOIs^S. 223
V
Here puttiug vx for y, '
x\l + v + v'')=7 (1);
x{2 + dv)=S (2).
Therefore, by dividing (1) by the square of (2) x" disappears, and we
*^^^'^' (2 + 3^~64'
whence, ^=2, or 18; and, from (2),
^i^-^'^)=^^ that is, i ^=^' and i 2/=^^=^ ;
or aj(2 + 54)=8; ( or a,'=t; ( or y=vx=2^.
249. III. When each of the two equations is symmetrical
with respect to x and y, put u+v for x, and u— y for j,
[Definition. — An expression is said to be symmetrical with respect
to X and y when these quantities are similarly involved in it. Thus,
each of the expressions,
x^-{-x^y'^ + y^, 4xy + 6x + 5y—ly 2x'^—3x^y—dxy'^+2y*t
is symmetrical with respect to x and y.]
Example. x^+y^=18xy\ (1)
x + y=12 [ (2).
Put u+v for X, and u—v for y ; •
then (1) becomes {u+vy+{u—vy = lS{u + v) (u—v),
or, u'-{-3uv'=9(u'-v') (3);
and (2) becomes {u + v)-{-{u—v)=12;
whence, u=6.
Putting this for u in (3), •
21Q-{-lSv' = 9{d6-v');
whence, t; = =h 2 ;
.-. x—u-{-v=6±:2=8oy4:;
and, ^='Z^— -^=6=^2=4 or 8.
Case III ;— Symmetrical Equations.
224 ELEMEKTAllY ALGEBRA.
250. The preceding are general methods for the solution
of equations of the kind referred to^ and will sometimes suc-
ceed also in other equations ; yet in many of these cases a
little ingenuity and experience will often suggest steps by
which the roots may be found more simply.
Ex. 1. Solve 3x'-2xy=16 ) (1)
2x + 3^=^:12 ) (2)
Multiplying (1) by 3, 9x''-6xy=4t6,
(2) by 2x, 4a;' + 6xy=2^;
', adding, 13cc' = 45-f 24^;, or 13.^^—240; =45, whence x=d or
~ly3^. Equation (2) gives y=^^ (12— 2:?;) =2 or 4if.
Ex.2. Solve x'+y'=26) (1)
2:?;^ =24 ) (2)
Here adding, x^ + 2x2/ ^y"^ =4:9, whence a; + ^=±7;
subtracting, x^ — 2xy-\-y''=: 1, whence iz;—^=:±l.
If x+y^ + 'l) or, if x + y^+'l)
and x—y=-\-l ) and x—y= — l )
then 22^=8, and x=4:, then 22;= 6, and x=3,
also 2y=6, and ^=3; also 2y=8, and ^=4.
Similarly, by combining the equation x-{-y= — 7 with
each of the two cc— ^=±1, we should get the other two
pairs of roots,
0;=— 4, y=—dy and x=—3, y=—4,
Ex. 3. Solve x+y = b\ {1)
x' + y'^^b) (2)
This may be solved by the method of Art. 249, and alstj
as follows :
Bv division, — -^ =-^ ;
x+y 5
that is, x" -xy + ?/ =13. (3)
SIMULTANEOUS EQUATIOIs^S. 225
From this equation, combined witli x+y=5, we can f:nd
X and 2/ by the first case, or we may complete the soluticn
thus :
x + y=5;
(4)
squaring, x^ + 2xy + 7/^=26;
also, (3) x''-xy+if=13;
therefore, by subtraction, 3:?;?/= 12;
or, xy=:4:;
and, 4zxy—16, (5)
Subtracting (5) from (4); x''—2xt/ + y^=9;
extracting the square root, x—y=:iz3.
We haye now to find x and y from the simple equations
x + y=:6, x—y=^±:3.
These give x=l or 4, y=^ or 1.
Ex.4. Solve x'+xy+tf = 19, x'+xy + y*=U3,
By division, __^:.__;
that is, x'^—xy-\^y^=7.
We have now to solve the equations
x'^ + xy + y'^=zl9, x''—xy-hy^=7.
By addition and subtraction we obtain successively
a;'+/=z:13, xy=6.
Then proceeding as in Ex. 2, we shall find
x=dzd or d=2, y=±2 or =t:3.
Examples — 57.
1. ^^{3x + 6y)+i{^x-dy) = 6U\ 2. x' + y'=2bl
3.T^-h2/ = 179) x-\-y= 1 )
22Q ELEMEKTARY ALGEBRA.
3. x'-\-y'=25) 4. 2{x-y) = ll\ 5. x'+xy=^m\
4^+3^=24) xy=20) x''-y'=.ll)
6. x-ij=2l- '7. x'=\^-y'-4:xyl
16{x'-y'):=:^16xy) x-y=2 )
8. xy^{x-^) (y + i) I 9, x +y = 6)
xY^(x' + d) (^^-4) ) x'-Vf^T2 )
10. 3^^ + 2a;+^=485 ) 11. x -y — \\
Zx=2y \ x^-f^l9^)
12. a:'+^' = 189) 13. x-^y^a\ 14. xy=a^ \^
x^y^xy''=.V^^) x'+y' = b') x-^y=b 3
15. Vx + Vy=Sl 16. x'-[-:.:i/=a')
x+y=9) y'' + xy=lf)
17. l^ + 9{x + y)=2{X'^y)\ ^-{x-y) = {x-y)\
18. x^—xy=a{x^-l)-\-h-^l, xy—y'^—ay^-l.
cc' Z> xy x^ If' ' xy
21. x^=^ax-\-'by, y'^—ay-\-'bx.
251. We shall now giye some problems, to be solved by
equations of the second degree, with more than one unknown
quantity.
Ex. 1. The sum of the squares of the digits of a number
of two places is 25, and the product of the digits is 12 : find
the number.
Let X, y be the digits, so that the number will be \^x^-y\
then 2:^+^^=25, and xy—\2, from which equations we get
a; =3, y — ^^ or a: =4, ^=3, and the number will be 34 or 43.
In this case both the roots give solutions.
Ex. 2. Find two numbers, such that their sum, their
product, and the difference of their squares may be all equal.
SIMULTAISTEOUS EQUATIONS. 22?
Here assume x+y and x—yioT the two numbers; [this
step should be noticed, as it simplifies much the solution ol'
problems of this kind :] then their sum = 2x, their product
—x^^y"^, and the difference of their squares =42;^;
.-. (1) 2x^4.xy, (2) 2x=x''~y'']
from (1), y=i-; from (2), 2x=x^—^;
whence, 2;= |-(2=fc v/5) ;
and, .\ x + y=i {3±V6), x—y=i{l±:V6),
the numbers required.
Ex. 3. A man sets out from the foot of a mountain to
walk to its summit. His rate of walking during the second
hal£ of the distance is half a 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 at a uniform rate, which
is one mile per hour more than his rate during the first half
of the ascent: find the distance to the summit, and his
rates of walking.
Let 2x denote the number of miles to the summit, and
suppose that during the first half of the ascent the man
X
walked y miles per hour. Then he took — hours for the first
X
half of the ascent and — — hours for the second.
^-2
Therefore, ^-|—L-=5i (1).
Similarly, ^==3f (2).
From (2), 2^=^(f/ + l);
15
therefore, x=—{y-\-l).
4
~ 8'
228 ELEMEJS'TAllY ALGEBEA.
From (1), x(2y-j^=~tj(^y-jy
Therefore, by substitution,
whence, 16{y + l) {4:y--l)—Uy{2y—l);
and, 28^'— 8% + 15 = 0.
5
From this quadratic equation we obtain y—3, or -^,
/Co
5
The value -^ is inapplicable, because, by supposition, y is
1 15
greater than — . Therefore, y = 3; and then x=:-^, so that
the whole distance to the summit is 15 miles.
Examples — 58.
1. The sum of the squares of two numbers is 170, and the
difference of their squares is 72 : find the numbers.
2. The product of two numbers is 192, and the sum of
their squares is 640 : find the numbers.
3. The product of two numbers is 60 times their differ-
ence, and the sum of their squares is 244 : find the numbers.
4. Find two numbers, such that twice the first, w^ith three
times the second, may make 60, and twice the square of the
first, with, three times the square of the second, may make 840.
5. Find two numbers, such that their difference multiplied
into the difference of their squares shall make 32, and their
sum multiplied by the sum of their squares shall make 272.
6. Find two numbers, such that their difference added to
the difference of their squares may make 14, and their sum
added to the sum of their squares may make 26,
EXAMPLES. 229
7. Find two numbers, such that their product is equal to
their sum, and their sum added to tlje sum of their squares
equal to 12.
8. The difference of two numbers is 3, and the difference
of their cubes is 279 : find the numbers.
9. A man has to trayel a certain distance, and when he has
travelled 40 miles he increases his speed 2 miles per Hour.
If he had trayelled with his increased speed during the
whole of his journey, he w^ould have arrived at his des
tination 40 minutes earlier, but if he had continued at his
original speed he would have arrived 20 minutes later : find
the whole distance he had to travel.
10. A number consisting of two digits has one decimal
place ; the difference of the squares of the digits is 20, and
if the digits be reversed, the sum of the two numbers is 11 *
find the number.
11. A person buys a quantity of wheat, which he sells so
as to gain 5 per cent, on his outlay, and thus clears £16. If
he had sold it at a gain of 5 shillings per quarter, he would
have cleared as many pounds as each quarter cost' him shil-
lings: find how many quarters he bought, and what each
quarter cost.
12. Two trains start at the same time from tw^o towns,
and each proceeds at a uniform rate toward the other town.
When they meet it is found that one train has run 108 miles
more than the other, and that if they continue to run at the
same rate they will finish the journey in 9 and 16 hours re-
spectively : find the distance between the towns, and the rates
of the trains.
13. A and B take shares in a concern to the amount, alto-
gether, of $2500; they sell out at par — A at the end of 2
years, B at the end of 8 years — and each receives, in capital
and profit, $1485: how much did each embark?
230 ELEMENTARY ALGEBRA.
14. Find three numbers, such that if the first be multi-
plied by the sum of tha second and third, the second by the
sum of the first and third, and the third by the sum of the
first and second, the products shall be 26, 50, and 56.
XXXIV. Eatio.
252. Ratio is the relation which one quantity bears to an-
other with respect to magnitude, the comparison being made
by considering what multiple, part or parts, the first is of
the second ; or, in other words, what fraction the first is of
the second. Thus, if one quantity be two-thirds of another
quantity, the former is said to be to the latter in the ratio of
2 to 3, for if both be divided into respectively equal parts,
the former will contain two, and the latter three of these
equal parts. And thus the ratio of 2 to 3 and. the fraction
f, express the same idea ; for f indicates that unity has been
divided into 3 equal parts, and two of them are taken.
253. The ratio, then, of one quantity to another, is repre-
sented by the fraction obtained by dividing the former by
the latter. Thus, the ratio of 6 to 3 is |, or 2 ; that oi aioh
is 7-; that of 15 to 40 is 77:, or -; that of 4a to 6Z> is ^, or
2a
^y. Of course the two quantities compared must be of the
same kind, or one could not be a fraction of the other. (See
Venable's Arithmetic, Art. 171.)
254. The ratio oi aio h is expressed, either by two points
placed between the quantities, as a : 5, or for shortness, by
its measure, -r-. The first of the quantities, a:!), is called
the antecedent term of the ratio, and the latter the consequent.
Ratio. Antecedent and Coneequent.
EATIO. J>31
255. A ratio is said to be of greater or less ijiequaliiy ac-
cording as the antecedent is greater or less than the conse-
quent.
25G. If the antecedents of any ratios are multiplied to-
gether, and also the consequents, a new ratio is obtained,
which is said to be compounded of the former ratios. Thus,
the ratio, ac : hd^ is compounded of the two ratios, a : 1),
and c : d.
When the ratio a : Z> is compounded with itself, the re-
sulting ratio is a^ '.h^\ this ratio is called the duplicate ratio
of (2 : Z> ; and the ratio a^ : V^ is called the triplicate ratio
of a:h,
257. Problems upon ratios are solved by representing them
by their corresponding fractions, and then treating these
fractions by the ordinary rules. Thus,
If the terms of a ratio he riiultiplied or divided hy the same
quantity, the ratio is not altered,
^ a ma
' h mo
Thus ratios are com/pared with one another by reducing the
fractions which measure these ratios to common denomina-
tors, and comparing the numerators; and they are com-
pounded by multiplying together the fractions which meas-
ure them. Thus, also, a ratio may be reduced to its lowest
terms by dividing the numerator and denominator of its
fraction by their g.c.d.
Ex. 1. Compare the ratios 5 : 7 and 4 : 9.
■^iis. 14, H; whence 5 : 7 > 4 : 9.
Ex. 2. Find the ratio of | : f . Ans. 4 -^ |-=| X |=:||.
Ex. 3. What is the ratio compounded of 2 : 3, 6:7, 14 : 15 ?
• • Ans. f X f X ii=TV or 8 : 15.
Ratio of greater or less inequality. Componnd, duplicate, and triplicate ra«
t7X)s. Solution of problems upon ratios.
232 ELEME^^TARY ALGEBRA.
Ex. 4. Reduce to its lowest terms, a''—x^ : a^ ■{■2ax-\'X^.
. (a—x) (a + x) a—x
Ans. 7 — 7—xr~-. — \= — ; — ? ^^ a—x:a+x,
{a + x) [a+x) a+x
258. If to botli terms of the ratio, a : b, the quantity x bo
added, that ratio will be increas'ed or diminished according as
a is less or greater than b.
For,
a . ^
^ a-\-x
b + x'
if
ab + ax .
b{b^x) ^
ab-hbx
"' < b{b+xy
that is.
if
ab-\-ax >
or < ab+bx;
that is,
if
ax >
or < bx;
or if
a >
or < b;
which shows the truth of the proposition.
Examples — 59.
1. Compare the ratios 3 : 4 and 4:5; 13:14 and 23 : 24 ;
3:7, 7 : 11, and 11 : 15.
2. Of a-\-b:a — b and a"^ + b'^ :a^—b^, which is greater,
supposing a>b?
3. What is the ratio b inches to c yards ?
4. Find the ratio compounded of 3 : 5, 10 : 21, and 14 : 15 ;
of 7 : 9, 102 : 105, and 15 : 17.
a ~\~ ax -f- x^
5. Find the ratio compounded of -^ ^ and
. a —ax + ax^—x
a^—ax-\-x^
a+x
6. Compound
x'-9x+20 :x^-6x and x'-13x+A2 :x^~5x.
RATIO. 233
7. Compound the ratios a-\-h : a—d, a^-{-b'^ : {a-hby,
{a'-by:a'-I)\
8. "What is the ratio compounded of the duplicate ratio
of a+b : a—b, and the difference of the duplicate ratios of
a : a and a : b, supposing a>b?
9. What quantity must be added to each term of the
ratio a : b, that it may become equal to the ratio c:d?
10. Show that a—b:a + b^a^—b^:a^ + b'^, according as
a: b is Si ratio of less or greater inequality.
11. Find two numbers in the ratio of 3 to 4=, such that
their sum has to the sum of their squares the ratio of 7
to 50.
12. Find two numbers in the ratio of 5 to 6, such that
their sum has to the difference of their squares the ratio of 1
to 7.
13. Find x so that the ratio x : 1 may be the duplicate of
the ratio Six,
14. Find x so that the ratio a —x :b--x may be the dupli-
cate of the ratio a : b.
XXXY. Proportion.
259. When two ratios are equal, the four quantities com-
posing them are said to be proportional to one another ; thus,
(t c
a\b~c\d\ that is, \i ——~, then a, b, c, d, are proportion-
als. Thus, four quantities are proportional when the first is
the same multiple, part or parts of the second, as the third
is of the fourth. This is expressed by saying a is jfo b as q
is to d, and denoted thus, a\b\\c\d\ or thus, a\b = c:d\
or thus, — — —,
b d
Proportion.
234 ELEMENTARY ALGEBRA.
The first and last terms in a proportion are called the ex-
tremes, the other two the means.
Problems on proportions, like those on ratios, are solved
by the use of fractions.
260. (1.) When four qua7iUties are proportionals, the
Xyroduct of the extremes is equal to the product of the means,
a c
For if — = — , then ad—dc,
h d
(2.) Hence, if three terms of a proportion are given, we
can find the other from the equation ad—lc. Thus
Ic .ad ad ^ he
"=!' ^=T' 'S^ ^=a-
(3.) If a : 1=^1 : d, we have ad—I'' ; that is, if the first
be to the second as the second is to the third, the product of
the extremes is equal to the square of the mean.
In this case a, l, d, are said to be in continued proportion,
261. If the product of two quantities he equal to that of
tivo others, the* four are proportionals, the factors of either
product heing the extremes, and of the other the means.
For, let ad— he,
dividing by M, , y=^7^
or a\l—c\d,
262. \i a\'b—c\d, and c\d=e:f, then a\h—e:f,
_, a c ^ c e
For _^_and^=-^;
therefore, -7-=-^, or a: J=e: /l '
263. li a:'b=c:d, and e:f—g:h, then ae:hf=cg:dh.
Extremes. Means. Demonstrate Art. 260 (1), (2), and (3). Demonstrate Art.
261; Art. 262; Art. 263.
PROPOitTiois". 235
1-1 a c ^ e a
For _=_and-^=^;
^ ae eg t.^ ^j-l
This is called comjpounding the two proportions. And so
we may compound any number of such proportions. Thus,
if a\l — c : d, a" : If—c" : d% &c.
2M. If four quantities le proportionals, they are propor-
tionals when talcen inversely. That is, if a:h=c:d, then
'b:a=d\c.
For (Art. 134, i), if | = |, l-^f=l--|;
that is, — = — , or Z>: a=:<^: c.
a c
265. If four quantities le proportionals, they are propor-
tionals when talcen alternately. That is, if a:d=c'.d, then
a : c=h: d.
For (Art. 134, u), since -r=-j, -yX — ^-yX — ;
d c d G
that is, ' — =-7» or a : c=d : d.
c a
266. If four quantities are proportionals, the first together
loitli the second is to the second, as the third together with the
fourth is to the fourth,
therefore (Art. 134, iii), —j—— , or a-\-'b : d=c + d:c,
c
267. Also, the excess of the first above the second is to the
second as the excess of the third above the fourth is to the
fourth.
Art. 264; Ai't. 265; Art. 266; Art, 267.
236 ELEMENTARY ALGEBRA.
(X c
For -7- — -7; therefore (Art. 134, iv), by subtracting 1
from each of these equals,
a—1) c—d 77 -, ,
—7 — = — 7-, OY a—o: o~c—d\d.
d
2G8. We have also (134, v),
a^l) d c:^d d adtzh c^d
hade a c
or a±Z> : a—G::^d : c, which, by inversion (264), gives
a'.a-±^l)=c\c±d.
269. When four quantities are proportionals, the sum of
the first and second is to their difference as the sum of the
third and fourth is to their difference.
For (Arts. 266 and 267),
ct-\-l) c-\-d -, a—h c—d
therefore,
that is,
d ^ d ' l ~ d
a+d ^ a—h_ci-d c—d^
a—h~c—d'^
or a-\-'h\a'-'b^c-^d\c—d.
270. If four quantities form a proportion, we may der.ye
from them many other proportions*, all equally true.
Thus, if — =— , then — 7=-^ or ma : m'h~c\ d,
d mo d
Similarly, ma : h — mc \ d, a\mJ)=c:md, a:d — mc\ md ;
and in like manner
ad , h d ^
— : — =c:dy a: — =c: - , &c.
mm mm
Art. 268; Art. 269 ; Art. 270,
PK0P0RTI02s\ 237
That is^ either the Jirst or fourth terms of any proi)ortion
may be multiplied or divided by any quantity, provided that
either the second or tliird be multiplied or divided by the
same.
Ct G
271. As'ain ; if a:I?=c:d, then — = — ;
° d
m a m c ma mc
and — X-7 = — X-T? or —.=—.;
n n d no nd
or ma : nh — mc : na.
Then, by the preceding Articles, or by Art. 134,
ma±7ib mc±;?f/^
7)ia '~ mc ^
madcnl) fnc^hnd
whence, also, = ;
a c
or ma dtznb: a = mc±nd: c,
A2:am(Art. 269); 7= •.
ma— no m 0—714
or ma-{-nI):ma—nd=mc + nd:mc—nd,
272. (1) In like manner, if a:b=c: d—e :/, &c., by which
a c 6
it is meant a : h=c : d, or a : h=^e : f, &c., so that —=1—=--:,
-^ h d f
&c. Then r^ : Z>=a + c+6 : l-^-d^-f, &c.
For, let — =ri?^=— =— ; i\\Qn a—hx, c—dx, e=fx\
h d f ^ J ^
therefore, a-^-c-^-e—lx-^dx -\-fx —{h + d 4-/ ) x,
.-. a; or — =^-— -— - , or a: o=a-\-c+e:d + d-}-f,
o-i-d-\-f
That is, if there be a7iy 7iumber of quantities in projjortioii,
as one antecedent is to its consequent, so is the sum of all the
antecedents to the sum of all the co7iseqttents.
Art. 271; Art. 272 (1), (2), (3).
238 ELEMEKTARY ALGEBRA.
(I C
Again (2) ; the equations above deriyed from — =a;=-y
=-^5 &c., give ma^=^mbx^ nc=7idx, pe=pfx, &c.
/. 7na-\-nc +2^c — (in h-\-nd -\-pf ^x\
., „ a ma + nc+pe
therefore, . x ot -:r-=^ 7-^;
• mo-\-na+pf
a _c
So also (3), since -.=-^-=:-, &c., r^=- = -;
Since d=- (Art. 260), this is ^=i.
, T r, a _a -\-c -{-e _ ma + nc -^pe
merelore, -_---— --^_^^^^_^^^^^^^ ;
and so on for any number of terms and any like powers.
Ex. 1. Find a fourth proportional to \, J, and \.
a ^ " ' I-
Ex. 2. Find a mean proportional to 2 and 8.
Since Vz=iaG (Art. 260), this is n/(2x8)= v/16 = 4.
Ex. 3. \i a\'b—c\d^ express (a^d) — {b-\-c) in terms of a,
Z>, c only.
Here {a^-d)-{l^c) = (a^^-^--{M-c)
^a^—ab—ac-\-'bc_{a—b) {a—c)
~^ a "' a '
Examples— 61.
1. Find a fourth proportional to 3, 5, 6 ; to 12, 5, 10 '
to h h h
2. Find a third proportional to 4, 6 ; to 2, 3 ; to |, \,
PROPOETIO^^ 2o9
3. Find a mean proportional to 4, 9 ; to 4, |-| ; to 1|, l-^^g-.
4. 1^ a '.!):'. h'.c, then, a^ -^If \a + c:\ a^—W : a — c.
5. If | = |, show that (a^l) (0 + (^)r=^((; + ^)'^r:r|^(a + ^)%
6. If a : Z> : : c : ^, and m:n\\p:q,
then ma-\-7id : ma—nb : :pc + qd :pc—qd,
7. If a\h\\l)\c, then a^— Z^'^ : «^ : : Z^"^ — c'^ : c.
8. li a\l)\:c\d:\e:f, then a— e : Z>— /: : c : d
9. If ^ : Z> : : Z> : (?,
then mo^—'nF \ma—nc\ ipa^-i-qH^ :pa-{-qc.
Solve the equations,
10. -Vx^ y/h'^ \/X'-\/'b=a:'b,
11. 2; + a: 2.?;— ^:=3a; + ^ :4a;— fl^.
12. a;+y + l::r+^+2=:6:7
y + 2:^; : ;z/-2a;=12a; + 62/— 3 : 62/'-12a;-l.
13. xi^'l—y.^—^-.x—y,
14. What number is that to which if 1, 5, and 13 be sev-
erally added, the first sum shall be to the second as the sec-
ond to the third ?
15. Find two numbers in the ratio of 2^ : 2, such that,
when diminished each by 5, they shall be in that of l-J- : 1.
16. A railway passenger observes that a train passes him,
moving in the opposite direction, in 2'^, whereas, if it had
been moving in the same direction with him, it would have
passed him in 30" : compare the rates of the two trains.
17. A quantity of milk is increased by watering in the
ratio of 4 : 5, and then three gallons are sold ; the rest being
mixed with three quarts of water, is increased in the ratio
of 6:7: how many gallons of milk were there at first ?
240 ELEMEA^TAEY ALGEBRA.
XXXVI. Arithmetical Progression.
273. Quantities are said to form a Series when they ^7-0-
ceed by a laio, i. e., when any one quantity may be obtained
from those which precede it by a rule, which is the Imv of
the series.
274. Quantities are said to form an Aritlimetical Series, or
to be in Arithmetical Progression, when they proceed by a
common difference.
Thus, the following series are in a.p. :
1, 3, 5, 7, 9,
12, 8, 4, 0, -4, ... .
a, a-\-d, a + 2d, a + 3d, ....
In the first and third the quantities increase as the series
proceeds ; in the second, they decrease ; the common differ-
ences being 2, —4, and d, respectively, which are found by
subtracti7ig any term from tlie term folloiving ; therefor.?,
when the progression is a decreasing one, the common differ-
ence is negative.
275. Given a, the first term, and d, the common difference
of an Arithmetical Series, to find 1 the n*^ term, and S the
sum of n ter^ns.
Since a is the first term, and d the common difference,
the second term is a + d', the third term is a-\-2d', the
fourth term is a + 3d; and so on, where the coefiicient of d
is less by one than the number of the term. So in the 71^^
term we shall have {7i—l)d; therefore, the 71"' term
l=a+{n-l)d. (1)
Again, the sum of the terms,
S=a'h{a+d) + {ai' 2d) + &c., -f {I- 2d) + {l-d)+l;
Series. Arithmetical Series, or Progression. Common difference, how found;
when ne^tive. How to find the last term, and the sum of the scricB.
AKITHMETICAL PROGKESSIOK^. 241
and by writing the series in the reverse order, we have also
8=l-{-{l-d)-{-{l-M)-\-kQ. + {a+M) + {a^cl)-\-a.
Therefore, by addition,
2/S'=(« + Z) + (^+?) + (a+Z) &c., to n terms;
.-. ^8^ {a + 1)71',
and since l=a-\'{n—l)d,
we have also, S=\^a-^{n—l)cT\—. (3)
The equation (2) furnishes the following rule :
The sum of any nmnber of terms in A. p. ^5 equal to the
product of the numher of terms into half the sum of the first
and last terms.
Ex. 1. Find the sum of 20 terms of the series 1, 2, 3, 4.
Here a=l, d—l, ^^=20; using formula (3),
;S^=r[2 + (20-l)lp/, or =(2 + 19)-2/=21XlO=210.
Ex. 2. Find the 9th term, and the sum of 9 terms of 7,
5i, 4, &c. -^
Here ^=7, ^=— |, 7^— 9;
.-. Z==7+(9-l)x-|=7-8x|=~5;
and ^=1(7-5)^9.
Ex. 3. Find the 13th term of the series -48, -44, —40,.
&c.
Here a= -48, ^==4, n=ld',
.-. Z==-48 + (13-1)4=0.
Ex. 4. Find the sum of seven terms of i+i + i, &c.
Formulas (1), (2), (3).
11
242 ELEMEiifTARY ALGEBRA.
Here a — ^, d=—^, 71=7; here, as in Ex. 1, we are not
required to find I; ,\ using formula (3),
^zz.(l + 6X-i)l-(l-l)J=0.
' The series continued to seven terms is i, i, ^, 0, —^, —i,
Examples — 62.
Find the last term and the sum of
1. 2+4 + 6 + &C. to 16 terms.
2. 1 + 3 + 54-&C. to 20 terms.
3. 3 + 9 + 15 + &c. to 11 terms.
4. 1 + 8+15 + &c. to 100 terms.
5. — 5— 3 — 1 — &c. to 8 terms.
6. 14-I + -I+&C. to 15 terms.
Find the sum of
7. f +t\ + tt + &c. to 21 terms.
8. 4-3-10~&c. to 10 terms.
9. i+f + 1 + &C. to 10 terms.
10. |_|_i^i_&c. to 13 terms.
11. l + 2|- + 4i + &c. to 20 terms.
12. |-|4— |i— &c. to 10 terms.
276. By means of the equations,
(1) l=a+{n~iyd, (2) S = {a + l)j,
and (3) S={2a+{n-l)d}^,
when any three of the quantities a, d, I, n, S, are given, we
may find the others.
We may also use these equations to solve many problems
in Arithmetical Progression.
ARITHMETICAL PROGRESSIOI^r. 243
• Ex. 1. The sum of 15 terms of an A. p. is 600, and the
common difference is 5 : find the first term.
Since S =600, n=15, and d=o, we have by (3),
600=(26^ + 14x5)-V-;
.-. 600=(a + 35)15; .*. a + 35 = 40; .-. a=5.
Ex. 2. What number of terms of the series 10, 8, 6, &c.
must be taken to make 30 ?
^^=30, ^=10, d=:-2; /. by (3),
30 = [20-2{n-l)]-;
.-. {22-27z)^r=:30;
that is, 7^' — 11^=— 30,
and the roots of this quadratic are 5 and 6, either of which
satisfies the question, since the 6th term is 0.
Ex. 3. How many terms of the series 3, 5, 7, &c. make
up 24?
Here ;S'=24, a=3, ^=2;
therefore, by (3), 24=r[6+2(^-l)]4^;
whence,, n =4,, or —6, of which the first only is admissible
by the conditions of the question.
277. Ex. 4. Find the Arithmetical Mean between two
quantities a and b.
Let X denote this mean ; then since a, x, and b are in A. P.
x—a=h—x;
1 a + h
\v hence, x=:———;
that is, the arithmetical mean between two quantities la
half the sum of the quantities.
Ex. 5. Insert five arithmetical means between 11 and 23.
244 ELEMENTARY ALGEBRA.
Here we have to obtain an A. p. consisting of seven terms,
beginning with 11, and ending with 23.
Thus, a=ll, Z=23, ^=7;
therefore, by (1), Art. 276, 23 = 11 + 6^;
.-. d=2.
Thus the whole series is 11, 13, 15, 17, 19, 21, 23.
278. Ex. 6. The sum of three numbers in a. p. is 21,
and the sum of their squares is 155 : find the numbers.
Let X = the middle number, and y the common dif-
ference; then x—y, x,x^-y, represent the three numbers;
then {x—y)-\'X-\-{x^-y)= 21)
and {x-^Jy^Vx'-\-(x■\■yy=zlbb )
or reducing,
^Xz=z 21 )
3:?;H2?/^=155 f
whence, 2;=7, ^=±2;
and the numbers are 5, 7, 9.
Examples — 63.
1. The first term in an A. p. is 2, the common difference
7, and the last term 79 : find the number of terms.
2. The first term of an A. p. is 13 j-^, the common differ-
ence — f, and the last term |: find the number of terms.
3. The first and last of 40 numbers in a. p. are 1^ and
If : find the other terms, and the sum of the series.
4. Insert 3 arithmetical means between 12 and 20.
5. Insert 5 arithmetical means between 14 and 16.
6. Insert 7 arithmetical means between 8 and —4.
ARITHMETICAL PROGRESSIO:^'. 245
7. Insert 8 arithmetical means between —1 and 5.
8. The first term of an arithmetical progression is 13,
the second term is 11, the sum is 40 : find the number of
terms.
9. The first term of an arithmetical progression is 5, and
the fifth term is 11 : find the sum of 8 terms.
10. The sum of four terms in arithmetical progression is
44, and the last term is 17: find the terms.
11. The sum of fiye numbers in arithmetical progression
is 15, and the sum of their squares is 55 : find the numbers.
12. The seventh term of an arithmetical progression is 12,
and the twelfth term is 7 ; the sum of the series is 171 : find
the number of terms.
13. A traveller has a journey of 140 miles to perform.
He goes 26 miles the first day, 24 the second, 22 the third,
and so on: in how many days does he perform the journey?
14. A sets out from a place and travels 2i miles an hour.
B sets out 3 hours after A, and travels in the same direction,
3 miles the first hour, 3i miles the second, 4 miles the third,
and so on : in how many hours will B overtake A,
XXXYII. Geometrical Progression.
279. Quantities are said to be in Geometrical Progression
when they proceed ly a comvion factor ; that is, when each
is equal to the product of the preceding by a common factor.
This common factor is called the common ratio, or simply
the ratio.
Thus the following series are in geometrical progression :
Geometrical Progression.
246 ELEMEKTARY ALGEBRA.
1, 3, 9, 27, 81 ... .
^> ^} h iQy it
__JL __4 ___l 6
a, ar, ar"^, &c.
The commo7i ratios being 3, ^, — f, and r, respectively. The
common ratio is found dy dividing any term hy the term
ivJiicli immediately precedes it; therefore, if the quantities
are alternately + and — , the ratio is negative,
280. Given a tJie first term and r the common ratio of a
geometrical series, to find 1 the nth term, and S the sum of
n terms.
Here, since a is the first term and r the common ratio, the
second term is ar, the third term is ar"^, the fourth term is
ar^, and so on; where the index of r in any term is less
hy one than the number of the term. Thus then the ?zth
term l—ar^~'^. (1)
Again, 8—a^-ar^ar'^-\-kQ,., +ar"-^;
and TS=.ar-{-ar''-\-ar^-\-kQ,,, +ar"-Har";
therefore, by subtraction,
rS—S^ar'^—a,
the other terms disappearing. Hence,
8— —z=a.^ ^ (2); ox B— (3), smce rl=iar\
Ex. 1. Find the 6th term, and the sum of 6 terms of
1, 2, 4, &c.
Here a—\, r=2, n—^\
.•.?=1X 2^=32; and /S'^^^-^ 63.
/C — 1
Ex. 2. Find the 8th term, and the sum of 8 terms of 81,
-27, 9, &c.
The common ratio, how found; when negative. To find the /ith term, and
the sum of n terms.
GEOMETRICAL PROGRESSJOIn^ 247'
Here a— SI, r= — l, n-.
therefore^
)re, ?=81x(^|-)'-3*X-^ = -|-3=:-^;
and
;S^=:Ii ^ = 60-2-0-
— 3— 1
Ex. 3. Find the sum of 8 terms of the series, 4, 2, 1, ^, (&c.
• Here a—^, r—^, n—S;
therefore, without finding I,
a ^V2^"V _ ^V^~2V _255 2_255
i-1 1-i 64^1"" 32*
Ex. 4. Find the sum of 1— f +-^— &c. to 4 terms.
Here a=l, r=— |, 71 = 4;
,(-iy-
4* . 4^-3*
-1 o-i-1
. ^-IX.^— ^^ -»^I — j^- 3 256-81
__175___25
- 7.3' ~ 27'
Ex. 5. Find the sum of 2^— 1+|— &c. to 5 terms.
Here ^=f? ^=-"1? n=6;
(-1)-
••^ -^- _|^i 2- -1-1 2- I
__5^ ^ 32 + 3125 _3157__
""2* 7* 5' ~14.5^~ ^*
Examples — 64.
Find the last term and the sum of
1. 1+4 + 16 + &C. to4terms. 2. 5 + 20+ 80 +&c. to 5 terms.
Formulas (1), (2), (3).
'ZiS ELEMEIs^TARY ALGEBRA.
3. 3 + 6 + 12 + &c.to6terms. 4. 2— 4 + 8- &c. to 8 terms.
5. l-4+16-&c.to7terms. 6. 1-2+2'- &c. to 10 terms.
Find the sum of
7. i + i + tV + &c. to 8 terms. 8. i + i + f + &c. to 6 -terms.
9. 3 +i_|.2 +&C. to 6 terms. 10. 3-i + -^j—&c. to 5 terms.
281. If the terms of a geometrical progression decrease
numerically as the series proceeds, then the common ratio r
is a proper fraction ; that is, r is less than 1. Therefore tho
powers r^, r^ r*, .... r'^, are still less than 1, and ar^ less
than a. Both the numerator and denominator of the
fraction S= z— are then negative, and we may write it
^ a—af a ar^
1— r "1— r 1— r
ISTow, the greater we take the number of terms n, the less
will be the value of ar"" ; and therefore, by taking n suffi-
ciently great, we may make as small as we please.
Hence by making n sufficiently great, we render the value
of S as near as we please.
1—r ^
This result we enunciate thus : In a Geometrical Progres-
sion in loliich the common ratio is a iwojper fraction, hy
talcing a sufficient number of terms the sum of the series can
be made to differ as little as ive please from> .
is said then to be the Limit of the sum of the series
1—r
a, ar, ar"^, &c., when r<l ; or we say, for shortness (but not
correctly), S=z is the sum of the series to infinity. Using
the same language, these series are called infinite Geometrical
Progressions.
' When the common ratio is a proper fraction. The limit of the sum oi a
series, when r<l. Infinite Geometrical Progressions,
GEOMETRICAL PROGEESSIOK". 249
It is common to denote the Limit of such a sum by 2.
Ex. 1. Find the Limit of the sum of the series 1 + i + i + &c.
Here a=l, r=i; .*. :s=- — -=— = 2; that is, the more
1— -2- Y
terms we take of tliis series, the more nearly will their sum
= 2, but will neyer actually reach it.
Ex. 2. Fmd the sum of 2i—^ + ^—&G. ad infinitum.
Here a=n, r^-^, :. ^=i3|z^= 1^^=1=2 A-
282. Eecurring Decimals are examples of infinite Geo-
metrical Progressions. Thus, for example,
.3333 .... denotes iV + yfo + 1A-0+&C:,
a G. p. of which the first term <^=yV ^^^ ^^ ^^2,^0 r=^.
Hence we may say that the Limit of this decimal is
Again; .3242424 .... denotes 33_+^|a_ + .^^_2a__+&c.
Here the terms after -^-^ form a G. p. of which the first
term = y 0^0-0^ ^^^ ^^^ common ratio is y^-g-. Hence the
Limit of this series is iQOQ - — ^2^_. Therefore the limiting
^ Too"
value of the recurring decimal is
3_ 24 _ 3x99 + 24 3 (100-1) + 24 _ 324-3 ^
10"^ 990 ~ 990 "" 990 ~ 990 '
and this value accords with the rule in Arithmetic. (See
Venable's Arithmetic, Eecurring Decimals.)
Examples — 65.
Find the Limits of the sums of the following series :
L 4+2 + 1+&C. 2. | + ^ + | + &c. 3. i— iV+A + <^^'-
Recurring Decimals.
11*
250 eleme:s'taky algebra.
4. I-H-I-&C. 5. l-i + i-&c. 6. l-| + ^-&c.
Find the limiting values of the following recurring
decimals :
10. .151515 11. .123123123 12. .4282828
283. By means of the equations of Geometrical Progres-
, „ 1 rv Ir—a af — a a
Bion, yiz., l—ar''-^. 8— -= -, :2=- , we may
solve many problems respecting series of this kind. It is not,
how^ever, generally easy to find 7i from the other quantities,
because it is an exponent. The method of logarithms will
serve to find it in all cases.
Ex. 1. Find a geometrical series, whose 1st term is 2 and
7th term -^j.
Here a=2, 1=-^, n=7; ,-, -^=2r% and r°=-g^, whence
r=dt:^, and the series is 2, ±1, i, ±J, &c.
Ex. 2. Given 6 the second term of a geometrical series and
54 the fourth, to find the first term.
54 dT^
Here 6=ar, 54=ar^: -'- -yr— — , or 9=r^:
6 ar
n
hence r=:dt3, a= — = ±:2,
r
Ex. 3. Insert three geometrical means between 2 and 32.
Here we have to obtain a geometrical progression con-
sisting of five terms, beginning wath 2 and ending with 32.
Thus, a=2, 1 = 32, n=5; therefore,
32=2r^; .-. r'=16; .-. r=2.
Thus the series is 2, 4, 8, 16, 32.
Ex. 4. How many terms of the series 2, —6, 18, &c., must
be taken to make —40 ?
GEOMETRICAL PROGRESSIO:^'. 251
Here a=:2; r=-3; S=-4:0;
therefore,
-3-1 '
hence 2(— 3)"=162;
/. (_3)^==:81.
But we know that 81=3*; therefore, n=4:.
Examples — 66, •
1. Insert 3 geometrical means between 1 and 256.
2. Insert 4 geometrical means between 5^ and 40-|-.
3. Insert 4 geometrical means between 3 and —729.
4. The sum of three terms in geometrical progression is
63, and the difference of the first and the third term is 45 :
find the terms.
5. The sum of the first four terms of a geometrical pro-
gression is 40, and the sum of the first eight terms is 3280 :
find the progression.
6. The population of a country increases annually in G. p.,
and in four years was raised from 10,000 to 14,641 souls : by
what part of itself was it annually increased ?
7. The sum of an infinite geometrical series is 3, and the
sum of its first two terms is 2f : find the series.
8. The sum of an infinite geometrical series is 2, and the
second term is — | : find the series.
9. A body moves through 20 miles in the first one-mil-
lionth part of a second, 18 miles in the second millionth
part of a second, and 16| miles in the third millionth part
of a second, and so on forever: what is the limit of the dis-
tance from its point of departure, which it can attain ?
252 ELEMENTAEY ALGEBRA.
XXXVIII. Harmonical Progression.
284. Quantities are said to be in Har7nonical Progression
when their reciprocals are in A. p. Thus, since 1, 3, 5, &c.,
^, —i, — f, &c., are in aiithmetical progression, their recip-
rocals, 1, ^, ^, &c., 2, —2, — f, &c., are in Harmonical Pro-
gression.
The term Harmonical is apj)lied to series of this character
from the fact that musical strings of equal thickness and
tension will produce harmony when sounded together, if
their lengths be as the reciprocals of the arithmetical series
of the natural numbers 1, 2, 3, 4, &c.
285. If three quantities, A, B, and C, are i7i Harmonical
Progression, tlien 2uiU A : : : A—B : B— C,
For, by definition, -j, ^ , and -^ are in arithmetical pro
gression; therefore,
1 2_A i.
B^A^G'S'
multiplying by ^^(7, AG-BG=AB-AO;
that is, G {A-B)=A (B-G) ;
therefore. A: G::A—B:B—G, wliicli loas to le ^proved.
This property is sometimes given as the definition of Har-
monical Progression, and the property gi^^en as the defini-
tion in Art. 284 deduced from it.
286. We cannot find a convenient expression for the sum
of any number of terms of a harmonical series ; but many
problems with regard to such series may be solved by invert-
ing the terms, and then treating these reciprocals as in arith-
metical progression.
Harmonical Progression. Demonstrate Art. 285. To fin(^ the sum of the
terms of a Harmonical Progression.
HAEMOKICAL PROGRESSION. 253
Ex. 1. Continue to three terms each way the H. p., 2, 3, 6.
Here, since -J-, i, ^ are in a. p. with common difference
— I", the arithmetical series continued each way is
1. h h h h h ^-h -i
therefore the harmonical series is
1,1,1,2,3, 6, infinity, -6 -3.
Ex. 2. Insert five harmonical means between f and -fj.
Here we have to insert five arithmetical means between |
and -1^^-. Hence by equation (1) Art. 275, -i/ =1 + 6(7; there-
fore, 6^— f, and d=-^-^''y hence the A. p. is
t 2.A 2_6 2 7 J.8. 2JL JL5_ •
^i 165 16, iti 165 165 8 ?
and therefore the H. p. is
287. The geometrical mean G between two quantities a
and l is the geometrical mean between their arithmetical
mean A and their harmonical mean H,
Eor, the arithmetical mean between a and l is (Art. 277),
A="^. (1)
h P
The geometrical mean 6^ gives --^=—; .*. G'^a'h and G—^/ab,
To find the harmonical mean ZTwe have ———^=-- ;
h H H a
'%ah
therefore, aE—al—al — lH. or H— — -7: (2)
a-\-o
multiplying (1) and (2), we get
AH———X ^ — al—G\
Therefore G—s/ AH, or G is the geometrical mean be-
tween A and H,
Demonstrate Art. 287.
254 ELEMEKTAKY ALGEBRA.
Examples— 67.
1. Continue the Harmonical Progression 6, 3, 2 for three
terms.
2. Continue the Harmonical Progression 8, 2, l^- for three
terms.
3. Insert 2 harmonical means between 4 and 2.
4. Insert 3 harmonical means between — and — .
o /vL
5. The arithmetical mean of two numbers is 9, and the
harmonical mean is 8 : find the numbers.
6. The geometrical mean of two numbers is 48, and the
harmonical mean is 46^2-g-: find the numbers.
7. Find two numbers, such that the sum of their arith-
metical, geometrical, and harmonical means /j 9|, and the
product of these means is 27.
8. Find two numbers, such that the product of their arith-
metical and harmonical means is 27, and the excess of the
arithmetical mean above the harmonical mean is 1|-.
XXXIX. Pekmutations and Combinations.
288. The Permutatio7is of any number of things are the
different arrangements which can be made of them by
placing them in different orders, taking either all the things,
or a certain number of them at a time, together.
Thus the 'permutations of a, Z*, c, taken all together, are
ale, achy hca, cia, cah, lac ; taken two together, are ac, ca, al^
la, Ic, cl.
289. Note. — Some writers on Algebra restrict the word permufa-
tions to the case where the things are taken all together, and call tlie
PERMUTATIOIsrS A^^D COMBII^TATIONS. 255
sets in all otiier cases Variations^ or Arrangements. But this distinction
is not always observed, and we shall use the word permutations in all
cases.
290. The number of permutations of n tilings, tciken two
together, is n (n— 1) ; taken three together, ^5 n (u — 1) (n— 2).
Let there be n different things, a, i, c, d, &c. Remove one
of them, a-, there will be ^—1 things, h, c, d, &c., left; now
place a before each of these n—1 things; we thus get n—1
permutations, n things taken tiuo together, in which a stands
first. JSText remove d from the n things ; there will remain
oi—l things, a, c, d, &c. ; and placing b before each of these
we get 71—1 permutations of n things tahen two together, in
which b stands first. Similarly placing c before each one of
the other letters, we find n—1 permutations, in w^hich c
stands first; and so on for the rest. Therefore, on the
whole, there are n{7i—l) permutations of n things talcen two
together, or ttuo and ttoo, as is the usual phrase. Therefore
there are also {n—1) {^^— 2) permutations of n—1 things
taken tioo together.
Let now a, one of the n things, be again removed ; the re-
•maining n—1 things, by what w^e have just proved, gives
{n — 1) {n—2) permutations when taken tivo together; put a
before each of these permutations ; we thus get {7i — l){7i—2)
permutations, each composed of three things, in w^hich a
stands first. Similarly, there are {7i—l) (^^— 2) permuta-
tions each, of three things in which b stands first ; similarly,
there are as many in which c stands first, and so on for the
rest. Therefore there are 72 {n — 1) {71— 2) per77iu tat ions of 7i
thiiigs talcen three together,
291. We observe at once that the second term of the -usi}
factor of the product which expresses the number of permu-
tations in each case of the preceding article, is numerically
less by one than the 7iumber of things taken together. From
Permutations. Demonstrate Art. 290.
256 ELEMENTARY ALGEBRA.
these cases we miglit infer by induction that this is a general
law, and that the number of permutations of 7i letters taken
r together is n{;n—l) {n—2) .... (?^— r+l), and this we
can now demonstrate.
For, suppose this law to hold for the number of permuta-
tions of n things, a, l, c, d, &c., taken r—1 together, which
would therefore be
71(71-1) (n-'^) .... (^^-(r-l)+l).
]^ow leaye out a\ there will be 7^— 1 things, h, c, d, &c.,
and the permutations of these, taken r—1 together, will be,
by the preceding result,
{71-1) {n-2) {7i-l-[r-l)+l);
that is, {71-1) {71-2) {7i-r+l).
Set a before each of these permutations; we get thus
(^—1) (n— 2) {7i—r-\-l) permutations taken r to-
gether, in wiiich a stands first. Similarly, we have as many
in which b stands first ; as many in wiiich c stands first, and
so of the rest ; therefore on the whole there would be,
n{7i-l) {71-2) {7i-r+l)
permutations of 7i things taken r together.^
If then the formula holds when the 7i things are taken
r—1 together, it will hold when they are taken r together;
but it has been proved true when they are taken 3 together ;
it holds, therefore, when they are taken 4 together ; and
therefore, when taken 5 together, and so on ;
that is, 7i{7i — l) {71—2) {7i—r+l)
represents the Ttwmher of per7nutations of ti tilings taken r
togetlie7% for all values of r (these values being limited only
by the definition).
General law for the number of Permutations of n letters taken r together.
PERMUTATIONS AND COMBUST ATIONS. 257
292. Hence, denoting by P^, P^, P^, &c., P^, the number
of permutations of n things taken 1, 2, 3, &c. r, together,
we have from the preceding formula,
P^^n, P^==n(n-1), P,=n{n-l){n-2), &c.
P,,=n{n—1) (^— r + 1).
29B. If T=7i, that is, if all the quantities are taken to-
gether, then the number of permutations (P) of n things is
n{n—l) (71—2) (n—n + l);
that is, n{n—l) {n—2) .' . . 1;
or reversing the order of the factors, we have,
P=1X2X3 . . . . Xn.
This result we may enunciate thus :
The number of Permutations of n things, taken all together,
is equal to the product of the natural numbers from 1 to n,
inchisive.
Thus the number of permutations of 8 letters, taken all
together, is 1x2x3x4x5x6x7x8.
294. For the sake of shortness, the continued product,
1.2.3.4 . . . . n, is often denoted by [^; thus [^ denotes
the product of the natural numbers, from 1 to ^ inclusive.
The symbol \n may be read factorial n.
Ex. [8, (read factorial eighty, denotes the product
1X2X3X4X5X6X7X8.
295. To find the number of permutations of n things,
which are not all different, taken n, i, e, all together.
Express the formula by the use of the symhols Pi, P21 Pz^ ^tc. The nuii>
ber of Permutations where r=:-n. The symbol \n. Demonstrate Art. 295
258 ELEMEis^TARY ALGEBRA.
Let there be n letters ; and suppose jp of them to be a's, q
of them to be ^'s, r of them to be c's, &c. ; the number of per-
mutations of them, taken all together, will be,
1.2.3 . . . . ^
1.2.3 .... ^Xl.2.3 .... ^X&c* , .
For let N be the number of such Permutations. Suppose
now that in any one of them we change the ^ a's into dif-
ferent letters; then these letters might be arranged in
1.2.3....^ different ways, and so instead of this one
permutation, in which p letters would have been «'s, we
shall now haye 1.2.3 . . . . p different permutations. The
same would be true for each of the iV^ permutations ; hence,
if the p a's were all changed to different letters, we should
have all together 1 . 2 . 3 .... j^X indifferent permutations
of n letters, whereof still q are 5's, r are c's, &c.
So, if in these the q b's were changed to different letters,
we should have 1.2.3 .... gXl.2.3 . . . .^XiV" differ-
ent permutations of n things, whereof still r would be c's ;
and so we may go on, until all the n letters are different.
But when this is the case we know that their whole number
of permutations ==1.2.3 . . . . n. Hence,
1.2.3 ... .i?Xl. 2.3 ... . gX&c.XiV'=1.2.3 . . . n,
1.2.3 . , . . n
and -^-12,3 i?Xl.2.3 .... gX&e.'
This value of iV^ may be written by the notation of Art.
294; thus,
\n
Ex. 1. How many changes can be rung with 5 bells out
of 8 ? How many with the whole peal ?
Here P, =38.7.6.5.4=6720; P=8.7.6.5.4.3. 2 1 = 40320.
PERMUTATIOJ^^S Al^D COMBIJsTATIOKS. 259
Ex. 2. How many differ tnt words may be made with all
the letters of the expression a^Jfc ?
12 3 4 5 6
Here are 6 letters, 3 a's, and 2 Z>'s ; .-. N= ' ' ' ' ' = 60.
JL./V.d X 1"V
Examples — 68.
1. How many changes may be rung with 5 bells out of
6, and how many with the whole peal ?
2. In how many different orders may 7 persons seat
themselves at table ?
3. How many different words may be made of all the let-
ters of the word laccalaureus ?
4. How many different words may be made of all the
letters of the word Mississippi?
5. How many different words may be made of all the
letters of the word Alabama?
6. Of what number of things are the permutations 720 ?
7. There are 7 letters, of which a certain number are a^% ;
and 210 different words can be made of them : how many a's
are there ?
8. If the number of permutations of n things taken 4
together is equal to twelve times the number of permu-
tations of n things taken 2 together, find n,
296. The Comlinations of any things are the different col-
lections or sets that can be made of them, without regarding
the order in which the things are placed. Thus the com-
binations of a, b, c, taken two together, are ab, ac, bc\ of a,
b, c, d, three together, are abc, abd, acd, bed.
297. It is readily seen that each combination which con-
tains r things will furnish 1.2.3 r permutations of
Combinations. Demonstrate Formula (Art. 298).
260 ELEMENTAKY ALGEBIcA.
•
these things taken all together. For we haye seen, Art. 291,
that^z things give 1.2.3 , . , , n permutations.
Thus the combination abc supplies 1.2.3, or 6 permuta-
tions, abc, act, lac, lea, cal, da,
298. The numler of comlinations of n different tilings,
talcen r together, is
n (^—1) (^—2) .... (?^— r+1)
1.2.S r *
For, each combination of r things will supply 1.2.3 . . . . r
permutations of r things ; hence, if Cr denotes the number
of combinations of n things, r together, we have
1.2.3 .... rxC^=number of permutations of n things, r
together,
= y^—n {n—\) (^--2) (n—r + l) ;
^ _n{n—l) {n—2) .... (n—r + l)
.-. G,- 1.2.3 . . . . r •
Therefore, (7i=— , C^= ^ ^^' , 0,=^ — ^23 ' ®^-'
where Ci, Co, C^, &c., express the number of combinations of
n letters taken one and one, two together, three together, &c.
«Aa rn\. ' n n{n-l){n- 2) (^-r+ 1)
299. The expression Cr — -^ \ 9,^ ^ ~^^
may be put in a yery conyenient form ; for, by multiplying
the numerator and denominator of the aboye fraction
by 1.2.3 .... {n—r), it becomes
n jn-l) {n-2) (n-r + l) X (n-r) 3.2.1
1.2.3 r X 1.2.3.... (^-r)
1.2.3 n lH:
"1.2.3 r X 1.2.3 (?z-r)~" [r |
PERMUTATIONS AND COMBINATlOjSfS. 201
■'■ <^ -7-^- (1)
\r \7i — r
300. The number of comUnations of n things taken r to-
gether is the. same as the ^lumber of combinations ofn things
taken n— r together.
For, to find the number of combinations of n things taken
n—r together,. we simply write n—r for r in formula (1).
We get thus
\n \n
'^''~ \n—r\7i—{ n—r)^\n—r\)r_^
which is equal to Cr , which was to be proved.
The truth of this proposition is also evident from a very
simple consideration, viz., that when we take r things from
^^ things, n—r things will be left; and for every different
collection containing r things there will be a different col-
lection left containing n—r things; therefore the number of
the former collections must be equal to that of the latter.
Ex. 1. Eequired the number of combinations of 20 things
taken 18 together.
Here the number of combinations of 20 things taken 18
together is equal to the number of combinations of 20 things
taken 2 together,
that is, (7is=C,-=^f^=:10xl9 = 190.
Ex. 2. Eind the number of combinations of 10 things, 3
and 6 together.
^ ^ 10.9.8 ^__ Ann 10.9.8.7 ^,^
Here C,=^-^j^=120, and C,= C= ^ 2.3.4 ^
Ex. 3. How many words of 6 letters might be made out of
the first 10 letters of the alphabet, with two vowels in each
word ?
state th** r^Tinciple explained in Art. 300.
262 ELEMEKTARY ALGEBRA.
In these 10 letters tliere are 7 consonants and 3 vowels;
and in each, of the required words there are to be 4 conso-
nants and 2 vowels : now the 7 consonants can he combined
four together in 35 ways, and the 3 vowels, two together, in
3 ways; hence there can be formed 35x3 = 105 different sets
of 6 letters, of which 4 are consonants and 2 vowels : but
each of these sets of 6 letters may be permuted 6.5.4.3.2.1
= 720 ways, each of these forming a different word, though
the whole 720 are composed of the same 6 letters ; hence the
number required=105X 720=75600.
Examples — 69.
1. How many combinations can be made of 9 things, 4
together ? how many, 6 together ? how many, 7 together ?
2. How many combinations can be made of 11 things, 4
together ? how many, 7 together ? how many, 10 together ?
3. A person having 15 friends, on how many days might
he invite a different p'arty of 10 ? or of 12 ?
4. Fintl the number of combinations of 100 things, taken
98 together.
5. Four persons are chosen by lot out of 10 : in how many
ways can this be done? on how many of these occasions
would any given man be taken ?
6. The number of combinations of ^+1 things, 4 to-
gether, is 9 times the number of combinations of n things, 2
together : find n.
7. How often may a different guard be posted, of 6 men
out of 60 ? on how many of these occasions would any given
man be taken ?
8. How many words may be formed, each consisting of
three consonants and a vowel, out of 19 consonants and 5
vowels.
Bi:ts"OMIAL TIIEOKE^.!. ^ 263
XL. Binomial Theorem.
301. The Bi7iomial Theorem is the name given to a rule
discovered by Sir Isaac JSTewton, by means of which any bi-
nomial may be raised to any given power much more expe-
ditiously than by the process of repeated multiplication given
in Involution.
302. To prove the Binomial Theorem when the index of the
poiver is a positive ivhole ntimher, (Bobillier's Proof.)
By actual multiplication the successive powers of the bi-
nomial a-\-x are found to be as follows :
{a-\-xy=a-{-xi
{a-\-xy=:a^-\-2ax-{-x^ ;
{a+xy=a^ + da'x-\-dax'+x'' ;
(a + a;) * = a' + 4a'a; + 6aV + 4:ax^ + x* ;
which, by dividing the first by 1, the second by 1.2, the third
by 1.2.3, the fourth by 1.2.3.4, and using the factorial nota-
tion of the preceding chapter to denote the continued pro-
ducts 1.1, 1.2, 1.2.3, &c., may be written thus :
(a + xy _a' x^
[1— |i + li'
(a + xy _ a^ g' eg' x^
{a^xy a^ a^ x" a' x^ x^
[3 ""]3 ^ l^'^"[ili^]3 '
{a + xy a' a^ x' ^^ a' x"" o^
in which a laio of formation is easily perceived in relation
to the exponent of the power of the binomial. The same
Binomial Theorem. Proof of the Binomial Theorem, when the index of the
power is a positive whole number.
264 ELEMENTARY ALGEBRA.
law of formation of the terms of the expansion is found to
hold for {a + xy, (a-\-xy, &c. Now the introduction of the
new factor « + a; in order to convert {a + xY~^ into (a + ic)%
inyolves precisely the same processes as the introduction of
the same factor {a + x) to conyert {a + xY into {a-\-xy. It
is reasonable then to assume that if the law is true for
{a-\-xy'~^, it is true for [a + xY] now we know by actual
multiplication it is true for {a-\-xy\ hence it is true for
{a-{-x)\ and hence for {a-{-xy, &c. Therefore the law holds
generally — viz., for any positive whole number exponent
we have
{x-\-aY _ cc** a x""-' a^ x"^"" a^ x""-^ a" ^
\n ~~\n'^\l |/^-"l"^]2~ j^^^ "^ |3 \n-3 "^"j^ '
which may be written,
\n \n
I. ix + aY=--x'' + TT-T^— ;^^''~' +-
[1 \n-l [2 1 ^-2
\n \n
+ ,^ , ^ 6^V-^ + + — J==— a'x^^. . . ,'\-a\
[3 1^—3 [r \n — r
Or by cancelling the like factors in the coefficients,
II. {x + aY=x''-{- nax""-' + — ^ — ^ a^x'"-' + —^ —^ ^
[± li
[r
303. In these expressions I. and II. the corresponding terms
\n
Ir In—
aV-' (1)
and nin-l){n-2) (n-r+l)^^^^^
are the same, and they express the term which has r terms
before it; that is, the (r + 1)*^ term. This term is called the
BINOMIAL THEOREM. 265
General Term; and both forms of it, (1) and (2), should be
carefully noted and remembered.
304. If the binomial is written {a-\-xy, the expansion I.
would be
\n \n
li r^~^ 1^ |y^— 2 ^
\n
the general term being , ^ ~_ — a^-V, the exponents of a
and X being interchanged. Similarly, II. becomes
{a-{-xy=a'' + na'^''x-\' ^\~ ^ a^-V+
+ — ^^ — -7 ^ — -^ a^-^a;^+ x\
And if «^=1, this last giyes
IIL(l + ^)-l+n^+^A'+ ^^^-^)^^-! )^'....
If l£.
w (w-1) (w-2) .... (n-r+l) ^, ^„
305. I. The law of the exponents of the terms in the ex-
pansion of the binomial formula is, that the exponent of the
leading letter of the binomial is in the first term n, and that
of the second letter is o ; and the former decreases by unity
and the latter increases by unity, in each successiye term to
the last term, in which the exponent of the leading letter is
0, and that of the second letter is n; and the sum of the ex-
pone7its of the tiuo letters in any term is always n, the ex-
ponent of the Mnomial. Moreover, the exponent of the
second letter in any term expresses the number of terms
which precede that term, and the exponent of the leading
letter expresses the number of terms which follow it. Thus
The law of the exponents of the terms.
12
260 ELEMEKTARY ALGEBRA.
we can easily write the exponents of the letters in any
required term.
II. The numerical coefficients of the first and last terms are
1 ; the coefficient of the second term is 7i, or the number of
combinations of 7i things taken singly ; the coefiicient of the
third term is the number of combinations of 7i things taken
2 and 2, &c., &c.; the coeflS.cient of the (r + 1)*^ term is the
number of combinations of n things taken r together. And
since the coefficient of the term which has r terms before it
is -] — , , and the coeflBcient of the term which has r
vr \n—T
\n
terms after it or n—r terms before it is -i — =^— , it follows
m—r [r
that the numerical coefficients of any tivo terms equidistant
from the heginning and end are the same,
306. From the above it will be seen that to find the coeffi-
cient of any term we may use either of the following rules :
KuLE I. — The coefficient of any term is the exponent of the
hijiomial, taken factorially, divided hy the product of the ex-
ponents of the ttvo letters in that term, taken also factori-
ally, i. e., divided hy the product of the number of terms tvhich
precede it and the number of terms lohich follow it, taken
factorially.
Examining Expansion II., Art. 302, we have for finding
the coefficient of any term from the preceding term,
KuLE II. — Multiply the coefficient of the preceding term hy
the exponent of the leading letter in that term, and divide the
product hy the numher of terms ivhich precede the required
term,
NoTE.-^The first rule is used always when we wish to find any terra
without finding the preceding terms.
Since from Art. 305 all the coefficients after the middle
term, or (first middle term when there are two), repeat
To lind the coeflScient of any term,— Rule I. and Rale 11.
BIK03<IIAL THEOREM. 267
tliemselves, after having found all tlie terms as far as the
middle, we may for the remaining terms simply write down
the coefficients already found, in an inverted order, as in the
following examples.
Ex. 1. {a + xy
=:««+ Jl_ a^a;+_[|_ aV+Ji_ aV + Jl__ aV + «&c.:
1^ m l\i liii
or,
/ , N8 8,8, 8.7 ,,, 8.7.6 . , 8.7.6.5 , . ,
= a' + Sa'x + 28aV + 5 6aV + 70a*x' + 5 6aV
+28a'x' + Sax''+x\
Ex. 2. {a+xy = a''+^ a'x+1^1 aV + ^l^^a*x'+&c.,
=a'-{' la'x+^laV + d6a'x'+36a'x'+21aV + 7ax' + x\
307. If the second term of the binomial is negative, the
second term of the expansion is negative, and every alternate
term also negative, as is evident by the rule of signs of
powers. Thus,
{a-xy = a'-la'x^2la'x^-3^aV-\-3ba'x'-2la^x'-\-lax'-x\
7.6 , 7.6^
= l--'ix-\-Ux'-3^x'-hdbx'-2lx' + W-x\
Ex.4. {3x-\yy
6 .. ..,, ^ . 6.5,. ,_, ,, 6.5.4,
Ex.3. (l-«jy = l-4-:c + f^rc^--|:^a;^ + &c.
=(3^)^- Y (3^)^(1^)+^ (3a:)* {^yy-^^xy a^)'+&c.
= 729a;''-6x243a;^Xi^+15x81:z:*Xiy'-20x27a;'Xi^y'
+ 15x92:^XTV^*-6x3a;X^^^* + A/
= 729a;' - 729a;> 4- ^-^^x'f - ^\^x^f + W^>* " -h^f + A^'-
Ex. 5. Find the 8th term in the expansion (x.-{-ay^
268 ELEMEIS-TARY ALGEBRA.
The exponents of 8tli term give x^a\
|11
Hence the term is -r^ — rr- x'a^ ;
or, cancelling out like factors,
Ex. 6. Find the middle term of {a—by\
The middle term is the 7th. Hence it is
112
|6 |6
Examples — 70.
1. {l + x)\ 2. (a-dxy. 3. {1-xy. 4. (a-xy,
5. (1+^)'^ 6. (1-2^)^^ 7. {a-3xy. 8. (2x+ay.
9. (2«-3^)^ 10. {l-ixy\ 11. {i-ixy\
12. Find the 8th term (independently of the rest) in
(a--xy.
13. Find the 98th term in {a-hy'\
14. Find the 5th term in {a'-by\
15. Find the middle term of {a-{-xy\
16. {x' + xyy. 17. {a'-x'y, is. {a' + ¥y.
308. The Binomial Theorem is true not only for n a pos-
itive integer, bnt for n negative or fractional. But the dis-
cussion in this case is not sufficiently elementary for this
book.
SCALES OF JS"OTATIOK. 269
XLI. Scales of Notation.
309. In the common system of Arithmetic numbers are
expressed by the use of 9 figures called digits, and one ci-
pher. This is effected, we know, by giving to each digit a
local, as well as its intrinsic, value. The local values of the
figures increase in a tenfold proportion in going from right
to left ; in other words, the local values of the digits pro-
ceed according to the poiuers of 10 froin right to left.
Thus, 4296 may be expressed by
4000 + 200 + 90 + 6, or 4XlO^ + 2XlO'^ + 9XlO + 6.
A system of notation is called a scale. In the common
system or scale, the number 10 is called the radix or base
of the scale.
310. It is purely conventional that 10 should be the radix ;
and therefore there may be any number of different scales,
each of which has its own radix. Notation is then the
method of expressing numbers by means of a series of
powers of some one fixed number, which is said to be the
dase of the scale in which the numbers are expressed. (We
use the word number here in the sense of whole number,)
If the digits of a number JSf, of n digits (including
among the digits for convenience), be a^, a^, a^ , . . a^^^,
reckoning from right to left, and r be the radix, N may be
expressed by the formula,
Obs. 1. — It will be noted that since the units figure does not contain
r, the highest power of r will be one less than the number of figures in
the number expressed.
Obs. 2.— In any scale of notation every digit is necessarily less than
r, therefore r-1 is the greatest digit, and r-1 expresses the number
of digits, and the number of figures used in any scale including is
equal to r.
Arithmetical Notation. Scale. Radix, or Base. The general formula for ex-
pressing numbers.
^70 ELEMElsTTARY ALGEBKA.
311. If r=2 the scale is called the Binary;
r=3 Ternary ;
r=4: Quaternary ;
T=6 Quinary ;
r=6 Senary ;
&c., &c.
T= 10 Denary ;
^=11 Undenary ;
r=zl2 Duodenary,
The digits, including the cipher, in the
Binary scale are 1, ;
Ternary ^,2,0;
Quinary 1, 2, 3, 4, 0;
&c., &c.
Nonary 1, 2, 3, 4, 5, 6, 7, 8, 0;
Denary 1, 2, 3, 4, 5, 6, 7, 8, 9, 0;
but in the duodenary scale we must have two additional
characters to express ten and eleven ; we therefore put t for
ten, and e for eleven.
.•. Duodenary digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, t, e,0;
also undenary 1, 2, 3, 4, 5, 6, 7, 8, 9, t, 0.
All numbers are supposed to be expressed in the common,
or denary scale, unless otherwise stated.
312. To express a given number in any proposed scale.
Let N be the number, and r the radix of the proposed
scale.
Then if «„, a^, a^, &c. be the unknown digits,
iV=:an-i^"~' + fl^^2^"~'+ . . ^-ay + ay + a^r+a^.
If now N be divided by r, the remainder is a^.
If the quotient be divided by r, the remainder is a^.
Names of different scales. To express a given number in any given scale.
SCALES OF 2s"0TATI0i^.
271
303...
...2
.•. 1st remainder ^^=2
50....
...3
2d " a=^
8
...2
3d " «,=:2
1
...2
4tli " a, =2
If the second quotient be divided by r, the remainder is a^ ;
and so on, until there is no further quotient.
Hence the repeated divisions of the given number N, by
the radix of the proposed scale, give as remainders the re-
quired digits of the number in the proposed scale.
Ex. 1. Express 1820 of the common or denary scale, in a
scale whose base is 6.
6 1820
6
6
6
6
.-. the number required is 12232.
This is easily verified, for
1X6'+2X6'+2X6' + 3X 6 + 2 = 1820.
This verification gives the method of transforming a
number from any other scale to the denary.
By the method of division given above, a number may be
transformed from miy given scale to any other of which the
radix is given. It is only necessary to bear in mind through-
out the process that the radix of the scale of the given num-
bers is not 10, but some other number. Or the same thing
may be done by first expressing the number (as by verifica-
tion above) in the denary scale, and then proceeding as in
Ex. 1.
Ex. 2. Transform 12232 from the senary scale to the
quaternary.
To transform a number from one scale to another.
4
12232
4
-2035..
..0
4
305..
..3
4
44..
..1
4
11..
..0
1..
..3
272 ELEMENTARY ALGEBRA.
(Observe, that in dividing 13 by 4, 12 does not mean twelm, but
1 x6 + 2=:8 ; so also, 23 is fifteen^ 32 is twenty^ and so on; i. e. we must
convert each partial dividend to the denary scale as we proceed.)
, 1st remainder ^^^^O
2d " a,=:3
3d " a,=l
4tli " ^3=0
5th " a,=0
.'. the number required is 130130.
This number transformed to the denary scale is,
1x4^ + 3x4*4-0x4^1x4^ + 3x4 + 0=1820.
Ex. 3. Transform 3256 from a scale whose radix is 7, to
the duodenary scale.
twelve 3256
twelve 166 4 .*. 1st remainder aj,=: 4;
11 1 .-. 2d " a=l]
.'. the number required is 814.
(Observe in this division that 33 is twenty-three, and the remainder,
eleven, is multiplied by 7 and added to the next figure, 5, giving eighty-
two for the next partial dividend, &c.)
Examples — 71.
t. Express the common number 300 in the scales of 2, 3,
4, 5, 6.
2. Express 10000 in the scales of 7, 8, 9, 11, 12.
3. Express a million in the duodenary scale.
SCALES OF IS'OTATION".
273
4. Transform 27z^ and 7007 from the undenary to the oc-
tenary scale.
5. If the number 803 is expressed by 30203, show that
the new scale is the quaternary.
6. The number 95 is expressed in a different scale by 137 :
find the base of this scale.
313. The common processes of Arithmetic are all carried
on with numbers expressed in any one of these scales as
with ordinary numbers, observing that when we have to find
what numbers we are to carry in Addition, &c., we must
not divide by ten, but by the base of the scale in which the
numbers are expressed.
Ex.1.
Addition,
r^A:
r=zl
32123
65432
21003
54321
33012
43210
22033
1444
31102
65001
332011 326041
Subtraction,
r=3 r=12
7^^348
he^t^
201210
102221
21212
1^864
Ex. 2. Multiply the numbers 1049 and 1^5 together in the
duodenary scale.
1049
lg5
51^9
e443
1049
202329 -duodenary.
.*. the product is
202329=2X12^ + 2X12' + 3x12^+2x12 + 9
= 501585~denary.
12^
274 ELEMEKTAKY ALGEBRA.
Ex. 3. Divide 234431 by 414 (quinary), and extract the
square root of 122112 (senary).
234431
41
414)234431(310
' 2302.
122112(252
4
122112
44
234340
423
414
45)421
401
122024
41
542)2012
1524
•
44
814. To find the greatest and least numders expressed hy a
give7i number of figures in arty proposed scale.
Let r be the base of the scale, and n the number of digits ;
then the number will be greatest when every digit is as great
as it can be, that is, =r— 1. Thus the number will be
(r-l)r"-^+(r-l)r"-2+ .... +(r-l)r' + (r~l)r+r-l;
or, (r-1) (r^Hr"-'+ .... ^r^^r-{-l).
But the quantity in the second parenthesis is the sum of
the terms of a geometrical progression, of which the first
term is r"~^, the ratio r, and the last term 1. This is equal to
7-^1
— - . We have then for our greatest number,
(r—1) -: or, r"— 1.
Again, the number will be least when the digit on the left
IS 1, and all the other figures 0, in which case it will be equal
to r^\
Ex. 1. In the denary scale the greatest number of 3 fig-
ures is 10'- 1:^999; and the least is 10', or 100.
To find the greatest and least numbers expressed by a given number of
figures in any proposed scale.
LOGAKITHMS. 275
Ex. 2. In the senary scale the greatest number of 3 fig-
ures is 555 = 6^ — 1 = 215, denary; and the least number of
3 figures is 100 = 6^=36, denary.
Examples— 72.
1. Extract the square root of 33224 in the scale of six.
2. Show that 144 is a perfect square, in any scale whose
radix is greater than four.
3. Show that 12345654321 is divisible by 12321 in any scale
greater than six.
4. Multiply the common numbers 64 and 33 in the binaiy
and quaternary scales, and transform each result to the
other scale.
5. Divide 51117344 by 675 (octenary), 37542627 by 42?f
(undenary), and 29^96580 by "2U^ (duodenary).
6. Extract the square roots of 25400544 (senary), 47610370
(nonary), and 32^75721 (duodenary).
7. Express in common numbers the greatest and least
that can be formed with four figures in the scales of 6, 7,
and 8.
8. Show that 1331 is a perfect cube in any scale of nota-
tion whose radix is greater than three.
XLII. LOGAEITHMS.
315. A geometrical progression whose first term is 1 and
ratio any number, as a, may be written
a\ a', a", a\ a\ a', a', a\ a^ &c., a^ ;
and the indices form the arithmetical progression
0, 1, 2, 3, 4, 5, 6, 7, 8 n.
27G elemein^tary algebra.
In this A. p. each term measures the order of the ratio of
the corresponding term in the geometrical progression to 1.*
Hence these indices are called the measures of the ratios of
the numbers in G. P. to 1, or the Logarithms of these
numbers.
316. From the rules established in the earlier chapters, we
know that to multiply or divide any two terms in the first
series we have only to add or subtract their indices — i, e,, the
corresponding terms in the second series; also, to raise a
number of the first series to a given power, we multiply its
index or corresponding term in the second series by the in-
dex of the power ; also, to extract a given root of any number
in the G. p., we divide its index or corresponding term in the
A. p. by the index of the root.
317. It is evident from the above that if a geometrical
progression can be formed which shall represent with a suffi-
ciently close approximation all numbers from 1 to 10,000.. and
the terms of the arithmetical progression corresponding to
this G. p., in the same manner as in Art. 315, be calculated,
and both series be recorded in a table, that much trouble
may be saved in arithmetical computation by operating
solely on the terms of the A. p., and finding from the table
the numbers of the g. p. corresponding to the results.
318. Such tables have been calculated, and are called
TaUes of Logarithms, To see how this may be efiected,
let a=10 m. the system (Art. 315) ; we have then
lo^ 10^ lo^ lo^ lo^ lo^ lo^ lo^ lo^ lo^ &c., (i)
and 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, (2)
are the logarithms of the corresponding terms of the first
series ; that is, in a system of logarithms whose base is 10,
* The ratio of a? -.1 is the duplicate ratio of a: 1.
The ratio of a^ : 1 is the triplicate ratio oi a\l.
And the ratio of a" : 1 is called n times the ratio of a ; 1.
Thus the indices Tneasure tJie ratios.
LOGAEITHMS. 277
O=log. 10" or log. 1;
l=log. 10^ or log. 10;
2=log. 10' orlog. 100;
3= log. 10' or log. 1000;
4=log. 10* or log. 10,000 ;
5=log. 10' or log. 100,000;
&c., &c., &c.
It is manifest now that the arithmetical mean between
any two terms of the series (2) will be the logarithm of the
geometrical mean between the two corresponding terms of
the series (1).
The arithmetical mean of and 1 is —^r— = .5.
The geometrical mean between 1 and 10 is
^/fxlO ==3.16227 + ;
and therefore .5 = the logarithm of 3.16227.
The arithmetical mean of .5 and 1 is .75.
The geometrical mean of 10 and 3.16227 is
Vl0x3.162"27= 5.62341 + ;
whence .75 = the logarithm of 5.62341.
The arithmetical mean of 1 and 2 is 1.5.
The geometrical mean of 10 and 100 is 31.62277+;
whence 1.5 = the logarithm of 31.62277.
And by repeating this process, with immense labor, the
inventor of logarithms, N"apier (a. d. 1618), and his suc-
cessor in these calculations, Briggs (a. d. 1624), calculated
tables of logarithms of natural numbers from 1 to 100,000,
But (he labor of calculating logarithms is much diminished
278 ELEMEl^TARY ALGEBEA.
by the use of series which cannot find place in an ele-
mentary work like the present ; besides, as will be seen, the
chief labor is with the prime numbers.
319. These logarithms in the common table of which 10 is
the base, are the indices (entire or fractional) of the powers
to which 10 is to be raised to obtain all natural numbers ap-
proximately. Thus, .30103 is the logarithm of 2, means that
2Q.3oio3__2. And this may be yerified by developing 10-^*^^°^
.30103
~ (1 + 9)ioooob by the binomial formula.
820. 10 is the most convenient base, but any positive
number except 1 may be taken as the base. E"apier took
2.71828 as his base. In general, by taking any positive
number (except unity) for a base, we may express any posi-
tive number as some power of it. And thus logarithms may
be defined to be the indices of the powers {entire or frac-
tional) to which ice raise a fixed number, called the hase, to
oUain the series of natural numhers. Each logarithm is the
representative of its corresponding natural number.
321. In the common system, of which 10 is the base, it is
clear that the logarithm of every number between 1 and 10
is a decimal fraction ; that of everv number between 10 and
100 is 1 with a decimal fraction annexed; that of any
number between 100 and 1000 will be 2 with a decimal frac-
tion annexed, &c. The integral part of a logarithm is
called the characteristic of the logarithm ; and the decimal
part is called the Mantissa, or " handful.'^ Thus is the
characteristic of the logarithms of numbers between 1 and
10 ; 1 is the characteristic of the logarithms of all numbers
between 10 and 100 ; 2 that of the logarithms of all numbers
between 100 and 1000, &c. And in general, the character-
istic of the logarithm of any number is alioays less ly unity
than the number of figures in the given number.
322. Tables of logarithms, arranged in convenient form,
are usually given in books on Trigonometry, and with them
LOGARITHMS.
279
explanations of the mode of finding in the table the loga-
rithms corresponding to a given number, or the number
corresponding to a given logarithm. The table below is a
portion of such a table of logarithms.
Logaritlims^ to hose 10, of all Prime Numbers f
yo7n
1 to 100.
No.
Logarithms.
No.
Logarithms.
No.
Logarithms.
No.
Logarithms.
2
0.3010300
19
1.2787536
43
1.6334685
71
1.8512583
3
0.4771213
23
1.3617278
47
1.6720979
73
1.8633229
7
0.8450980
29
1.4623980
53
1.7242759
79
1.8976271
11
1.0413927
31
1.4913617
59
1.7708520
83
1.9190781
13
1.1139434
37
1.5682017
61
1.7853298
89
1.9493900
17
1.2304489
41
1.6127839
67
1.8260748
97
1.9867717
323. We will now show more fully the properties of loga-
rithms, which render them so useful in diminishing the
labor of arithmetical calculations.
324. In the same system, the sum of the logarithms of two
numbers is the logarithm of their product ; and the difference
of the logarithms of two numlers is the logarithm of their
quotient.
Let m and n be the two numbers ; lei x = log. m, and
y = log. n; let a be the base of the system; then a''=7n,
on
and a^—n\ hence a'^'^—mn, and a'^^——\ or x-^y is the
m
log. mn, and x—y is log. — ; that is, log. m + log. ^ =:log.
n
7)1/
mn ; and log. m — log. n — log — .
n
Ex. 1. Log. 6=log. 2+log. 3 = .3010300 + 4771213
=.7781513.
Ex. 2. Log. mnp=log. mn + log. p=log. m + log. n-\-log,p.
Ex. 3. Log. 5=log. 10-log. 2=:l-log. 2=.6989700.
Ex. 4. Log. |=:log. 7-log. 5=.1461280.
280 ELEMENTARY ALGEBRA.
Ex. 5. Log. .07=log. T^=log. 7-log. 100:=:. 8450980 -2,
which is written thus, 2.8450980; it being understood that
in this position of the negative sign it belongs only to the
characteristic 2, and not to the mantissa, which is still
positive.
Ex. 6. Log. ^\=.4771213-1.9867717 = -1.5096504,
which maybe written thus: -2 + (1 -.5096504) =2.4903496.
S25. If the logarithn of a number be multiplied by m, the
product is the logarithm of that number raised to the m.th
fower.
Let iV^be the number whose logarithm is x\ then a^=N\
therefore a'^'^=N'^\ that is, mx is the log. of N''^\ or log.
N'^—mx^m log. N,
Ex. 1. Log. (13)^=5xlog. 13 = 5X1.1139434=5.5697170.
Ex. 2. Log. b^—%j log. b.
Ex. 3. Log. 4= log. 2=^ = 2 log. 2 = .6020600.
Ex. 4. Log. (a''-xy=^2 log. (a + a;)+2 log. {^a-x),
Ex. 5. Log. (a"'Z>"c^. . .)=m log. a^-n log. b-Vp log. c-f-. . .
326. If the logarithm of a number be divided by m, the
quotient is the logarithm of the mth root of that numMr
Let a;=log. iV, or a'^—N^
X log. N
then arfi — N^^ or log. iV^»n=
m m
^ ^ r K^ %-^ .6989700 ,^,^,^^
Ex.1. Log. 5*=—^= J- — =.1747425.
Ex. 2. Log. y -|-=— log. a-~ log. b,
Ex. 3. Log. Va^—x^=i log. {a + x)-{-i log. (a—x).
LOGAEITHMS 281
Ex. 4. Given log. 128=2.1072100 to extract the 7th root
of 128.
Log. VlM=\ log. 128=1 (2.1072100) =.3010300=log. 2.
.•.V'l28=2.
Ex. 5. Log. -VTt^^ log. 1-1 log. 71= -I log. 71
=4-X ~L8512583=| (-2+(l-.8512583))=4- (2.1487417).
]^ow in order to diyide this log. by 7, we place it under
the form 7 + 5.1487147, so that the negative characteristic
may become a multiple of 7 ; then
4- ( 7 + 5.1487147) = f.7355306.
327. We can now see how the work of computing a table
of logarithms is facilitated by the application of the above
properties of logarithms. For the logarithms of the com-
posite numbers are all found by adding together the loga-
rithms of their prime factors.
328. While we are at liberty to take any number except 1
as the base of a system of logarithms, we can now under-
stand the great advantages of the system which has 10 as a
base ; that is, the advantages of having the base of the scale
of notation the same as the base of the system of logarithms.
For, 1st, the characteristic of the logarithm of any whole
number is always one less than the number of figures in the
given number. Hence when the number of figures is given
we know the characteristic ; and when the characteristic of
the logarithm is given, we know the number of figures in the
required number.
2d. By every multiplication or division of a number by
10, the characteristic of its logarithm is increased or dimin-
ished by unity.
For log. 1280= log. (128X10)
=log. 128 + log. 10=log. 128 + 1,
282 ELEMENTARY ALGEBRA.
log. 128u0=log. 128 + log. 100=:log. 128+2,
log. 128000= log. 128+log. 1000=log. 128 + 3,
log. 12.8=.log. (128^10)=log. 128-log. 10=log. 128-1,
log. 1.28=log. (128 -f- 100)= log. 128-log. 100=:log. 128-2,
log. .128=log. 128-log. 1000=log. 128-3, &c.
From the tables of logarithms, log. 128=2.1072100.
Therefore, log. 1280 =3.1072100,
log. 12800 =4.1072100,
log. 128000 =5.1072100,
log. 1280000=6.1072100,
log. 12.8 =1.1072100,
log. 1.28 =0.1072100,
log. .128 =1.1072100,
log. .0128 =2.1072100.
We* observe that the logarithms of all numbers which
contain the same significant figures, arranged in the same
manner, have the same mantissa, or decimal parts ; that is,
the mantissa of the logarithm remains the same however we
may change the corresponding number, by annexing ciphers
or by inserting a decimal point in it, changing the position
of its decimal point to the right or left.
Moreover, the logarithm of a decimal fraction has a nega-
tive characteristic greater by unity than the number of O's
between the decimal point and the first significant figure of
the number.
329. To find a fourth proportional to three given numbers,
using logarithms.
Let the numbers be a, b, and c; let x = required fourth
be
proportional. Then a : b=c:x; .*. x=—.
LOGARITHMS. 283
Therefore log. x=log, J + log. c— log. a. Hence tlie rule :
From the sum of the logarithms of the second and third
terms subtract the logarithm of the first term : the remainder
will be the logarithm of the fourth proportional The fourth
proportional may then be found from the tables.
Note. — In the following examples use the table of logarithms given
in Art. 322.
Examples — 73.
1. Find the logarithms of 8, 9, 12, 20, 25, 60.
2. Find the logarithms of |, i, f, .03, ■^\, .0033.
3. Eequired the logarithms of 168, 1.04, and 3690.
4. Given the logarithms of 3 and 7 : find the logarithm of
14700.
5. Find the logarithm of 83349, from the logarithms of
3 and .21.
6. Determine the logarithms of V^-^% and \/l.625, by
means of those of 2, 3, 5, and 13.
7. Find a fourth proportional to the quantities 1.3, .0104,
and 2.375, by logarithms.
8. Find by means of logarithms the number of figures in
the results of the involutions of 2^° and 3'^
9. Find the logarithm of V^J" X V^f" X Vf.
ANSWERS TO EXAMPLES.
Examples— 1.
I. 22, 2. 26. 3. 89. 4. 564. 5. 274. 6. 10.
7. 6. 8. 6. 9. 34. 10. 39. 11. 6. 12. 5. 13. 9.
14. 5. 16. x + a + b, 17. x-}-x-^a + x + a + b. 18. x—a.
19. aS'-^. 20. x~a + b, 21. 2: + a:+2+2:-f2+3.
22. 4xlO + 5+yV 23. 100a;+10^ + ^, 100;^ + 10y + .'?;.
24. X7i + bn. 25. lla;+5. 26. ~. 27. — . "^
^* 'a; a;4-3'
iV— r
28. — y^.
Examples — 2.
1. 55.
2. 81.
3. 94. 4. 8.
5. 27.
6. 81
7. 13.
8. 11.
9. 31. 10. 15.
•11. 10.
12. 3
3. 3.
U. 127.
answers to examples. 285
Examples — 4.
1. 5. 2. 16. 3. 9. 4. 224. 5. 459. 6. 7.
7. 74. 8. 12. 9. 8. 10. 238. 11. 420. 12. 144.
13. 43. 14. 15.
Examples — 5.
1. 2a+2b. 2. 2a, 3. 2a-2b. 4. 2a, 5. 2a+2^.
6. 2+m + n. 7. 7m— 1. 8. 4:xy-h4:X. 9, p—q-^-S,
10. eah-bc, 11. 15a-9^. 12. dx'-df,
13. 9« + 95+9c. 14. 4:z;+2y + 4^. 15. a—b.
16. 3a;-3a-2Z>. 17. 2a+2b. 18. a + ^ + c.
19. --2« + 2J + 2^. 20. 2a;' - 2x' - 8x + 10.
21. 5^* + 4a;' + 3^' + 2^- 9. 22. 4:a' + 2a'h-4.a¥+b'-n\
23. a'x + da\ 24. 6aJ-9a'a; + 7aa;' + ^x^ 25. 5a;^
26. 10a;^ + 83/'4-12a;+12.
Examples — 6.
1. 3^ + 45. 2. 4:a+2c, 3. a+5b + 4:C + d.
4. 2.T^-2a;--4. 5. 3a;* -a;' -14a; +18. 6. x'-ax + 2a\
7. -5a;y-5a;^4-2i/' + 7/^. 8. 3a;' + 13a:y-16a;2;-y'-13y;2.
9. 2a'-6a'b + 6ab'-2b\ 10. 3a;'' 4- 4a; + 16, a;H8a;'.
286 eleme^'tary algebra.
Examples— 7.
1. 4.a-4:X. 2. 4:a'-4.a'c. 3. x'-3y'-3z\
4. 2ax'' + 2by'+2cz\ 5. a'-dd'+3c\ 6. 2ab-i-U\
7. 0. 8. -3x-y-h4.z. 9. 8a;-8. 10. -4:C + 4:d.
Examples — 8.
1. {a-h+c)x'-{I)-c-{-d)x^-{c + d+e)x, 2. 2{ax-dy).
3. {a-\-d)x^—{a—5b)xy-}-{a—c)y\
4. 2{ax-\-cy), 2b{x+y).
5. — («— 5J):r+(2(^+3Z> + c)y, (aj— 45 — c)^+(«^— 35— 2^)?/,
(5— c):c+(3a— c)y.
6. {6a—b)x—{2a-db'-6c)y, —{a-^-c)x+{a—b+2c)y,
(4.a-b-c)x-{a-2b-7c)y.
Examples — 9.
1. abx^y% —mnx^, 2a'cx'^y, ab'^c^, d^bc^^ -~^y«
2. x^—x'y^-xy'', —a^x-\-d^x^—aQ^^ —abx^-\-(^bx^—ab'^x^
x'y-3x^y^-\'3xY-xy\
3. 20" ■\-1ab-\-W,2ac-bc-Ud+3bd.
4. ^x' + lZxy + Qy^ Mb'-ab'-Ub*.
5. x'+Qx'-rlx-^e, x'-ex'-^Ux-G.
6. a*+a'-2a'+3a-l, a*-a'-8a'4-a+l.
^ VNIYER"
AKS^RS TO ;gXAMPLES. 287
SAUfQ :
7. Six' -if. 8. a''\-S2b\ 9. x'-ia'x+da'.
10. 27a' + ¥ + S-lSab, 11. x'-y'+z' + Sxyz
12. a'-l. 13. a'-8Z>'-27c'-18a^c. 14. a'-h2aW-{-b\
15. a;'— (a+c)cc'' + («^c + ^)^ — be;
x*—(a''—b+c)x' + a{b + c)x—bc
16. 1 — (a— l)a;— (a— d + l)^' + (fl^ + Z>-c)^' — (^>+c)2;* + c:?;'.
Examples— 10.
1. a'-2ax+x\ l-^ix^+4:x\ 4a* + 12a' + 9>
9a;''-24a;y^-lG?/^
2. 9 + 12:^: + 4a;^ 4:i;^-12:ry + 9/, a*-6a'c?;+9aV,
b'x*'-2bcx'y + c'xy.
3. 4^^^-!, 9aV-Z>^ a;*-l.
4. x'+4.x + d, x'+3x'-4:, a'b'-qb-Q, 4.aV-Sabx+db\
5. x'-5aV + 4:a\ 6. mV-13mV2;y + 36^y.
7. 4^1 8. x'+4.y\ 4.a*-6a'b'-}-b\
9. a' + 2r/^»+^>^-c^ «'^^>''+2^c + c^ a'-b'-2bc—c\
10. «'-2aJ + ^>''-c^ -a^ + 2«5-^>•'+c^ -^'+Z>'-2Z^c + c'.
11. 4ta'-b' + 6bc-9c', -4:a' + 12ac-hb'-9c\
12. 4a^-Z>''-6^^c-9c^ -4a'-{-4.ab-b' + dc\
13. a=+2ac+c'-J'-2^>^-^^ a'' + 2ac?+6Z'-^'-2^^c-c',
288 ELEMENTARY ALGEBRA.
+ ^ab - ib% a' + 6ac + 9c'- W + Ud- d\ .
Examples — 11.
I. 5a;'. 2. -3a^ 3. ^xy. 4. -MhW
5. ^a'Wy\ 6. x^-'^x^L 7. -aM-4a-5.
8. x^-Zxy^A.f, 9. 5«^P+a^-4.
10. 15a'^'-12«JH9aJc'-5c\ 11. x-L 12. a;- 8.
13. a;^ + a; + 3. 14. 3ii;'^-2a; + 4. 15. 3:z;'' + 2:z; + l.
16. x^-Zx^l. 17. 2;^ + a;* + 2;' + ^'+i?;+l.
18. a^-\-ab-l\ 19. a;^ + 3^>+9^^''+272/'.
20. x^-x'y + xy\ 21. i?;*+ii^> + i^y + :2^^'+y'.
22. a*-2a^Z> + 46^^Z>^-8t^Z>' + 16^\
23. 2a'-Mb + l^aV-mh\ 24. a;''+^y+^'.
25. a;' + 2:r2/ + 3/. 26. x^^^x-{-2. 27. a;'-3:c-l.
28. ic'-5a;+6. 29. :z:'-4a; + 8. 30. a;'' + 5:i;+6.
31. x-c, 32. a;'-j9a; + 5'.
33. y'-{m-l)y^-{m-n-l)y''-{m-l)y + l.
34. a' + Z>' + c' + fl^&-flJc + Z>c, a' + ^'+c' + a^ + «c-2><:.
35. a—ax-{-ax^ — ax^-{- -^ — , l + 5a:+15a:' + 45a;' + - — ^
l-{-x l—6x
36. l + 2a: + 3a;^ + 4a;H , ^ , .
1— 2a;4-ar
AKSWlrKS TO EXAMPLES. 289
Examples — 12.
1. v.—x,a^-\-a^x^ a^x^ + ax^ + a?^ w'-a^x + a^x^-d^x^ + ax*' — x^,
2. 3rc + l, 5:c-l, 2ic-3. 3. Zmn-h, hn^-n^,
4. l-2z + 4^', 92:' + 3^ + l, l--2a;+4^'-8a;'.
5. ^' + 3a;>+9:?;2/^ + 27^', a*-2^^Z> + 4a'^Z>'-8a^^^ + 16^>*,
x''-x'Y + xhf-xY + x'y^-y'\
6. ia'-iaZ^^-^^ x\f-xYz-\-xyz'-z\
7. «+Z> + c, a+^— ^.
8. (o^ + y)''— (a; + ^)2; + 2;^=a;'' + 2a:?/+^''— ir^;— y^ + 2;^
Examples— 13.
1. (l-2:r) (l+2.r), (a-32:) (<^+3a;), (3m-27^) (37?i+2^),
a;X5a-2) (5a + 2), xy{4:X-6ij) (4a; + 5?/).
2. (^+^) (a;^_:r^ + 2/^), (^-7/) (x'+xtj + f), (l + xy)
(l-xy+xY), (^-1) (^+1) (^' + 1). ^y((^y-^") {cty^-x"),
2«Z''c(<^-2c) («+2c).
3. x\bx-a) {bx-\-a), a\a-3I)') {a + SI?'), (2x-S)
(4^^ + 6.'?;+ 9), (a-2h) (a' + 2ad + 4.b'), x'y{a + dy)
{a''-3ay + 9y"),
4. {x+2){x'-2x'-\-4.af-Sx+16), x'{a + dx) {a' -3ax+9x'),
{2x'+f) {4.x'-2xY+/), {ab'+c') {ab'-c") {a'h'^-c^
abc(a-\-cy.
290 ELEMEl!fTARY ALGEBKA.
5. {3x-l) (3a; + l) (dx' + l), {x^2) (x+2) {x' + 2x + 4:)
{x'-2x + 4.), x\x-'b)\ x\x-ay (x+a)\
6. (4a;— 5) (2a; + 1), {a+^h) {a-b), 7{x-y){x+v).
7. {x-yy{x+7/)% {c + a-I)){c-a + I?), Sal?,
8. (x+yY, mn{m--7i), 61){a—b).
9. 2{x+y){4.x-y), 2{x-y){4.y-x), ^y(x-\-y),
10. {a-V'b){a' + ah^l)''), (a-by, 0.
Examples— 14.
1. (x + l) {x + 6), {x + 4:) {x + 5), {x-2) {x-d), {x-3)
(x-5), (a;+l)(a;+7), {x^l){x-9),
2. {x + 3) {x-2), {x-3) {x + 2), {x-d) {x+1), {x+5)
{x-3), {x+S) {x-1), {x-9) {x+1).
3. {2x+3){2x + l), (4a; + l)(a; + 3), (4a;-l)(a;+3), (2a;-3)
(2a;+l), (3a;-2) (a;+2), (3a; + 4) (2a;-l).
4. (4a; + l)(3a;-2), 2(6a;~l)(a;-l), (4a; + l)(3a;-l), (a; + 4)
{x-3), (3a;-5)(a;+l).
5. a\x—a) {x—2a), a{a—Sx) (a + 2a;), db{Za—2li) (a+5),
(2«+a;)(2a-a;)(3a'+a;').
6. xy{2x'\-y) {x-\-2y), 3y\3x-+2y) {x-y), a\3ax-l)
{2ax-\-l), x\2b-dx){db+x).
ANSWERS TO EXAMPLES. 291
Examples — 15.
1. 2x'{a + xy. 2. x\a+x)\ . 3. ab{a-l))\
I. 2(^-1). 5. x\x^l), 6. 2(a;+a),
7. a\x-\-l), 8. 3(6?:?; + 2). 9. x-l, 10. iz; + 5.
11. a;-10. 12. x'-x+l, 13. a; + 3y.
Examples— 16.
1. 3x-2. 2. 2a; + 3. 3. dx+6, 4. 8a;'' + 14a;- 15.
5. 4a;-5. 6. x' + 2x-d.
Examples — 17.
1. 3:r~2. 2. dx-2, 3. 2(:rH2:r+l). 4. y-2.
5. :^;-2«. 6. x+d. 7. 3{:?;-|-3). 8 x' i-y'.
9.'a{a-{-h). 10. a{a'-V). 11. a;^~2:?;^-i'y'.
12. a;'+4a; + 4
Examples — 18.
1. 12aWc, S6x^y% ax^y—axy\ ab^—ad^.
2. 120a*b% 10a'b\ ISOOaV.
?92 ELEMENTARY ALGEBRA.
3. 6{a'-b'), 12a{a'-l), 120xy{x'-y'),
4. 24.a'b\a'-I)'), d6xf{x'-y').
5. (^ + l)(^H-3)(2;-4). 6. {x + 2){x + 4:){x' + 3x + l),
7. x{2x + l){dx-l){4.x + 3),
8. {x'-5x+6){x-l){x-4:).
9. (^' + 3a; + 2)(:?:-3)(:r+5).
10. {x'+x + l){x'+l){x + l){x-l),
11. 36a^Z^V. 12. 120{a + by{a-hy,
13. 24(a-^)(a'+Z>^). ^ 14. 106ab'{a+I)){a-'b),
15. ^'-1. 16. 0:^-1. 17. x''-l.
18. (.T + 1) (:?:+2)(^ + 3).
Examples — 19.
1. 3:;^f -^. 2. 4ac+7r. 3. 2aH- — . 4. 2x--f-
3. ^4. _A^. 6. 2a; ^. 7. a;^ + 3aa; + 3a^+- ^^^'
a; + 3* * x—3' * ' x—2a
8. ^-1 — , \ . 9. x'+x'i-x-hl-^-^.
x—x + l x—\
ANSWEES TO EXAMPLES. 293
a-^b ' '• 3{a + by x' + l
Examples — 20.
^ 2a^x ^^ a-i-b „ a+b . 2ax
"%"• ~2b~' ~^:-b' H^-^'y""
h[a—b) ' a—b ' ' x + 6'
c cc + T rt CC4-3 ^^ i?;+5 ^^ x—b
CK— 5 ic— 7 x-hc x + G
iA 3:r— 4 -„ x-\-a—b—c ^. x + 3
IZ. —. ^. Id. — — 1 • 14.
4^;— 3' ' x-\-b—a—c ' x^ — 2x-\-b
X~'~0 ^ g^ X -IT u ^ jy X
•O- '_ m-Ul-
.-r:--! x^ + a' a'+g'y + y a:'- 5a: a-5
"a "' z' ' a^ + J' ' x + b ' «+&•
"■ 12a;" 12a;" 12a;"
4(a;-l) 3(a;- l) ix
^'' 40«"-l)' 4(a;^-l)' 4(a;''-l)*
a(a; + «) — a;(a;+«) a;' — aa;
.294 ELEMENTARY ALGEBRA.
fa(a+d) l{a-h) ab V \ _a{^h)(^y^
^^' {x-iy{x+if {x-^iy{x+iy' {x-iy{x+iy'
4:{x-iy 6{x-l){x+l)
{x-iy{x-{-iy' {x-iy{x+iy'
^^ a{x^ + ax+a^) a^—x^ ax
an x^+ax-{-a^ x^—ax+a^ a^
1
x' + a'x' + a'' x' + a'x' + a'' x' + a'x' + a*'
Examples — 21.
a'4-2>' da''—ab+2¥ 26a -20b
2
• 2{a-{-b)b' 6{a-b)b ' 12
ab a'+b' a^±b^ g'-gh + b^
g-y g-^r a'-b'' a'-b' •
ANSWERS TO EXAMPLES. 295
4. -^, 0. ' ^
6 "
"• 4a'-
■r
2a;
a-hhx ^ l + x^ + x""
b + ax ' x^{x'^ + iy' ' x+y
x^ — y^ ' * a'{a + x)'
X —y a\x—a) x-\-y
x-^x^-\-Zx^ l-\-2x-\'':^x^
Examples — 22.
1 '^
5^ x{a-{-b) — ab
{x—a){x—by
(«— a)(«— 5)*
3 ^
4 ^-^-^
d* v/'*
6 ^
7. 1.
c(c—a) {c—b)
Examples — 23.
1. 1^. 2. 1.
4.
1
296 ELEMENTARY ALGEBRA.
5.
x—a.
®- ab •
8.
ax
9 i^+yy
x^^-f
a'-xr
11.
X
abc
x-y
14.
x" a"
y'
A. 15. 1.
y
7.
x^c
X-\-l)
^« x'' — ((jr''^oJ'x—a^
Id. ~"T"3~^
1.
4.
d{a-iy
b{a+b)
7.
a + x
cc+y
10.
1
13.
5.^-1
ifi
aj'-Ga"
2.
Examples — 24.
9cV ^ 1
16a V x+y
^ x{a-^2x) 2x
a x—y
o. • "•
a;— a' * c^-a—V
^-1\' 19 /-^'
:ca
2/
6^^+a^ + l .. fa^ + a^)(^Ha ')
17« . lo. •
Examples— 25.
1. -. 2. 1. 3. -^. 4. x^l.
X x^\
ANSWERS TO EXAMPLES. 297
5. 1.
6. ^-f.
x—5
7. A.
8. 0.
9. i.
10. 2f.
11. 0.
12. 0.
13. a.
Examples — 26.
1. w,-27«w^||^, -^f.
2. a;' + 6a;' + 12^ + 8. 3. :?;'-8^' + 24:c'*-32a' + 16.
4. iz;' + lore' + 90:?;' + 270:?;' + 405a; + 243.
5. l + 10a; + 40a;' + 80a;' + 80a;' + 32a;'.
6. Sm'-12m' + 6m-l.
7. 81a;* + 108a;' + 54a;' + 1 2a; +1.
8. 16a;*-32^a;' + 24aV-8a'a;+a\
9. 243a;' + 810^a;* + 1080a'a;' + 720a'a;' + 24.0a'x + 32a\
10. 64^' -1446^'^* + 108^5' ~27Z>'.
n. aV-3a'xy + daxtf-y\
12. aV + 4f^V + 6aV + 4aa;'+a;^
13. 32a'm'-80a'm'+80a'm'-40a'm'+10am'-m'^
U a'-da'b+3a'c-h3ad'-6a'bc + dac'-b'+db'C'-ddc'+c\
15. l-3a; + 6a;'-7.^'' + 6a;*-3a;'^ + a;^
298 elementabt algebea.
Examples — 27
1. 4:{aO + ad-\-U-^cd). 2. 2(a' + 2«c + c'+Z>' + 2M+6Z').
3. l+2ir+3^'+2^' + a;\ 4. l-2aj + 3a;'-2a;N-a;*.
5. l + 'Zx-x'-^x' + x'. 6. l + 6^+13.T' + 12r^;' + 4:?;*.
7. l-6a; + 152;'-18^' + 9a:\ 8. 2(4+25^^+16^*).
9. l-^x + Zx^-x' + ^x^-^-x'.
10. l-\-^-{-10x'-\-2Qx'-\-^bx'-^'il4.x' + l%x\
Examples — 28.
1. ±:2a¥c\ dtiWy\ ±10a*Z>V.
.3^^ ^-j^ +^^y
^- 5^ ^ 8a ^ 4a&^'
aV^ _^' 4^' Ga^g*^
"^^ 2 ^ 3a;^^ 5a* ^ 7 *
Examples — 29.
1. x'+x+l. 2. l-a; + 2a;^ 3. a;'+3.T + 8.
4. a;'*-2a;-2. 5. l-2a;+3aJ^ 6. 2a;*-i?;'-2.
7. x'-ax + 2a\ 8. ic''-a2;+S^ 9. a;' - 6a;' + 12a;- 8.
13
ANSWEKS TO EXAMPLES. 299
10. x'-\'2ax^-Wx-a\ 11. l-x + x^-x^-^-x'.
12 |5_^^|^. 13. 1-x, a-2. 14. 2a-35.
15. x''—xi/ + y\
Examples — 30.
1. 421, 347, 69.4, 737, 1046, 4321.
2. 2082, 20.92, 1011, 20.22, 129.63.
3. 1.5811, 44.721, .54772, .17320, 10.535, .03331, .06324,
.07071.
Examples — 31.
1. x + 2y. 2. a-S, 3. a; + 4. 4. 2^-35.
5. a+Sb, 6. 2x—7y. 7. m—4:nx, 8. ao?— 5Z'2;,
9. a' + 2a + l, 10. a;'-4:z; + 2. 11. a'-ab + b'.
12. a-5+c;.
Examples — 32.
1. 21, 23, 25, 32, 4.7, 48, 64, 9.6.
2. 114, 11.7, 125, 108, 1.41, 192.
8. 1.357, .5848, .2154, L587.
300 ELilMENTABY ALGEBRA.
Examples—
33.
1. 5.
2. 2.
3. 3.
4. 4.
5.
-h
.. d—a
to. .
7. 3.
8. 1.
9. 4.
10.
—\a.
1. -4.
12. |.
13. -|.
14. ^'.
15.
x=5.
Examples — 34
1. 42. 2. 12. 3. 12. 4. 5. 5. 7. 6. 4.
7. 5. 8. f. 9. 7. 10. ^(25a-18^).
11. 7. 12. If 13. 11. 14. 5. 15. 2^. 16. 3.
17. 2. 18. 4. 19. 2.
Examples — 35.
1. 10. 2. 8. 3. 12. 4. 6. 5. -7.
6. 16. 7. 5. 8. 31 9. -6. 10. 5.
11. 8. 12. J. 13. 3. 14. 2. 15. 7.
16. If 17. i. 18. 1. 19. 17. 20. 2.
21. 4. 22. 2. 23. 18. 24. 8. 25. x=2.
26. a;=-||. 27. a;:^-7. 28. :r=4. 29. a:=-L
30. 20. 31. 3. 32. 5. 33. a- 5.
ANSWERS TO EXAMPLES. 301
34.
b—a, 35.
a-\-b
37.
ah
38. ^«*,.
a + b
a+h—c
40.
a^l-Vc^d
41. c.
43.
i{a + b + 3).
Examples — 36.
39.
42.
a-hb '
a-hb
b—a
1. 12. 2. 9. 3. 120. 4. $1.75.
5. 35, 13. 6. 513, 466. 7. 15. 8. 31, 18.
9. 15. 10. 90, 60. 11. IsToYember 20tli.
12. 16. 13. 37, 30, 20. 14. 20. 15. 41.
16. 88. 17. $36, $12, $16. 18. 5.
19. £45, £57, £63, £65. 20. 15, 5.
21. 98f miles from B ; lOf hours.
22, 10, 14, 18, 22, 26, 30. 23. 28, 14. 24. 88, 44.
25. 5, 6. 26. 22, 7, 12 gallons. 27. 3000.
28. 18, 3, 3. 29. 24000. 30. £140.
Examples — 37.
1. 45 gallons. 2. 2450, 196, 98. 3. 84.
4. 15 feet by 11 feet. 5. 20 lbs., 15 lbs., 15 lbs.
302 ELEMENTAKY ALGEBEA.
6. $240. 7. 3i days. 8. 75. 9. 1504.
10. 1540, 880, 616. 11. 10 lbs.
12. 18, lOf, 6i days. 13. $1.05, $1.17. 14. 6f oz.
15. 654. 16. 76,30. 17. 21-3^ hrs., lOJ^lirs. 18. 12,16.
19. 10, 15, 3, 60. 20. 240, 180, 144 days. 21. 12.
22. 20, 80. 23. 5^^. 24. 240. 25. 24. 26. 60.
27. 25. 28. 7 hours, 5-^', 6 hours, 16^'.
29. 40 minutes past eleven. 30. $100000000,
31. 7, 15, 48. 32. 189.
Examples — 38.
^ mna « m{nb—a) n{a^ml)
m+n * n—m ^ n—m
ma na
4.
m-\-n m + n
mpa npa nqa
mp+np + nq^ mp-i-np + nq^ mp-hn^J-^nq
^ ml—na ^ dbc „ d
5. . 6. -1 TT-. 7.
n—m ' ab + ac + bc ' b + c
be
b + c'
ANSWEES TO EXAMPLES. 303
EXAMPLES-~39.
1. 10; 7. 2. 17; 19. 3. 2; 13. 4. 4; 1.
5. 5; 5. 6. 21; 12. 7. 19; 2. 8. 38i; 70.
9. 6; 12. 10. fif; IM- "• 5; 7. 12. 2^; 1.
13. x=l, y=ri. II. ic=10, y=24
15. a;=144, 2/=:216. 16. .2; .2. 17. 10; 8.
18. 12; 3. 19. 3; 2. 20. a; J. 21. a\ I.
22. -^; -^,. 23. ^; .. 24. ^gr; -gj.
25. -^; -^. 26. -^; 0. 27. «; I,
a + b^ a + b a+b
Examples — 40.
1. x=l, y=2, z=d. 2. x=7, y=10, z=9.
3. x=6, y=6) z=:7, 4. x=4:, y=z—6, z=6.
5. a;=3-5, y=6, z=-2, 6. a;=l|, i/=2f, ^=:-12.
7. x=:2, y=-dy z=:L 8. a:=12, y=12, z=l2.
9.x=6,y=:7,z=-3. 10. |; f; f.
11. a;=i(J + c--a), &c. 12. x=%{a + b + c)-'a, &c.
13. a:=K^ + c),&c. 14. x=y=z=:- ^^^
^ab + bc+ca
304 ELEMENTARY ALGEBRA.
Examples — 41.
1. ^5g. 2. 48. 3. 108 sq. ft. 4. 4 liours, 6 hours.
5. 20, 30, 60. 6. 24, 72. 7. 49; 21.
8. 45; 63. 9. |. 10. (24-1)20.
11. 1 ; 2. 12. 50 yards; rates 4 and 5 yards per minute.
13. 11, and 5, gallons. 14. A. D. 1752. 15. 50; 75.
16. 90; 72; 60. 17. 4; 2. 18. 8; 5.
19. 4 miles walking, 3 miles rowing, at first.
20. 30 ; 50 miles per hour.
21. 60 miles; passenger train 30 miles per hour.
22. 150; 120; 90. 23. rr=40, ^=160, ^^=480.
a-\-b a—b
24.
25.
771C — all + am . {n—h) ^ i7i—nc + dn{a—7n)
rob— an ^ mh—an
l + a* a^—h 2n 2n
2a ' 2a ' ' m—V m + 1*
Examples — 42.
1. x^-\-x^, +2:^, +a:3; ah"^ + ah^ + ah"" -^ a^.
2. ah^-\-a^-Va'b^+ah^) ah'''\-al)'+ah' + ah\
ANSWERS TO EXAMPLES. 305
a'b-' + 3a'd-' + 6ab-'+4.a-'b + 2a-^b^ ;
1 1 A _i_ _A_
a'^ b''^ c''^ a-'b^ ab-^'
1 3 5 4 2
a-'b''^ a-'b^ a-'b'^ a'b-^^ a'b-''
1 4 2 1
^^ 3a-^Z>^c^ ^ a'b c-' '^ ab-'c~' ^ 3abc '
1 2 3 5
'^ 3.-, o o ' . S.7I.2+ ,4
5. v/^+2V«'^+3Va'+4V6x + Va',
Va V{a:'b) 2V(ac') V{Fc') Vjbc ')
Vb''^ 2v/c "^ 3Vb' "^ 4Va "^5V^''
^' a'^ b^'^abc'^ aW ' Va''^ Vb' "^ ^fa' ^Vb''
a 6> be a vb vb Va
Examples — 43.
1. i. 2.1 3.^. 4.100. 5.^.
6. ^-^ 7. a^ 8. ^-^ 9. a-\ 10. a^"^-
11. .7:^-^1. 12. a-b. 13. .T^+2.'z;^ + a:-4
S06 ELEMENTARY ALGEBRA.
14. x'+l+x-\ 15. a-'-l. 18. a'-3a^ + da-^-a--\
17. a'' + 2ahi + ab-x^yK 18. x^+x^y^ + x^y^ + y^.
19. a^ + t«^^^+^^ 20. lex-^-nx-^y-^^+dy-^'
21. a;+2/. 22. a^-ah^+hK 23. a*+2>^-c*-
24. a;* + 2iz;M+3a;*a+2A^-}-^'.
25. x^-2x"K 26. ^r-2-ar\ 27. aJ"* + 1 + a-^Z».
Examples — i4.
1. 64^ 81* {i)K {i)\ {i)K 8i
2. 25^, (V)*, (K)^ (K)i, {i(^^+2a^+J^)}i;
125^ (1^)4, (^a^)^, ftW)^ {i(a^+3a^^+3«&»+J')}i
6561-^ (n^3^)-% (a«)-^ (^)"*.
4. v/125, v/3, v/12, v/|, x/i v/320.
5. V54, V256, V2048, V3, V|, V^.
6. v/(4a), v/(98a^a;), j/^.
7. v/(2a5), v/(6a':.), |/||;, ^^, .^{^'-^r').
ANSWERS TO EXAMPLES. 307
8. 3v/5, 5v/5, 36v/3, 3V5, 18V2, iv/G, V12, V54, 6.
9. 4V2, 8V2, 6V48, fv/2, ^v/2, fV2, ■|v/21, |V150,
V375.
10. 2v/3, 15v/3, |v/3, ^n/3, iv/3, ^VB.
Examples — 45.
1. v/108, v/112; V81, V80; V120, V128, V135.
2. 7v/2. 3. 9V4. 4. |v/3. 5. '^.
9. 2+|n/6.
10. iCv/S+v/S+v/S), iv/6+iV33+iV120.
11. ^(2y2+v/3), v/5 + 1, v/5-v/2, 4+v/2, i(7+3v/5).
Examples — 46.
1. v/3+1. 2. 3+v/2. 3. v/5-v/3.
4. 2V6-3V2, 5. 4v/2-3. 6. iV6-l.
7. 2-^|v/3.
308 ELEMENTARY ALGEBRA,
Examples — i7.
r 4 2. 50. 3. 25. 4. H. 5. {a--h)\
(,. a. 7. -^^. 8. -3^3^.
Examples — 48.
1. ±3.
2. ±3. 3.
±1.
4. ±4.
5. ±i.
6. ±2^. 7.
±|.
8. ±5.
9. ±3.
10. ±5. U.
±2.
12. ±2.
13. ±%/3.
14. £B=d=3.
Examples — 49.
1. 4, -2.
2. -1, -9.
3.
20, -6.
4. 7, 5.
5. 8, -40.
Examples— 50.
6.
10, -110.
1. 1, -8. 2. 17, -4. 3. —5, -20.
4. -1, -12. 5. 1, -20. 6. 25, -136.
answers to examples. 309
Examples — 51.
I. 0, -51. 2. 6, -ii. 3. 8|, -10.
4. 14, -lOf. 5. 12, -12tJj. 6. 13, -11^
Examples— 52,
1. 10, 2. 2. 3, -1. 3. 2, -I
4. U, -i|. 5. If, -n. 6. 7, -li.
7. 3, i- 8. i(-9±3v/3). 9. 3, if.
10. 3, -f 11. |(37±V57). 12. 2, -3.
Examples — 53.
1. G, 3^V 2- 6, -4|. 3. 1, lOf. 4. 3, -8,'^.
5. 5, -l^-V 6. 5, li. 7. 5, -li.
8. 31, 0. • 9. a±-. 10. (a±by.
a
310 ELEMENTARY ALGEBRA.
Examples — 54.
1. x'-'ke-21=
:0. 2. 6a;''+5a;-
-6=
0.
3. cc^'+llx+SO:
=0. 4. Sx^-Sx--
=0.
5. a;'-100=0.
6. a;'-3aa;+fl'-
-4=0.
a
7. a;'' + 2a;-l=0.
EXAMPLES — 55.
•
1. ±2, ±3.
2. 49. 3. 4
4. ±4,
5. 5, -3.
C. 3, -2.
7.
12, -3
8. 9, -12,
9. ±3. 10. 2.
11.
4.
12. 16.
13. 1, |. 14. 4.
15.
3a'.
16. 0, ±5.
17. 0, ±V2.
18.
2, ±1.
19. 0, ±v/(a5).
20. a, -2flr, —2a.
EXAMTLES— 56.
1. 3, 4, 5. 2. 36, 24. 3. 30, 24. 4. 18, 12, 9.
5. 196. 6. ±12, ±15. 7. 24.
8, 15 yards, 25 yards. 9. 4550. 10. 40 yds. by 24
ANSWERS TO EXAMPLES. 311
11. 16. 12. 4 yards, 5 yards. 13. £60, or £40.
14. 8d. 15. Equal.
Examples — 57.
1. a;=7, ?/=±4. 2. x=4:, y=-dj
a;=-3, y=4. ^
S. a;=4, ^=3,) 4. cc=:8, ^/^^^^
5. x=:6, y=5, ) 6. a:=5, y=^,)
7. z=:5, y=3j 8. cc=3, y=4:J
9. a;=4, 2/=^J 10- ^=10, y=15,^
x=2, y=4.5 a;=-.10|, y=-16i.\
11. a;=3, ^=2,^ 12. a:=5, 2/=4,)
t;=4, y=5.$
13. a;=i{aiFv/(25'-a'')},i
14. x=i{±V{W + b') + I>},-i
U. x=8,y=l,-i ,„ ^ «'
=1, y=.8.i ^^' ''=^T777;rn?i' y=^-
x=l, i/=8. $ v'(a'' + 5^)' ^~ ^{a' + d'Y
312 ELEMENTARY ALGEBRA.
18. fl^ + Z> + l, --7—; I?,-
a-hl ' ' a + 1
19. ±-|; ±3Z>. 20. ±1-; ±2^.
d 4:
21. 0, a-Vh, h{a-l))^hV {{a-Vdh){a-l))}.
Examples — 58.
1. 11; 7. 2. 8; 24. 3. 10; 12. 4. 18- 8: 6; 16,
5. 5; 3. 6. 4; 2. 7. 2; 2. 8. ?; 4.
9. 60. 10. 6, .4. 11. 160; £2.
12. 756; 36; 27. 13. £275, £225. 14. 2, 5, 8.
Examples — 59.
*• ia? io ^ irf^
TFt 5
mz> ihh, AW
2. «+J.
*'-"363-
^- 4 .4
- »'— lla; + 28
4. A, |.
7. 1.
ANSWERS TO EXAMPLES. 313
8. -('iti)" 9. i^l^. IJ. c. 8.
o{a — o) o—d
12. 35, 42. 13. 4. 14. — ^.
a-\-o
Examples — 61.
1. 10, 4|, 2tV 2. 9, ^, If. 3. 6, 1|, If.
10.
^(^1) ' ^^ a + h,OY h{a-h). 12. .^' = l, y-=i\
13. rr:=±9, //=±3. 14. 3. 15. 25, 20.
16. 8:7. 17. 6.
Examples — 62.
1. 32, 272. 2. 39, 400. 3. 63, 363.
4. 694, 34750. 5. 9, IG. 6.-1, 0. 7. -28.
8. -275. 9. 16i. 10. -m. 11. 336f.
12. -84.
Examples— 63.
1 12. 2. 20. 3. I^Vt^ 1-^V^ &c., &c. ; 6 = 60.
4. 14, 16, 18. 5. 141 14|,... 6. 6J, 5,...
314 ELEMENTARY ALGEBRA.
7. -h h'- 8. 10, 4. 9. 82. 10. 5, 9, 13, 17.
11. 1, 2, 3, 4, 5, 12. 18, 19. 13. 7. 14. 5.
Examples — 64.
1. 64, 85. 2. 1280, 1705. 3. 96, 189.
4. __256, -170. 5. 4096, 3277. 6. -512, -341.
T. tV^. 8. 1H|. 9. 4,VV 10. 2f-||.
Examples — 65.
1. 8. 2. H. 3. i 4. ^. 5. I
6. 4. 7. tV 8. 1. 9. 1/,. 10. h.
11. iV^. 12. lU.
Examples — 66.
I. 4, 16, 64. 2. 8, 12, 18, 27. 3. -9, 27, -81, 243.
4. 3, 12, 48; or 81, -54, 36. 5. 1, 3, 9, . .
14
ANSWERS TO EXAMPLES. S15
6. ^^. 7. 2 + | + f + &c. ; or 4-| + |-&c
8. 3_|4.|_&c. 9. 200 miles.
Examples — 67.
1. iil. 2. f, A, 2. 3. 3,i^.
4. A. tV. a. 5. 6, 12. 6. 36, 64
7. 1, 0. 8. 3, 9.
Examples— 68.
1. 720, 720. 2. 5040. 3. 19958400.
4. 34650. 5. 210. 6. 6. 7. 4. 8. 6,
Examples— 69.
I. 126, 84, 36. 2. 330, 330, 11. 3. 3003, 455.
I. 4950. 5. 210, 84. 6. IL
7. 50063860, 5006386. a 116280.
Examples — 70.
1. l + Gx+lox' + 20x' -^iDx' -\-6x' -hx',
2. a'-loa*x + dOaV'-270a'x''^4:06ax*-24Saf.
316 ELEMENTARY ALGEBRA.
4. a'-9a'a; + 36aV~84aV + i2GaV-126aV-f 84aV
-36aV + 9ax''-x^
5. 1 + 12x-{-66x'-{-220x' + ^9dx' + 792x + 924a;" + 792a;'
'\'4.mx'+220x' + 6Gx'' -\-12x'' +x'\
6. l-20a;-fl80a;^-9G0a;^4-33G0x*-8064a;^ + 13440a;^
-15360a;' + 11520a;''-5120a;' + 1024a;^°.
7. ^•-18r/'a; + 135c'^V-540aV + 1215aV-14o8aa;*
+ 729a;".
8. 25Ga;^ + 1024r/a;' + 1792aV 4- 1792aV + 1120aV
+ 448a'^a;' + 112aV + lea'x-^-a',
a 128a'-1344a«a; + 6048r^V-15120aV + 22680«.V
--20412r^V + 10206aa;''-2187a;^
10. l-^6x■i■\^-x''--16x' + ^^x'-\^x'' + ^%^-x''-^x'' + ^,d'
U. l-i^x + %^x^^^i-x' + ^\^x'-ii^x'^iUx'-¥A^'
12. 36aV. 13. -l^^XAVy.
110
14. 4t05a"b'. 13. t^=^V,
[5 |o_
answers to examples. 317
Examples — 71.
1. 100101100, 102010, 10230, 2200, 1220.
2. 41104, 23420, 14641, 7571, 5954.
3. 402854. 1 511, 22154. 6. a
Examples— 72.
1. 152. 4. 100001000000 (binary) == 201000 (quat.).
5. 57264, 95494, eltS. 6. 4112, 6543, 62/^.
7. 1295, 216; 2400, 343; 4095, 512.
Examples — 73.
1. .9030900, .9542426, 1.0791813, 1.3010300, 1.3979400,
1.7781513.
2. r.5228787, 1.3979400, 1.6020600, 2.4771213,
2^5228787, 375185140.
3. 2.2253093, .0170334, 3.5670265.
318
ELEMENTARY ALGEBRA.
4. 2 + log. 3 + 2 log. 7.
5. Gh-2 log. 3 + 3 log. .21.
6. 2 log. 2-t log. 3 + f log. 5-1, and I log. 13 -| log. 2
7. .019. 8. 4 and 6. 9. 1.8035700.
' 1
_</x
^.
)
A*
^-
P"^^ V J
V .^^
x./ f rn ^ If
U. C. BERKELEY LIBRARIES ^ %^
iiiiiiiiii nil nil nil III II nil mil mil III III! ^.v^ ^.
^\.
s %,