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EATON'S
ELEMENTARY ALGEBRA
DESIGNED FOR THE
USE OF HIGH SCnOOLS AND ACADEHES.
WILLIAM F. BRADBURY, A. M.,
■OPKIXS MASTER Vf THE CAMBRIDGE niGH SCHOOL ; AOTUOR OF A TREATI8K OK
TsmoNOiurKr Aia> soanrufG, and of an kuementart oecxetbt.
BOSTON:
THOMPSON, BIOELOW, AND BROWN
26 & 29 CORMBILU
EATON AND BRADBURY'S
used with unexampled success in the best schools and
academies of the country.
Eaton's Primary Arithmetic
Eaton's Intellectual Arithmetic.
Eaton's Common School Arithmetic.
"•: .•'•.EatoSj's" ffi^fa* School Arithmetic.
►V. '» './* ^iTok's'.ELi^^i^Ta £)p Arithmetic.
Eaton's Grammar School Arithmetic.
Bradbury's Eaton's Elementary Algebra.
Bradbury's Elementary Geometry.
Bradbury's Elementary Trigonometry.
Bradbury's Geometry and Trigonometry, in one volnme.
Bradbury's Trigonometry and Surveying.
Keys of Solutions to Common School and High
School Arithmetics, to Elementary Algebra, Geom-
etry, AND Trigonometry, and Trigonometry and Sur-
veying, for the use of Teachers.
Entered according to Act of Congress, in the year 1868,
BY WILLIAM F. BRADBURY AND JAMES H. EATON,
in the Clerk's Office of the District Court of the District of Massachusetts.
University Press : Welch, Bigelow, & Co.,
Cambridge.
CAJORl
'^
PREFACE
It was the intention of the author of Eaton*s Arith-
metics to add to the series an Algebra, and he had com-
menced the preparation of such a work. Although its
completion has devolved upon another, the author, as far
as practicable in a work of this character, has followed
the same general plan that has made the Arithmetics so
popular, and spared no labor to adapt the book to the
wants of pupils commencing this branch of mathematics.
A few problems have been introduced in Section II., to
awaken the pupiPs interest in Algebraic operations, and
thus prepare him for the more abstract principles which
must be mastered before the more difficult problems can be
solved. Special attention is invited to the arrangement
of the equations in Elimination ; to the Second Method of
Completing the Square in Affected Quadratics ; and to the
number and variety of the examples given in the body
of the work and in the closing section.
The Theory of Equations, the Explanation of Negative
Results, of Zero and Infinity, and of Imaginary Quantities,
are omitted, as topics not appropriate to an Elementary
Algebra. It may also be better for the younger pupils to
Qi 1 ?59 7
IV PREFACE.
pass over the two theorems in Art. 74, until they become
more familiar with algebraic reasoning.
While the book has not been made simple by avoiding
the legitimate use of the negative sign before a parenthesis
or a fraction, the difficulty which is caused to beginners
by the introduction of negative indices in simple division
has been obviated by deferring their introduction to the
section on Powers and Roots, where they are fully ex-
plained.
The utmost conciseness consistent w^ith perspicuity has
been studied throughout the work. It is hoped the book
will commend itself to both teachers and pupils.
W. F. B.
Cambbidge, Mass., May 17, 1888.
CONTENTS.
SECTION I.
Taom
Defixitioxs A!n> Notation . 7
The SigoA 7 I Axioms 9
SECTION II.
Alqeobaic OpzKAnoxs 10
SECTION III.
Dfn5Tno:fs asi> Notahon (Contioaed from Section I.) 14
SECTION IV.
Additiox 19
SECTION V.
SCBTRACnOM 25
SECTION VI.
MoiTirucATiox 81
SECTION VII.
Dnnsiox 87
SECTION VIII.
Dbxoxstkatiox or Tokobzms 43
SECTION IX.
Factorixo 47
SECTION X.
OuATBST Comiox DirisoR 54
LXAST COXXOK MOLTtPlI
SECTION XI.
Pa.^cTioxs
8BCTI0N XII.
fl.«n«ral Prinriple* 6.3 To inultiply a Fraction by an Integral
Picrn« of Fr.irtions 04 QimnMty 78
To n'liKc a Frirtion toit^lnwpst tprm« 05 To multiply an Int«^n^l Quantity by
To re lure FrartionB to equivalent Frac- a Kmrtlon . 75
tioti* hiving a ('omnion Denominator C* To <livi le a Fraction by an Integral
To all Fnetlorw 68 | Quantity 76
To subtract Fmotions 09 To dirlde an Integral Quantity by a
To reduce a Mixed Quantity to an Im- j Fraction 77
proper Fraction 70 To multiply a Fraction by a Fraction . 77
To rwluce an Improper Fraction to an To divide a Fraction by a Fraction . . 79
Integral or Mixed Quant'ty ... 72
VI CONTENTS.
SECTION XIII.
Equations op thk First Degree containing but one Unknoww Quaittitt . . • 82
Definitions 82 ' Reduction of Equations 86
Transposition 83 Problems 91
Clearing of Fractions 85 1
SECTION XIV.
Equations of the First Degree containing two Unknown Quantities . . . , 104
Elimination by Substitution .... 105 I Elimination by Combination .... 108
Elimination by Comparison .... 106 | Problems • . 112
SECTION XV.
Equations of the First Degree containing more than two Unknown Quan-
118
SECTION XVI.
Powers and Roots 125
Negative Indices 125
Multiplication and Division of Powers .
of Monomials 126
Transferring Factors from Numerator
to Denominator, or Denominator to
Numerator of a Fraction .... 127
Involution of Monomials 129
Involution of Fractions ...... 130
Involution of Binomials 131
Square Root of Numbers 139
Cube Root of Numbers 142
Evolution of Monomials 147
^-quare Root of Polynomials .... 148
To find any Root of a Polynomial . . 152
SECTION XVII.
Radicals 154
Definitions 154 To add Radicals 160
To reduce a Radical to its Simplest Form 154 To subtract Radicals 161
To reduce a Rational Quantity to the To multiply Radicals 162
form of a Radical 157 To divide Radicals 163
To reduce Radicals having different In- To involve Radicals 155
dices to equivalent ones having a To evolve Radicals 166
Common Index 158 Polynomials having Radical Terms . 167
SECTION XVIII.
Pure Equations which require in their Reduction either Involution or Evo-
170
SECTION XIX.
Affected Quadratic Equations 178
Completing the Square 178 I Third Method of Completing the Square 185
Second Method ofCompleting the Square 182 I Problems 193
SECTION XX.
Quadratic Equations containing two Unknown Quanttties 196
SECTION XXI.
Ratio and Proportion 207
SECTION XXII.
Arithmetical Progression 215
SECTION XXIII.
Geometrical Progression 225
SECTION XXIV.
MlSCXLLAinOUS EZAMFLZS
ELEMENTARY ALGEBRA.
SECTION 1.
DEFINITIONS.
!• Mathematics is the science of quantity.
2« Quantity is that which can be measured ; as distance,
time, weight.
3* Arithmetic is the science of numbers. In Arithmetic
quantities are represented by figures.
4. Algebra is Universal Arithmetic. In Algebra quan-
tities are represented by either letters or figures, and their
relations by signs.
NOTATION.
5. Addition is denoted by the sign +, called plus ; thus,
3 + 2, i. e. 3 plus 2, signifies that 2 is to be added to 3.
6. Subtraction is denoted by the sign — , called minus;
thus, 7 — 4, i. e. 7 minus 4, signifies that 4 is to be sub-
tracted from 7.
7i Multiplication is denoted by the sign X ; thus, 6X5
signifies that 6 and 5 are to be multiplied together. Be-
tween a figure and a letter, or between letters, the sign X
is generally omitted ; thus, Ca6 is the same as 6 X a X 6.
Multiplication is sometimes denoted by the period ; thus,
8 . 6 . 4 is the same as 8 X 6 X 4.
8 ELEMENTARY ALGEBRA.
8. Division is denoted by the sign -^ ; thus, 9 -r- 3 sig-
nifies that 9 is to be divided by 3. Division is also indi-
cated by the fractional form ; thus, f is the same as 9 -i- 3.
9. Equality is denoted by the sign = ; thus, $1 = 100
cents, signifies that 1 dollar is equal to 100 cents. An ex-
pression in which the sign = occurs is called an equa-
tion, and that portion which precedes the sign = is called
the Jirsl member, and that which follows, the second mem-
bef.l .','".' * ' ; , ■
10. Inequality is denoted by the sign > or <, the
smaller quantity always standing at the vertex ; thus,
8 > 6 or 6 < 8 signifies that 8 is greater than 6.
11. Three dots .•. are sometimes used, meaning hence,
therefore.
12. A Parenthesis ( ), or a Vinculum , indicates
that all the quantities included, or connected, are to be
considered as a single quantity, or to be subjected to the
same operation ; thus, (8 + 4) X 3 = 12 X 3, or = 24
+ 12 = 36 ; 21 — 6 -=- 3 = 15 -f- 3, or = T — 2 = 5.
Without t^e parenthesis, these examples would stand
thus :8 + 4X3 = 8 + 12 = 20;21 — 6-^3 = 21
— 2 = 19; the sign X. in the former, not affecting 8j
nor the sign -i-, in the latter, 21.
Examples.
1. g + t — 3 + 4 = how many?
2. (9 + 15) -r- 3 = how many?
3. — — — X 14 = how many ?
4. (14 + 13) X (5 — 2) = how many?
5. 10 + (7 — 4) -^ 3 X 4 = how many ?
6. 25 — (6 + T) = how many?
T. 150 -^ (18 — 11) = how many?
DEFINITIONS. 9
8. Prove that 175 + 8 — 49 = 14 + 190 — 64 — 16.
9. Prove that 216 — 44 + 14 > 144 + 13 — T5.
10. Place the proper sign (=, >, or <) between these
two expressions, (247 + 104) and (546 — 195).
11. Place the proper sign (=, >, or <) between these
two expressions, (119 — 47 + 16) and (317 — 104).
12. Place the proper sign (=, >, or <) between these
two expressions, (417 + 31) — (187 — 72) and (127 +
179).
AXIOMS.
IS. All operations in Algebra are based upon certain
self-evident truths called Axioms, of which the following
are the most common : —
1. If equals are added to equals the sums are equal.
2. If equals are subtracted from equals the remainders
are equal.
3. If equals are multiplied by equals the products are
equal.
4. If equals are divided by equals the quotients are
equal.
6. Like powers and like roots of equals are equal.
G. The whole of a quantity is greater than any of its
parts.
7. The whole of a quantity is equal to the sum of all
its parts.
8. Quantities respectively equal to the same quantity
are equal to each other.
a*
10 ELEMENTARY ALGEBRA.
SECTION II.
ALGEBRAIC OPERATIONS.
14. A Theorem is something to be proved.
15. A Problem is something to be done.
18. The Solution of a Problem in Algebra consists, —
1st. In reducing the statement to the form of an equa-
tion ;
2d. In reducing the equation so as to find the value
of the unknown quantities.
Examples for Practice.
1. The sum of the ages of a father and his son is 60
years, and the age of the father is double that of the son ;
what is the age of each ?
It is evident that if we knew the age of the son, by-
doubling it we should know the age of the father. Sup-
pose we let X equal the age of the son; then 2x equals
the age of the father; and then, by the conditions of the
problem, a:, the son's age, plus 2^7, the father's age, equals
60 years; or 3a: equals 60, and (Axiom 4) x, the son's
age, is ^ of 60, or 20, and 2x, the father's age, is 40.
Expressed algebraically, the process is as follows : —
Let X = son's age,
then 2x =z father's age.
a; + 2 a; = 60,
3 a: = 60,
X = 20, the son's age.
2 X = 40, the father's ago.
DEFINITIONS. 11
2. A horse and carriage are together worth $450 ; but
the horse is worth twice as much as the carriage ; what
is each worth? Ans. Carriage, S150; horse, $300.
All problems should be verified to see if the answers
obtained fulfil the given conditions. In each of the pre-
ceding problems there are two conditions, or statements.
For example, in Prob. 2 it is stated (Ist) that the horse
and carriage are together worth $450, and (2d) that the
horse is worth twice as much as the carriage ; both these
statements are fulfilled by the numbers 150 and 300.
3. The sum of two numbers is 12, and the greater is
seven times the less ; what are the numbers ?
4. A drover being asked how many sheep he had, said
that if he had ten times as many more, he should have
440 ; how many had he ?
5. A father and son have property of the value of
$8015, and the father's share is four times the son's;
what is the share of each ?
Ans. Father's, $6412; son's, $1603.
6. A farmer has a horse, a cow, and a sheep ; the horse
is worth twice as much as the cow, and the cow twice
as much as the sheep, and all together are worth $490 ;
how much is each worth ?
OPERATION.
Let X = the price of the sheep,
then 2 a: = " " " " cow,
and 4 a; = ^* '' " *' horse ;
and their sum Tar = 490,
a; = 70, the price of the sheep,
and 2 a: = 140, " " " " cow,
and 4x = 280, " '* " " horse.
12 ELEMENTARY ALGEBRA.
7. A man has three horses which are together worth
$540, and their values are as the numbers 1, 2, and 3;
what are the respective values ?
Let a:, 2 a:, and 3x represent the respective values.
Ans. $90, $180, and $2*70.
8. A man has three pastures, containing 360 sheep,
and the numbers in each are as the numbers 1, 3, and 5 ;
how many are there in each ?
9. Divide 63 into three parts, in the proportion of 2,
3, and 4.
Let 2x, 3 a;, and 4 a: represent the parts.
10. A man sold an equal number of oxen, cows, and
sheep for $ 1500 ; for an ox he received twice as much
as for a cow, and for a cow eight times as much as for
a sheep, and for each sheep $ 6 ; how many of each did
he sell, and what did he receive for all the oxen ?
Ans. 10 of each, and for the oxen, $ 960.
11. Three orchards bore 812 bushels of apples ; the first
bore three times as many as the second, and the third
bore as many as the other two ; how many bushels did
each bear?
12. A boy spent $4 in oranges, pears, and apples; he
bought twice as many pears and five times as many apples
as oranges ; he paid 4 cents for each pear, 3 for each
orange, and 1 for each apple ; how many of each did he
buy, and how much did he spend for oranges ? how much
for pears, and how much for apples ?
. (25 oranges, 50 pears, and 125 apples.
( Spent for oranges, $0.75 ; pears, $2 ; apples, $1.25.
13. A farmer hired a man and two boys to do a piece
of work ; to the man he paid $ 12, to one boy $ 6, and
to the other $ 4 per week ; they all worked the same
time, and received $ 264 ; how many weeks did they
work ? Ans. 12 weeks.
DEnXITIOXS. IS
14. Three men, A, B, and C, agreed to build a piece
of wall for S 99 ; A could build 7 rods, and B 6, while
C could build 5 ; how much should each receive ?
15. Four boys, A, B, C, and D, in counting their money,
found they had together $1.98, and that B had twice as
much as A, C afi much as A and B, and.D as much as
B and C ; how much had each ?
Ans. A 18 cents, B 36, C 54, and D 90.
16. It is required to divide a quantity, represented by
a, into two parts, one of which is double the other.
OPERATION.
Let X = one part,
then 2x = the other part .
Sx = a,
X = -, one part,
o
2x = -—, the other part.
o
17. If in the preceding example a = 24, what are the
required parts ?
^""- 3 == T = ^' ^""^ T = T = •"•
18. It is required to divide c into three parts so that
the first shall be one half of the second and one fifth of
the third. . c 2r , •'>^
Ans. -, — , and — .
19. Divide n into three parts, so that the first part
shall be one third the second and one seventh of the
third.
20. A is one half as old as B, and B is one third as
old as C, and the sum of their ages is p ; what is the
age of each ? ^^^ ^,^ P g,^ 2p_ ^^^ ^'s '^.
14 ELEMENTARY ALGEBRA.
SECTION III.
DEFINITIONS AND NOTATION.
[Continued from Section I.]
17. The last letters of the alphabet, x, y, z, &c., are
used in algebraic processes to represent unknown quanti-
ties, and the first letters, a, b, c, &c., are often used to
represent known quantities.
Numerical Quantities are those expressed by figures, as
4, 6, 9.
Literal Quantities are those expressed by letters, as
a, X, y.
Mixed Quantities are those expressed by both figures
and letters, as 3 a, 4 a;.
18. The sign plus, -f-' i^ called the positive or affirm-
ative sign, and the quantity before which it stands a 'pos-
itive or affirmative quantity. If no sign stands before a
quantity, -|- is always understood.
19. The sign minus, — , is called the negative sign, and
the quantity before which it stands, a negative quantity.
20. Sometimes both + and — are prefixed to a quan-
tity, and the sign and quantity are both said to be am-
biguous; thus, 8 zi= 3 = 11 or 6, and a z^h ::= a -\-hf
or a — 6, according to circumstances.
21. The words plus and minus, positive and negative,
and the signs -f- and — , have a merely relative signifi-
cation ; thus, the navigator and the surveyor always rep-
resent their northward and eastward progress by the sign
+, and their southward and westward progress by the
sign — , though, in the nature of things, there is nothing
to prevent representing northings and eastings by — ,
and southings and westings by +• So if a man's prop-
DEFnaiiONS. 16
erty is considered positive, his gains should also be con-
sidered positive, while his debts and his losses should be
considered negative; thus, suppose that I have a farm
worth $6000 and other property worth $3000 and that
I owe SIOOO, then the net value of my estate is S5000
+ $3000 — $1000 = $7000. Again, suppose my farm
is worth $5000 and my other property $3000, while
I owe S 12000, then my net estate is worth $5000
_|_ S3000 — $12000 = — $4000, i. e. I am worth
— $4000, or, in other words, I owe $4000 more than I
can pay. From this last illustration we see that the sign
— may be placed before a quantity standing alone, and
it then merely signifies that the quantity is negative,
without determining what it is to be subtracted from.
22. The Terms of an algebraic expression are the quan-
tities which are separated from each other by the signs
+ or — ; thus, in the equation 4a — b =i Sx -{- c — 1 y,
the first member consists of the two terms 4a and — 6,
and the second of the three terms 3 x, c, and — 7 y.
23. A CoEFFiCTEXT is a number or letter prefixed to a
quantity to show how many times that quantity is to be
taken; thus, in the expression ix, which equals x -\- x
-\- X -{- X, the 4 is the coefficient of x ; so in dab, which
equals ab -\- ab -\- ab, 3 is the coefficient of a 6; in 4 a 6,
4 a may be considered the coefficient of b, or 4 6 the co-
efficient of a, or a the coefficient of 45.
Coefficients maybe numerical or literal or mixed; thus,
in 4 a 6, 4 is the numerical coefficient of ab, a is the lit-
eral coefficient of 4 6, 4 a is the mixed coefficient of 6.
If no numerical coefficient is expressed, a unit is un-
derstood ; thus, X is the same as Ix, be as 16c.
24. An Index or Exponent is a number or letter placed
after and a little above a quantity to show how many times
that quantity is to be taken as a factor; thus, in the ex-
16 ELEMENTARY ALGEBRA.
pression b^, which equals 6 X & X &, the 8 is the index or
exponent of the power to which b is to be raised, and it
indicates that b is to be used as a factor 3 times.
An exponent, like a coefficient, may be numerical, lit-
eral, or mixed ; thus, ar^, x^, x^"^, &c.
If no exponent is written, a' unit is understood ; thus
b = b^, a =z a\ &c.
Coefficients and Exponents must be careful)}'- distin-
guished from each other. A Coefficient shows the num-
ber of times a quantity is taken to make up a given
sum ; an Exponent shows how many times a quantity is
taken as a factor to make up a given product ; thus
4:X'=x-\-X'\-x-^x, and x^=zxy,xy^xy(^x.
25* The product obtained by taking a quantity as a
factor a given number of times is called a power, and
the exponent shows the number of times the quantity is
taken.
26. A Root of any quantity is a quantity which, taken
as a factor a given number of times, will produce the
given quantity.
A Root is indicated by the radical sign, \/, or by a
fractional exponent. When the radical sign, \/> is used,
the index of the root is written at the top of the sign,
though the index denoting the second or square root is
generally omitted ; thus,
^/ X, or x^y means the second root of x ;
^Ic, or a:^ " '' third '' " x, &c.
Every quantity is considered to be both the first power
and the first root of itself.
27t The Reciprocal of a quantity is a unit divided by
that quantity. Thus, the reciprocal of 6 is -, and of a;, -.
DEFINITIONS. 17
28« A Monomial is a single term; as a, or 3a*, or
5bxy.
29. A Polynomial is a number of terms connected
with each other by the signs plus or minus ; sls x -\- y,
or da -\- 4:X — *lahy.
30* A Binomial is a polynomial of two terms ; as
3x + 3y, or a: — 2/.
31* A Residual is a binomial in which the two terms
are connected by the minus sign, as a: — y.
32t Similar Terms are those which have the mme powers
of tJie same letters, as x and 3 a:, or 5ax^ and — 2ax'^.
But X and ar, or 5 a and 5 b, are dissimilar,
33. The Degree of a term is denoted by the sum of
the exponents of all the literal factors. Thus, 2 a is of
the first degree ; 3 a^ and 4 fl 6 are of the second de-
gree ; and 6 a' x* is of the seventh degree.
34. Homogeneous Terms are those of the same degree.
Thus, 4a^ar, 3a6c, xry, are homogeneous with each other.
3^. To find the numerical value of an algebraic expres-
sion when the literal quantities are known, we must sub-
stitute the given values for the letters, and perform the
operations indicated by the signs.
The numerical value of 7 a — b* -\- c^ when a = 4,
6 = 2, and c = 6 is 7 X 4 — 2* + 6^ = 28 — 16
+ 25 = 37.
Examples.
Find the numerical values of the following expressions,
when rt = 2, 6=13, c = 4, d = 15, m = 5, and n = 7.
1. a + b — c + 2d. Ans. 41.
2. a* + Sbc — 2cd. Ans. 40.
3. i^^lt^* Ans. 219.
18 ELEMENTARY ALGEBRA.
4. m^ — 2mn + n^
6. ^ - (^ - ^')-
6. (a" _ c + 6) (wi + n).
Y. ^' ~ "•'' X {d — m + n). Ans. 86Y.
\/c 4" V^^ + c — \/bm. Ans. 1.
9. 3a\/& — c X 4.71 \/2bm,
10. , ^ . Ans. 4.
d — 2 c
11. \^d — n + ^/1 n.
12. (6 — fl) (cZ — c) — m. Ans. 116.
13. 13 (4.d) + Ad — la.
14. 4a6 + \/lOOc — ^d — n. Ans. 122.
15. 4a s^QOl + ba'b^
16. 6 — a — (d — n). Ans. 3.
17. b — a — d — n. Ans. — 11.
18. (b — a) {d — n). Ans. 88.
19. (b — a) d — n. Ans. 158.
20. a + b^lO {d — m) + 14 V'c.
36* Write in algebraic form : —
1. The sum of a and b minus the difference of m
and n. (m > n.)
2. Four times the square root of the sum of a, b, and c.
3. Six times the product of the sum and difference of
c and d. {c >- d.)
4. Five times the cube root of the sum of a, m, and n.
5. The sum of m and n divided by their difference.
6. The fourth power of the difference between a and w,
ADDITION.
19
SECTION IV.
ADDITION.
37. Addition in Algebra is the process of finding the
aggregate or sum of several quantities.
For convenience, the subject is presented under three
cases.
CASE I.
38. When the terms are similar and have like signs.
1. Charles has 6 apples, James 4 apples, and William
5 apples ; how many apples have they all ?
6 apples,
4 apples,
6 apples.
OPERATION.
or, letting a
represent -
one apple,
6 a
4 a
5 a
It is evident that just as
6 apples and 4 apples and
6 apples added together
make 15 apples, so 6 a and
15 a
cr naake 15 a.
15 apples,
In the same way — 6 a and — 4 a and — 5 a are
equal together to — 15 a.
Therefore, when the terms are similar and have like
RULE.
Add the coefficients, and to their sum annex the common
letter or letters, and prefix the common sign.
(2.)
(3.)
(4.)
(5.)
(6.)
0')
b ax
3a2
4x
6y
— Zj^
— bby
Sax
4a«
X
lOij
— 2ar'
— 26y
4ax
la'
5x
y
— Ix"
- hy
2ax
3a«
3x
2y
— 4.x'
- hy
19 ax
13 X
-^16x»
20
ELEMENTARY ALGEBRA.
8. What is the sum of arc^, Saa^, 2ax^, and 4:asP?
Ans. 10 a x^.
9. What is the sum of 3 5a:, 4 5a:, 6bx, and bx?
10. What is the sum of 2a:y, 6xy, lOxi/, and Sxy?
11. What is the sum of — Ixz, — xz, — 4a:z, and
— xz? Ans. — 13x2;.
12. What, is the sum of —2 5, —3 5, —6 5, and
— 35?
13. What is the sum of — a5c, — 3a5c, — 4a5c,
and — a be?
CASE II.
39. When the terms are similar and have unlike signs.
1. A man earns T dollars one week, and the next week
earns nothing and spends 4 dollars, and the next week
earns 6 dollars, and the fourth week earns nothing and
spends 3 dollars ; how much money has he left at the
end of the fourth week ?
If what he earns is indicated by -[-, then what he
spends will be indicated by — , and the example will
appear as follows : —
OPERATION. Earning 7 dollars and
then spending 4 dollars,
the man would have 3
dollars left^ then earn-
ing 6 dollars, he would
have 9 dollars; then
spending 3 dollars, he
would have left 6 dol-
lars ; or he earns in all 7 dollars -j- 6 dollars =13 dollars ; and spends
4 dollars -|- 3 dollars = 7 dollars; and therefore has left the differ-
ence between 13 dollars and 7 dollars == 6 dollars; hence the sum
of -f- 7 d, — 4 d, -{- G d, and — 3 d is -\- 6 d.
Therefore, when the terms are similar, and have unlike
signs :
+ 7 dollars,
-\-*ld
• — 4 dollars.
or, letting d
— 4:d
+ 6 dollars,
. represent ^
+ 6d
— 3 dollars,
one dollar,
— 3d
+ 6 dollars.
+ ed
ADDITION. 21
RULE.
Find the difference between the sum of the coefficients of
the positive terms, and the sum of the coefficients of Uie neg-
ative termSf and to this difference annex the common letter
or letters, and prefix the sign of the greater sum.
(2.)
(3.)
(4)
(5.)
3xy
^y"
13o6c2
U^y
xy
-2i/»
6a6c» —
13x2 1/
— 5a:t/
ly"
— ahc"
lOx^y
Ixy
-31/^
— Sabc' —
^r^y
— 2x1/
Uy"
4a6c2
24x^3/
4x1/
Uabc"
(6.)
(T.)
25x1/2
8(x + i/)
—
- 60 X 1/ z
-4(x + i/)
10x1/2
n^ + y)
—
-6txi/2
-3(x + i/)
8x1/2
- (^ + 2/)
— 74x1/2 T (x + y)
8. Find the sum of 8 xV, — Hx^y", ITx^y*, and — x^^.
9. Find the sum of T(x+y), 8(x + y), — (x + y),
and 4 (x + y). Ans. 18 (x + y).
10. Find the sum of — a x^, + a x*, — 10 a x^ + 25 a x^,
and — 13ax2.
11. Find the sum of 21 a b, — 34 a i, — 150 ab, 21 a b,
and — 13 a i. Ans. — 143 a b.
12. Find the sum of ax*, — 14 a x", IT ax*, — ax*,
44 a X*, and — a x*.
13. Find the sum of 17 (a + b), — (a + b), (a + b),
and — 13 (a + 6). Ans. 4 (a + 6).
22 ELEMENTARY ALGEBRA.
CASE III.
40. To find the sum of any algebraic quantities.
The sum of 5 a and 6 & is neither 11a, nor \lh, and
can only be expressed in the form of b a -\- Qb, or 6 5
-j- 5 a ; and the sum of 5 « and — 45 is 5a — 45; but in
finding the sum of 5 a, 6 5, 5 a, and — 4 5, the a's can be
added together by Case I., and the 5's by Case II., and
the two results connected by the proper sign ; thus, 5 a
4-65 + 5a — 45r=10a + 2 5.
1. Find the sum of 6 c?, —25, x, Bi/, 5 x, —3 5, 35c
+ 4 c?, 5x, T 5 + 2 X, and — 3 5 c.
OPERATION. Yor convenience, simi-
^d — 25-1- X -\- Sy -{- Sbc lar terms are written un-
4.d—Sb^6x —3 be ^^^ ^^ch other ; then by
+ Y5-|-5a:
-\-2x
Case I. the first column
at the left is added; the
second by Case II., and
10c?-l-2 5-f 13a; + 3^^ so on ;+ 35c and — 35c
cancel.
This case includes the two preceding cases, and hence
to find the sum of any algebraic quantities :
RULE.
Write similar terms under each other, find the sum of
each column, and connect the several sums with their proper
signs.
(2.)
4a;— Ya-t-3y — 45-t-3z
6a— ij -\- 4:b — 2z
4:a — 2i/-\-Sb— z
— 3a — 85 — 10 c
4x —10c
ADDITION. 23
(3.)
— 36+ 3c — n \/x + y
— 10c-|-8\/^ — y
7 a 4- 6— 10c + 6\/x
4. Add together *l s/~x, — 8.r, 7 x'*, — 6\/^ 4a:*.
— 8 a:, 4 a:, and 7 a:'*, — ^ \^ x.
Ads. 18 ar*— \2x — *l\/x.
5. Add together Zax — 4a6 + 2a7y, *l ah -\- bx — 4a,
*l xy — 3 a + 4 X, and -{- ahc — ax -\- 6a: y.
6. Add together 7a: — 3ay — 5a6-|-4c, 3aa: + 4ar
-}- 5 a i — 5 c, and 3c — 3ax-f- *?y4"<^-
Ans. llx — 3ay + 3c-|-7y.
7. Add together 5 a — 32 + 7x + 4aa: — 3a^ bah
— 5a + 22 — 4aa: + 4, and 6 — 2 a6 + 3a: + 4y -f
4 ax.
8. Add together ^xy -\- Qxz — 6mn-4-4n, 4mn —
3xy-|-2n — 8m n, — 6a:z + 47i — 3a:y4"6, and 10 win
— lOn + 3 — 9. Aim. 0.
9. Add together 8am+19na:— 55 3 + c, — 19u +
14 6 — 16c+y, and 18 n a: — 44 am + 15 u — 4y.
10. Add together 17 aa:« + 19 aar^ — 14 aa:* + 16 aa:^,
13 ax' —5 ax* + 6 aar» — 10 ax\ and Uax* + 17 a x^
— 3ax'^+15ax^ Ans. 71 ax^ + lOax"** — 5ax\
11. Add together m -^ n — 4a + 6c — 7y, 8c — 4m
+ 3n — 5a + 3c, 7a— 17c + 7y— 10m— 6n, and
14n — 8a — 7c+ lOy — 8 m.
12. Add together 8axy+17 6xy — 16cxy — 9axy,
16 6xy — 18 cxy + lOaxy — 14 ax 2, 16cxy + 25axy
— 7 6xy + 25cxy, and lOaxar + Zhxy — lOcxy +
4ax2. Ans. 34axy + 296xy — Sexy.
24 ELEMENTARY ALGEBRA.
13. Add together S{x -\- i/), — 4:{x -\- y), and 1 (x -{- y).
Ans. 6 (x -f- y).
14 Add together 5 (2 x + y — 3 z), and —2(2x-\-y
— 3z).
Note. — If several terms have a common letter or letters, the
sum of their coefficients may be placed in parenthesis, and the com'
mon letter or letters annexed; thus,
6a:-f8a; — 5ic=(6 + 8 — 5)a:;
ax -^ dh X — 2 c X = (a -\- ^h — 2c)x;
hcxy-\-adxy — ac xy = {h c -\-ad — a c) xy.
16. Add together ax — bx -\- Zx, and 2ax -\- 4:bx — x.
Ans. (3a + 36 + 2) x.
16. Add together by — Scy -\-6ay, and cy -\-4:by
— 2 ay.
11. Add together 2xy — axy, and 6xy — Saxy.
Ans. (8 — 4 a) xy.
18. Add together 1{Bx-{-5y)-\-Sa — 6x + Sab, 3x
+ 5 (3a: +5y) -{-la —bab, and 8x +2(Sx -\- 5y)
^la — Sab. Ans. 4Yx+ TOy + 3a.
41. From what has gone before, it will be seen that
addition in Algebra differs from addition in Arithmetic.
In Arithmetic the quantities to be added are always con-
sidered positive ; while in Algebra both positive and neg-
ative quantities are introduced. In Arithmetic addition
always implies augmentation ; while in Algebra the sum
may be numerically less than any of the. quantities added ;
thus, the sum of 10 a: and — 8 a; is 2x, which is the
numerical difference of the two quantities.
SUBTRACTION. 25
SECTION V.
SUBTRACTION.
42. Subtraction in Algebra is finding the difference
between two quantities.
1. John has 6 apples and James has 2 apples ; how
many more has John than James ?
Let a represent one apple, and we have
6a
2a
4a
., or 6a — 2a = 4a.
2. During a certain day A made 9 dollars and B lost
6 dollars ; what was the difference in the profits of A
and B for the day ? If gain is considered -|-> ^^^^ loss
must be considered — , and letting d represent one dol-
lar, it is required to take — 6 c? from 9 d.
OPERATION.
9rf
— 6d
It is evident that the difference be-
tween A's and B's profits for the day
is 9 c? + 6 c/ = 15rf; that is, 9 d —
I5d (-6<0 = 9rf+6rf=15rf.
Hence it appears that, as addition does not always im-
ply augmentation, so subtraction does not always imply
diminution.
Subtracting a positive quantity is equivalent to adding an
equal negative quantify; and subtracting a negative quan-
tity is equivalent to adding an equal positive quantity.
Suppose I am worth $1000; it matters not whether a
thief steals $400 from me, or a rogue having the author-
ity involves me in debt $400 for a worthless article ; for
2
26 ELEMENTARY ALGEBRA.
in either case I shall be worth only $600. The thief sub-
tracts a positive quantity ; the rogue adds a negative quan-
tity.
Again, suppose I have $1000 in my possession, but
owe $400 : it is immaterial to me whether a friend pays
the debt of $400 or gives me $400 ; for in either case I
shall be worth $1000. In the former case the friend
subtracts a negative quantity ; in the latter, he adds a pos-
itive. Or, to make the proof general :
1st. Suppose + 5 to be taken from a -\- h
the result will be a ;
and adding — h to a -\- h v7Q have a -\- h — h,
which is, as before, equal to a.
2d. Suppose — h to be taken from a — b
the result will be a ;
and adding -\- b to a — b we have a — b -\- b,
which is, as before, equal to a.
3. Subtract b -\- c from a.
OPERATION. I subtracted from a gives
a — (b -X- c) :=. a — b — c a — b; but in subtracting b
we have subtracted too small
a quantity by c, and therefore the remainder is too great by f, and
the remainder sought is a — b — c.
4. Subtract b — c from a.
OPERATION. In subtracting b from a we
a — (b — c)=a — 5-j-c subtract a quantity too great
by c ; therefore the remainder
(a — b) would be just so much too small, and the remainder sought
is a — 6 -|- c.
43* By examining the examples just given it will be
seen that in every case the sign of each term of the.
subtrahend is changed, and that the subsequent process
is precisely the same as in addition ; hence, for subtrac-
tion in Algebra we have the following
SUBTRACTION. 27
RULE.
Change (he sign of each term of (he subtrahend from -f-
io — , or — to -f-> or suppose each to he changed, and then
proceed as in addition.
(!•)
(2.)
(3.)
(4.)
(5.)
(6.)
0-)
Min.
9
9
9
9
9
9
9
Sub.
9
6
3
— 3
— 6
— 9
Rem. 3 6 9 12 15 18
In examples 1-7, the minuend remaining the same while
the subtrahend becomes in each 3 less, the remainder in
each is 3 greater than in the preceding.
(8.) (9.) (10.) (11.) (12.) (13.) (14.)
Min. 9 6 3 0—3—6—9
Sub. 9 9 9 9 9 9 9
Rem. —3 —6 —9 —12 —15 —18
In examples 8-14, the minuend in each becoming 3 less
■while the subtrahend remains the same, the remainder in
each is 3 less than in the preceding.
(15.) (16.) (17.) (18.) (19.) (20.) (21.)
Min. 9 6 3 —3 —6 —9
Sub. 9 6 3 —^ — ^ —9
Rem.
In examples 15-21, both minuend and subtrahend de-
creasing by 3, the remainder remains the same.
(1.) (2.) (3.) (4.) (5.) (6.)
Min. 26x 27axy — 13a6 —18c 49xy —4386
Sub. 10a: — 4axy 4a6 — 6c — 25xy 27 6
Rem. 16x Zlaxy —llab — 12tf
28 ELEMENTARY ALGEBRA.
(1.) (8.) (9.) (10.) (11.) (12.)
Min. lOx — 4:axy 4,ah — 6c — 2bxy . 21h
Sub. 26x 21axi/ —ISab — > 18 c 4:9x1/ —438^
Rem. — 16a: — Slaxi/ 11 ab 12c
(13.) (14.)
Min. 6 a:— 14^ + 3 2r 1 a+18b — 10c
Sub. 3a:+3y+;2 -_25 5+6c — 8</
Rem. Sx—l1t/-j-2z 1a-\-4:3b — 16c + Sd
15. From 28 a: take — 11 x. Ans. 45 a:.
16. From — 34Y a take 223 a. Ans. — 570 a.
11. From — T6y take — 33 y. Ans. — 43 y.
18. From Ub take —150 5. Ans. 194 5.
19. From — 411 c take 984 c.
20. From — 84;^ take — 117 z.
21. From 17 a a: — 18 5c + 44a:y take 25 6c — 14 a: 3^
4- 20 a. Ans. 17 a a: — 43 J c + 58 a:y — 20 a.
22. From 384 a:— 74y-f- 18 c take 118 a: + 743^ — 27 c.
Ans. 266 a:— 148 3/ + 45 c.
23. From x^ — f + x^ — 10 a^ take x'^ + 4.f — x^-^
4.7?. Ans. 2a:* — 6a:^ — 5y.
24. From Qaby — 4a:y-j-3a:2; take — 4,aby — 3a:;3
<— 4 a: y.-^
25. From a:^ -f 2 a: ^^ + ^2 ^^^ ^"^ _ 2xy -{- y"^.
26. From x^ -\- 2 x y -{- y"^ take —x^-\-2xy — y\
44. The subtraction of a polynomial may be indicated
by enclosing the polynomial in a parenthesis and prefix-
ing the sign — .
Thus, a? -\- i^ — z^ taken from a:' — s? may be written
SUBTRACTION. 29
When a parenthesis with the sign minus be/ore it is re-
moved, the sign of each term within the parenthesis must be
changed according to the Bute for subtraction.
Thus, ar^ — 23_(^_j_y»_2«) = 2r'' — z3_^_^_j.
And conversely,
A polynomial, or any number of the terms of a polyno-
mial, can be enclosed in a parenthesis and the minus sign
placed before the parenthesis without changing the value of
the expression, prodding the signs of all the terms are
changed from plus to minus or from minus to plus.
Thus, a' ^ Ir + c" + d — X = a^ — {ty" — c"" — d -j- x).
Note. — When the sign of the first term in the parenthesis b
plus, the sign need not be written. (Art. 18.)
According to this principle a polynomial can be writ-
ten in a variety of ways.
Thtis, a:*— 3 a:2y4- 3 a:y2— y' = ar' — (3 x2y — 3 a:y2 -{_ y3)
= ar^ + 3x7f-(3x'f/-\-^')
Remove the parenthesis, and reduce each of the follow-
ing examples to its simplest form.*
1. a^ — (2ab-{- c^). Ans. a^ — 2 a ^ — c«.
2. x^ — 6ax -\-r^ — 6x^y—{x'-\-6ax + 2^ — ex^y).
Ans. — 12 ax,
3. m^ — n^-\-2x—(4:m^-\-'Sn^ — 4:c).
4. 16a:y+ Uc — lSy— (— Uc + 27y — 16a:y).
Ans. 32xy + 28c — 46 y.
6. 4:3^1/— (Sxf—lx'f + Sa^i/),
6. — (-a:«+7-25xy + /).
80 ELEMENTAEY ALGEBRA.
Place in parenthesis, with the sign — prefixed, without
changing" the value of the expression,
1. The last three terms of 1 x'^ — 14 xy — Ssr + ^y.
Ans. 1 x''—{Uxi/-\-Sz — 4.y).
2. The last three terms of x^ ■}- y^ — Zxy -\- ^c.
Ans. x'^ — i^xy — y^ — 4 c).
3. The last four terms oi4.a — 1h — Qc — ^d-\-x'^.
4. The last four terms of a^ -\. h^ -\- c"" — d"" + a\
5. Write in as many forms as possible by enclosing
two or more of the terms in parenthesis, a^ — b^ -f~ ^^
^d\
45. In subtraction, when two quantities have a com-
mon factor their difference is the difference of the coef-
ficients of the common factor multiplied by this factor.
Thus, ax — hxr=z {a — V) x.
1. From ax^ take cx^ — dx^. Ans. (a — c -\- d) x^.
2. From 4 \/ a; take a\/ x -{- h^/ x.
Ans. (4 — a — h) \/ x.
3. From ax^ take hx^ — hx^. Ans. {a — h) x^ -\- h x^.
4. From 4 a:^ — Qx take ax"^ -\- hx.
Ans. (4 — a) x^ — (6 -|- b) x.
6. From 6 a^ -|- 4 a^ — a take a^ x — a^y -\- az.
Ans. (6 — x)a'-{- (4 +y) a" — {I + z) a.
6. From ah — he take ^h -\- ex.
1. From a"^ — bx -\- c a^ x take hx^ -\- c x — d^J x.
8. From xy'^-^x' — x^y'' take y"" + x'^y — x'^y^.
MULTIPLICATION. 81
SECTION VI.
MULTIPLICATION.
46* Multiplication is a short method of finding the
sum of the repetitions of a quantity.
47. The multiplier must always be an abstract num-
ber, and the product is always of the same nature as the
multiplicand.
The cost of 4 pounds of sugar at 17 cents a pound is
n cents taken, not 4 pounds times, but 4 times ; and
the product is of the same denomination as the multi-
plicand 17, viz. cents.
In Algebra the sign of the multiplier shows whether
the repetitions are to be added or subtracted.
1. (+a)X(+i) = + ia;
i. e. -\- a added 4 times is -\-a-{-a-{-a-\-a = -\-4ia.
2. (+a)X(— 4) = -4a;
i. e.-\- a subtracted 4 times is — a — a — a — a = — 4 a.
3. (-a)X (+4) = -4a;
i. e. — a added 4 times is — a — a — a — a = — 4 a.
4. (--a)X(— 4) = + 4a;
i.e. — a subtracted 4 times i3-{-a-\-a-\-a-{-a = -\-4:n.
In the first and second examples the nature of the
product is -j- ; in the first, the -[" sign of 4 shows that
the product is to be added, and + 4:a added is + 4a;
in the second, the — sign of 4 shows that the product
is to be subtracted, and + 4a subtracted is — 4a. In
the third and fourth examples the nature of the product
ij — ; in the third, the + sign of 4 shows that the prod-
uct is to be added, and — 4 a added is — 4 a ; in the
82 ELEMENTARY ALGEBRA.
fourth, the — ■ sign of 4 shows that the product is to be
subtracted, and — 4 a subtracted is -\- 4:a.
48. Hence in multiplication we have for the sign of
the product the following
RULE.
Like signs give -\- ; unlike, — .
Hence the products of an even number of negative fac-
tors is positive, of an odd number, negative.
49. Multiplication in Algebra can be presented best
under three cases.
CASE I.
50. When both factors are monomials.
. 1. Multiply 3 a by 2 5.
OPERATION.
3aX25 = 3X«X2x^ — 3X2X«X^ = 6aJ.
As the product is the same in whatever order the factors are
arranged, we have simply changed their order and united in one
product the numerical coefficients.
Hence, when both factors are monomials,
RULE.
Annex the product of the literal factors to the product
of their coefficients, remembering that like signs give -|-
and unlike, — .
2. Multiply a^ by a\
OPERATION.
a^X«^=(«X«X«)X(«X«) = «XaX«X«X« = a^
As the exponent of a quantity shows how many times it is taken
as a factor, a' = a X « X « I and a^ = a y, a', and a^ X «^ = ot X
a X a X « X «» and this is equal to a^ (Art. 24.) Hence,
Powers of the same quantity are multiplied together by
adding their exponents.
(3.)
Zah
(4.)
5x^y\
Zxtf
15 rV
MULTIPLICATION.
(5.) (6.)
7 a 6 _- 14 wn^
— %an 6an*
33
a.)
— 40=6
I2abxy
— 66 a^ i^
— 84 a m n'
4a<^»^
8. Multiply x^ by ar*.
9. Multiply x^ by a/.
10. Multiply a^ by — a.
11. Multiply —a* by ct".
12. Multiply — c« by — c^
13. Multiply 8 a:/ by Taar.
14. Multiply 504 a^ fr'' by — 8 a« h.
15. Multiply — 25a:y2 by AiXyz,
16. Multiply — 411 a 6 c2 by — 3 a 5» c.
17. Multiply together 444 xy, 3^^/, and — 2 2;.
Ans. — 2664ar5y*2r.
18. Multiply 4 a2 ^2 ^ c? by — 4 a J c^ rf2.
19. Multiply — 5 a:^ by — 6 a:*. Ans. 30 ar^+".
20. Multiply together 14 a 6 c-, — 5 d^ h c, and — 4 a J'^.
21. Multiply 25 \/ay by 3 6a:. Ans. 1bhx\/ay.
22. Multiply 4 (a: + y) by 3 (x + y).
Ans. 12(x + y)-.
Note. — Any number of tenng enclosed in a parenthesis may
be treated as a monomial.
23. Multiply — 12 (a^ — ^) by — 4 {a" — P).
Ans. 48 (a« — H^K
24. Multiply (a — x)* by (a — x)^.
25. Multiply 4 (a + 6)"* by 2 (a + b)\
Ans. 8 (a + 6)""+".
26. Multiply a» (x + z)» by a 6« (x + «).
2* C
34
ELEMENTARY ALGEBRA.
CASE II.
51* When only one factor is a monomial
1. Multiply 8 + 5 by 3.
OPERATION.
8+ 5
8 +
5]
8 +
5
8 +
5
24 +
15.
I or
13
3
24 + 15 = 39
2. Multiply 8 — 5 by 3.
In this example, not the
sum of 3 repetitions of 8
only, but of 8 and 5, is re-
quired ; the sum of 3 repeti-
tions of 8 = 24 ; of 3 repeti-
tions of 5 = 15. Hence, the
sum of 3 repetitions of 8 -[-
5 = 24 -f- 15.
OPERATION.
8 —
6]
8 —
5
8 —
5
24 —
15
- or
— 5=3
3 3
24
15 =9
The sum of 3 repetitions
of 8 = 24 ; but it is not the
sum of 3 repetitions of 8 that
is required, but of a num-
ber 5 units less than 8 ; 24,
therefore, will have in it the
sum of 5 units repeated 3
times, or 15, too much; the product required, therefore, is 24 — 15.
Therefore,
The product of the sum is equal to the sum of the prod-
ucts, and the product of the difference to the difference of
the products.
3. Multiply X -\- 7/ — z by a,
OPERATION.
X U- y z "^^^ ^^^ ^^ *^® repetitions
of a; a times, of y a times, and
— . . - of — z a times is a x -\- ay
- az.
ax
ay
az
RULE.
Multiply each term of the multiplicand by the multiplier,
and connect the several results by their proper signs.
MULTIPLICATION. 35
(4.) . (5.)
3ax — 4a:
12 ao:' — 24 aar' + 42 aary — 4 a^x + 8 a 6a; — 4 ^r'a:.
6. Multiply 5 rn n + 4 m'' — 6 n^ by 4 a 6.
7. Multiply IGa^ar — 8arz + 4y by —Zxy.
8. Multiply 6a^ — c x= + rfa: by — ar*.
9. Multiply — 63 a:y — 14 a: — 6 ^ by — 4 z.
Ans. 252 ary z + 56 X z + 24 r».
10. Multiply 14 a* — 13 a'' + 12 a^ — 11 a by 4 a**.
11. Multiply a: — 2 a + 14 by a x.
12. Multiply 17aa: — l-kby -\- llcz by — 4:abcxyz.
13. Multiply 21 aH^ — 3 x/ _ 4 5 c by — 9 a a;y.
CASE III.
52« When both factors are polynomials.
1. Multiply 7 + 4 by 5 — 3.
OPERATION. Multiplying 7 -f 4 by 5 is
»y I 4 =11 taking the multiplicand 3 too
R q o many times ; therefore, the
true product will be found by
35 + 20 — 21 — 12 =22 subtracting 3 (7 -f 4) from
5(7+4).
2. Multiply X — y by a -\- b.
OPERATION. a times x-^y = ax — ay\ but
X y X — y is to be taken, not a times
^ I J only, but a-\-b times; therefore,
a (a: — y) is too small by ft (a: — y)\
ax — a y + 6 X — by and the product required is a x —
ay -\-hx — by. Hence,
RULE.
Multiply each term of the multiplicand by each term of
the multiplier, and find the sum of the several products.
86 ELEMESTAEY ALGEBEA.
(3.) . (4.)
a^ -\- 2 a X -\- x^ x^ -\- a b
a — X xy — ah
c^ -\-2'C?x-\- ax^ 7?y^ '\- ahxy .
— a^ X — 2ax^ — x^ — abxy — a^y^
a^ -\- a^ x — ax^ — x^ x^y"^ — a^ IP-
5. Multiply a2 _|_ j2 _ ^2 ^^ ^2 _j_ ^2,
Alls, a" + a^ 52 + 52 c" — c\
6. Multiply x"^ — 2xy -\- y"^ by x"^ -\- y'^.
Ads. x^ — 2x^y^2 x'y'' — 2xy^ + y\
1, Multiply 4 a^ — 2 an + 3 a2 52 by 2a'' — 2b\
Ans. 8 a^ — 4 a^ — 2 a* ^>2 _|. 4 ^3 J3 _ g ^2 54^
8. Multiply x^ + 2x^ + Sx''-]-2x+lhjx'' — 2x+l.
9. Multiply x^-{-y^-\-z^ — xy — xz — yz hy x -\-y -\-z,
Ans. x^ + 5/^ + 2'^ — 3 X y z.
10. Multiply 4 a;5 — T ic^ + 10 ar' — 13 a;2 by 3 a: — 2.
11. Multiply a^ + a2 — 1 by a" — 1.
12. Multiply x^+lax—Ua^ by a.- —la.
Ans. ar^ — 49a2a:— 14a3a; + 98a*.
13. Multiply X -\- y — a by x — y -{- a.
14. Multiply a" + 5'* by a"* — b"".
Ans. a "'+" + a*" 6" — a" 6"* — 5'" + ^
15. Multiply Ixy— 14. x" y"^ -\- 21 x"" f by Gary — 3.
Ans. — 21 xy + Ux"" f — 14.1 a? f + l2Qx^y\
16. Multiply 6 an — 9 a S2 __ 12 aH^ by 2 a & — 3 5^.
Ans. 12 a^ }? — 36 a^ 6^ _ 24 a^ Z.» + 2Y a i* + 36 a" b\
V\. Multiply x^ — x^ -\- x^ — x -\- \ by a; + 1.
Ans. x^ + 1.
18. Multiply x^ — x^-\-x} — x-\-\ by a: — 1.
Ans. x^ — 2x''^2x^ — 2i?-\-2x—\.
19. Multiply X* + ar^ + a;2 + a: + 1 by ar + 1.
20. Multiply a;* + a^ + a:^ + a; + 1 by cc — 1.
DIVISION. 87
SECTION VII.
DIVISION.
53. Division is finding a quotient which, multiplied by
the divisor, will produce the dividend.
In accordance with this definition and the Rule in
Art. 48, the sign of the quotient must be -|- when the
divisor and the dividend have like signs ; — when the
divisor and the dividend have unlike signs ; i. e. in di-
vision as in multiplication we have for the signs the fol-
lowing
RULE.
Like signs give -f- > unlike, — .
CASE I.
54* When the divisor and dividend are both monomials.
1. Divide 6ab by 2b.
orERATiON. Thg coefficient of the quotient must
6ab-7-2b = Sa be a number which, multiplied by 2,
the coefficient of the divisor, will give
6, the coefficient of the dividend ; i. e. 3 : and the literal part of
the quotient must be a quantity which, multiplied by &, will give ab\
i. e. a: the quotient required, therefore, is 3 a.
Hence, for division of monomials,
RULE.
Annex the quotient of the literal quantities to the quotient
of their coefficients, remembering that like signs give -\- and
unlike, — .
38 ELEMENTARY ALGEBRA.
2. Divide a^ by a\
OPERATION.
a« -T- a^ = a^ For a' X a' =^ a^ (Art 50.) Hence,
Poivers of the same quantity are divided by each other by
subtracting the exponent of the divisor from that of the
dividend.
(3.)
21x^y''
9 xy
(5.)
•^2lQa?y
4.Qx''y
3xy
= — 6a;
(4.)
4:8a^xy
— 16aa;
^-S
ay
(6.)
— 16a2
x^yz__
— 4 axyz
*
f-
Ans. —
■Sxy.
2iahx.
Ans.
6ab.
1. Divide S4:aby^ by 2 ay.
8. Divide 297 icV by — 99a:y2.
9. Divide — 14,xy^z by 2a:2/.
10. Divide —UiaH^x by — 24«5a:.
11. Divide ax* by czar'.
12. Divide 8 a:« by — 8 x\
13. Divide — 210a;''y by ^2 x^ y.
14. Divide —210abx by — 135 a i a;.
15. Divide —4t14:aH^c^ by 158a53c.
16. Divide x"" by ar". Ans. a:"""".
11. Divide 14 a"* a:" by — 1 a"" x. Ans. — 2 a"* ""a;"
18. Divide — UT a^ b* c^ d' by SSaHc d\
19. Divide 12 (x -\- yf by 4.{x-\-y). Ans. 3 (a: + 2^).
20. Divide —21 {a — by by 9(a — 5)^.
21. Divide {b — cy by (b — cy. Ans. (5 — cf.
22. Divide 14 (x — yf by t (a: — yy.
— 1
DIVISION. 39
CASE II.
55» When the divisor only is a monomial.
I. Divide ax -{- at/ -{- az by a.
OPBRATiox. In the multiplication of a poly-
a)ax-^ay-\-az nomial by a mononiial, each term
jp I -. I ^ of the multiplicand is multiplied
by the multiplier ; and theiv fore
we divide each term of the dividend ax -\- ay -\' az by the divisor
a, and connect the partial quotients by their proper signs. Uence,
RULE.
Divide each temi of the dividend by the divisor, and con-
nect the several results by their proper signs.
(2.) (3.)
3 a ) 6 a x^ — 24: aa? — 5 a:« y ) — 15 ai^y — 25 ar^y
2 2r»— 8ar» 3x + 5
4. Divide 12 a ar* — 24 a a:^ + 42 a a:y by 3 ax,
6. Divide — 4:a^x-\-Sabx — 4 6^ar by — 4a:.
6. Divide 6 a^x* — 12 a^x^ + 15 a*ar« by 3 a^x\
Ans. 2 a;* — 4 a x -[- 5 a* a:*.
1. Divide 12a*y« — 16 a^y* + 20 a«y< — 28 aV by
4aV.
8. Divide — 6 ar» + 10 x^ — 15 x by — 5 x.
9. Divide 273 (a + xy — 91 (a + x) by 91 (a + x).
Ans. 3(a + x) — l = 3a + 3x— 1.
10. Divide 20a6c — 4ac -f 8acd— 12 a^ c^ by -^ 4: a c.
II. Divide 16 a^x*— 32«2x^ + 48 «*x* by 16 a* x*.
12. DivideT2x«y« — 36x«y»~64x»y«2«by — 18x2y^
Ans. --4 + 2x?/+3x2«.
13. Divide 18ax»y— 54x^2/*+ 108cx^y« by 9x*y.
14. Divide 40 a*b^ + 8 a' 6^ — 96 a» i«x« by 8 a* b\
15. Divide 39x*2« — 65aa:«2* + ISx'z* by — ISx^z^.
40 ELEMENTARY ALGEBRA.
CASE III.
56. When the divisor and dividend are both polyno-
mials.
1. Divide :t? — Zx'y -\-Zxy' -^ f by x^ — 'lxy-^- y^.
OPERATION.
7? — 2 a; ^ -f- ^^) ic^ — Zx^ y -\- ^xy^ — ^ (a^ — y
a^ — 2 x'^ y -\- X y"^
— x"^ y -{- 2 X y^ — if
-- x''y + 2xy^^f
The divisor and dividend are arranged in the order of the powers
of X, beginning with the highest power, a;^, the highest power of x
in the dividend, must be the product of the highest power of x in
x^
the quotient and x^ in the divisor ; therefore, —^ = x must be the
highest power of x in the quotient. The divisor x^ — 2xy -\- y^ mul-
tiplied by X must give several of the partial products which would
be produced were the divisor multiplied by the whole quotient.
When (x' — 2xy -\- if) X = x^ — ^x"^ y -\- xif is subtracted from
the dividend, the remainder must be the product of the divisor and
the remaining terms of the quotient ; therefore we treat the remain-
der as a new dividend, and so continue until the dividend is ex-
hausted.
Hence, for the division of polynomials we have the fol-
lowing
RULE.
Arrange the divisor and dividend in the order of the
powers of one of the letters.
Divide the first term of the dividend by the first term of
the divisor ; the result will be the first term of the quotient.
Multiply the whole divisor by this quotient, and subtract
the product from the dividend.
Consider the remainder as a new dividend, and proceed
as before until the dividend is exhausted.
DIVISION. 41
Note. — If the dividend is not exactly divisible by the divisor, the
remainder must be placed over the divisor in the form of a fraction
and connected with the quotient by the proper sign.
2. Divide a:* + 4y* by a:« — 2ary + 2y'.
x*—2x'y
+ 2^1/'
2x>y
4y
ixf
2x»y' —
2r'y» —
ixy' +
4xy» +
4y
4y
Note. — By multiplying the quotient and divisor together all the
terms which appear in the process of dividing will be found in the
partial products.
3. Divide a:* — 1 by a: — 1.
X — 1) X* — 1 (x^' + x' + x+l
x»
—
1
• 3^
—
7^
x"
—
1
3?
—
X
X
—
1
X
1
4. Divide ax — ay -{-hx — by -\- z by x — y,
ax — ay
ax — ay
x^y)ax-^ay-\-hx^hy-\-z{a + h-\- j-^
hx — hy
hx — hy
5. Divide 2by^2}ry — Zlryz-\-Qb^y-\-byz—yz
by 2 6 — 2. Ans. 3 h^y — by-{-y.
42 ELEMENTARY ALGEBRA.
6. Divide c^ -\- x^ by c -\- x. Ans. c^ — ex -\- x^.
1. Divide a^-{-a^-{-a^x-\-ax-\-Sac-\-Schya-\-l.
Ans. a^ -\- a X -{- 3 c.
8. Divide a -\- b — d — ax — bx-\-dx by a-{- h — d.
9. Divide 2 a* — 13 a^ 2/ + 11 a^y^ _ g « ?/ -f 2 ?/ by
2 or — ciy -\- y'^. Ans. a^ — 6 a y + 2 ?/l
10. Divide a^ — Z an -\- Z ah'' — b^ hy a^ — 2 ah -\- h\
11. Divide 2 a;3— 19x2+ 26a:— 19 by a: — 8.
Ans. 2a;2 — 3a;4-2 ?-r
' X — 8.
12. Divide ar^ + 1 by a: + 1.
Ans. X* — x^ -\- x' — a: -|- 1.
13. Divide a;« — 1 by a; — 1.
Ans. a:^ 4" ^* ~1~ ^^ + ^^ "t" ^ ~l~ 1-
14. Divide x^ — 1 by a: -j- 1.
15. Divide -x^ — ^ by x — y.
Ans. x^ -\- x^ y -\- x^ y'^ -\- X ^ -\- y^.
16. Divide a^ — x^ by a — x.
17. Divide m^ — n^ hy m' -\- m n -\- n^.
Ans. w^ — m^n -\- mn^ — n*,
18. Divide 4 a* — 9 a^ + 6 a — 3 by 2 a^ + 3 a — 1.
19. Divide a:* + 4x2^2 _|_ 3^4 |3y ^_^ 2 y.
20. Divide a^ — a^ x"" -\- 2 a a? — x"" by a^ — ax-{- x\
21. Divide x^ + 2 x^ y^ -\- y^ hy x'^ — xy + y\
22. Divide 1 — a^ by 1 + « + a^ + al
Ans. 1 — a.
23. Divide 10 x^ — 2Qx''y-\- 30 ^ by a; + y.
24. Divide 7 a a;* + 21 a x^ + 14 a by a: + 1.
Ans. 7 a a;^ + 14 a ar^ — 14 a a: -|- 14 a.
25. Divide 27 a^ y^ — 8 a^y by 3^/^ — 2 ay.
DEMONSTRATION OF THEOREMS. 4S
SECTION YIII.
DEMONSTRATION OF THEOREMS.
57. From the principleB already established we are pre-
pared to demoustrate the foUowiug theorems.
THEOREM I.
The sum of two quantities plus their difference is twice
the greater ; and the sum of two quantities minus their dif-
ference is twice the less.
Let a and b represent the two quantities, and a ^ & ; their sum
is a -\-b] their difference, a — b.
PROOF.
1st. {a + b) + {a — b)=a-{-b-\-a^b = 2a;
2d. (a-\-b) — (a — b)=a + b — a+ b = 2b.
Therefore, when the sum and difference of two quanti-
ties are given to find the quantities,
RULE.
Subtract the difference from the sum, and divide the re-
mainder by two, and ice shall have the less ; the less plus the
difference will he the greater.
In the following examples the sura and difference are
given and the quantities required.
1. 16 and 12. Ans. 14 and 2.
2. 272 and 18.
3. 456 and 84.
4. Sum 2 X and difference 2 y.
Ans. x-\-y and x — y.
5. Sum Ix'^ + Zy and difference bx'^ — Zy.
6. Sum 2 a — 8 ft and difference 10 a + 14 b.
Ans. 6 a + 3 6 and — 4 a — 11 ft.
44 ELEMENTARY ALGEBRA.
THEOREM II.
58. The square of the sum of two quantities is equal to
the square of the first, plus twice the product of the two,
plus the square of the second.
Let a and h represent the two quantities ; their sum will be
a-\-b.
PROOF.
a +h
a +b
a''+ ah
+ ab + h''
a^-\-2ab + b''
According to this theorem, find the square of
1. X -{- 1/. Ans. a:^ -j- 2 a: y -f- y •
2. 2x + 2y. Ans. 4a:2_|-8x^/ + 4/.
3. a:+ 1.
4. 4: + X.
5. 2x + St/. Ana. 4:x^+l2xy + 9f.
6. Sa-\-b.
THEOREM III.
59. The square of the difference of two quantities is
equal to the square of the first, minus twice the product of
the two, plus the squar-e of the second.
Let a and h represent the two quantities, and a ]> 6 ; their dif-
ference will be a — h.
PROOF.
a — b
a — b
a^ — ab
— ab + b^
a2 _ 2 a 6 + 63
DEMONSTRATION OF THEOREMS. 45
According to this theorem, find the square of
1. X — y. Ans. x^ — Ixy '\- y^.
2. 2x — 4y.
3. X— 1. Ans. x^ — 2a:+ 1.
4. Tx — 2.
THEOREM IV.
60. The product of the sum and difference of two quan-
tities is equal to the difference of tJieir squares.
Let a -^ b he the sum, and a — b the difference of the two
quantities a and b.
PROOF.
a +b
a —b
a-' + ab
--ab — b'
According to this theorem, multiply
1. X -\- y hy X — y. Ans. x^ — y^.
2. 2 3:+ 1 by 2a:— 1.
3. ar» + t/2 by x* — y^. Ans. x* — y*.
4. 3 X + 4 by 3 a: — 4.
6. 3xy-["4a6by3xy — 4a J.
61 • This theorem suggests an easy method of squaring
numbers. For, since a^ = {a — b) (a -\- b) -\- If^y
992 = (99 — 1) (99 + 1) + 1* = 98 X 100 + 1 = 9801.
In like manner,
96«= 92 X 100 + 16 = 9216.
998^ = 996 X 1000 + ^ = 996004.
4972 = 494 X 500 + 9 = 247 X 1000 + 9 = 247009.
46 ELEMENTARY ALGEBRA.
In accordance with this principle find the square of
1- 98. 4. 493. 7. 888.-
2. 89. 5. 789. 8. 999.
S. 45. 6. 698. 9. 1104.
Miscellaneous Examples.
1. Find the square of 3 a: — 6y.
Ans. 9x^ —Mxy-\-S6f.
2. Find the square of 4:axy -\- 7 abx.
Ans. 16a^x^y^-[- 5Qa^bx'^i/ + 4:9an^x^.
3. Multiply 1 x-\-l by T:r— 1. Ans. 49 x^ — 1.
4. Required those two quantities whose sum is Bx-\-2a
and difference x — 2 a. Ans. 2 a; and x -\- 2 a.
6. Expand (rr^ — 4)2.
6. Multiply 4:ab -}- S by 4aJ— -3.
T. Find the square of 14: aH^ + 10 a?^ 2^.
8. Find the square of 4 a — b.
9. Multiply 10 a: 4- 2 by 10 a; — 2.
10. Find the square ofSaa; — ^axy.
Ans. 9a'x^ — 4.^a''x'y-\-Ua^x^f.
11. Find the square of 2 a + 5.
12. Find the value of (6 a + 4) (6 a — 4) (36 a^ + 16).
Ans. 1296 a^ — 256.
13. Find the square of 10 a^ _ 5 52
14. Expand (3 a^a: + 4 bfy.
Ans. 9 a* a:2 _|_ 24 ^2 j^y8 _j_ jg ^2^^«^
15. Find the product of a}'' + 1, «» -f 1, a^ _|_ j^ ^2 _j_ |^
a+l'^nda— 1. An^. a^^ — I.
16. Find the product of a -[- i, a — 5, and a^ — 5^
FACTORING. 47
SECTION IX.
FACTORING.
62. Factoring is the resolving a quantity into its fac-
tors.
63. The factors of a quantity are those integral quanti-
ties whose continued product is the quantity.
Note. — In usin^ the word factor we shall exclude unity.
64* A Prime Quantity is one that is divisible without
remainder by no integral quantity except itself and unity.
Two quantities are mutually prime when they have no
common factor.
%5» The Prime Factors of a quantity are those prime
quantities whose continued product is the quantity.
66. The factors of a purely algebraic monomial quan-
tity are apparent. Thus, the factors of arbxyz are
67. Polynomials are factored by inspection, in accord-
ance with the principles of division and the theorems of
the preceding section.
CASE I.
68. When all the terms have a common factor.
1. Find the factors of ax — ah -\- ac.
OPERATION. As a is a factor of
(ax — ah -\- ac) z= a {x — 6-j-c) each term it must be
a factor of the poly-
nomial ; and if we divide the polynomial by a, we obtain the other
factor. Hence,
48 ELEMENTARY ALGEBRA.
RULE.
Write the quotient of the poly-nomial divided by the com-
mon factor^ in a parenthesis, with the common factor pre-
fixed as- a coefficient.
2. Find the factors oi ^ xy ^12 xf -\- l^ ax^ f.
Ans. Qxy{l — l2y-\-^axy^).
Note. — Any factor common to all the terms can be taken as well
as 6 a; y ; 2, 3, a:, ?/, or the product of any two or more of these quan-
tities, according to the result which is desired. In the examples
given, let the greatest monomial factor be taken.
3. Find the factors of a: -|- x"^. Ans. x {\ -\- x).
4. Find the factors of 8 a^ a;^ + 12 a^ ar^ — • 4 a a:y.
Ans. 4 a a; (2ax -\- S a^ x^ — y).
6. Find the factors of 5 x^ y'^ + 25 ax^ — 15 x^f.
Ans. 5 a^ (xy"^ -\- 5 ax^ — 3 y).
6. Find the factors of 7 a a: — %hy -^-l^x^.
T. Find the factors of 4 x^ y^ — 28 x'' y'^ — 44 ?:* y'^.
8. Find the factors of 55 a^ c — 11 a c -\- ^^ a^ c x.
9. Find the factors of ^^ a" x^ — 2^1 a^ x"" y\
10. Find the factors o^ Ibd'^ cd — ^ ah'' d^ -\-l%a^(?d\
CASE II.
69. When two terms of a trinomial are perfect squares
and positive, and the third term is equal to twice the
product of their square roots.
1. Find the factors oi a" + 2 ah -\-h\
OPERATION. Y^Q resolve this into
a^ -\- 2 ah -\- y^ z= {a -\- h) {a -\- h) its factors at once by
the converse of the
principle in Theorem II. Art. 58.
FACTORING. 49
2. Find the factors of a' — 2 a J + ^.
OPERATION. 'VVe resolve this into
a* — 2 a i + 6' = (a — b) (a — b) its factors at once by
the converse of the
principle in Theorem IIL Art 59. Hence,
RULE.
Omitting the term thai is equal to twice the product of
the square roots of the oUier two, take for each factor the
square root of each of the other two connected by the sign
of the term omitted.
3. Find the factors of x* — 2a;y +5^.
Ans. (x—y) (x — y).
4. Find the factors of 4: a^ c^ -{- 12 acd + 9 d\
Ans. (2ac + 3d) (2ac-}-3d).
6. Find the factors of 1 — 4: x z -\- 4: x^ z^.
Ans. (1 — 2xz) (1 — 2x2).
6. Find the factors of9a:2_g^_j_l
Ans. (3a:— 1) (3a:— 1).
1. Find the factors of 25 a:* + 60 a; + 36.
8. Find the factors of 49 a^ — 14 a a: + ar».
Ans. (la — x) (la — x).
9. Find the factors of 16^2—16 0*^ + 4 0*.
10. Find the factors of 12 ax + 4ar' -f 9 a^
11. Find the factors of 6 x + 1 + 9 x».
CASE III.
70. When a binomial is the difference between two
squares.
1. Find the factors of a^ — ^.
OPERATIOX. We resolve this into its fac-
q2 if^ z=z (a -[- b) (a b) ^^ ** ^"^^ ^^ ^'*® converse of
, the principle in Theorem IV.
Art 60. Hence,
S D
50 . ELEMENTAEY ALGEBRA.
RULE.
Take for one of the factors the sum, and for the other
the difference, of the square roots of the terms of the hi-
nomial.
2. Find the factors of oiP — y^.
Ans. (x+y) (a; — y).
3. Find the factors of 4 a^ _ 9 b\
Ans. (2a + 3 52) (2 a — 3 J^).
4. Find the factors of 16 x"^ — c^.
5. Find the factors of a^^^c^ — oc^y^.
6. Find the factors of 81x^ — 49/.
1. Find the factors of 25 0^ — 4 c*.
8. Find the factors of m^ — w^*^.
Note. — When the exponents of each term of the residual factor
obtained by this rule are even, this factor can be resolved again by
the same rule. Thus, x^ — ?/* = (oi^ -\- y^) {a? — y^) ; but 7? — ^ =
(x -f- 2^) {x — y) ; and therefore the factors of x^ — y^ are a:^ -|- ^,
X -\-y, and x — y.
9. Find the factors of a^ — h\
Ans. (a2 -f 6=2) (« + h) {a — h).
10. Find the factors of a;^ — y'^.
Ans. {x^ + y') (x^ + f) {x + y) (a: - y).
11. Find the factors of a* — 1.
12. Find the factors of 1 — x^.
Ans. (1 + x') (1 + a:2) (1 _|_ ^) (i __ ^y
13. Find the factors of a' — a^.
Ans. a^{a-\- 1) (a — 1).
14. Find three factors of x^ — x^.
71. Any binomial consisting of the difference of the
same powers of two quantities, or the sum of the same
odd powers, can be factored. For .
FACTORING. . 51
I. T?ie difference of the same powers of tv30 quantities
is divisible by the difference of the quantities.
Ivct a and b represent two quantities and a ]> 6, and by actual
division we find
^ = -' + -^ + ^'
and so on.
II. The difference of the same even powers of two quan--
tities is divisible by the sum of the quantities.
a -f 6
0* — 6*
= a — h,
a +6
~~^ = a^ — aH -\- a^V" — a^l^ + ah'' — ¥,
and so on.
It follows from the two preceding statements that
The difference of the same even powers of two quantities
is divisible by either the sum or the difference of the quan-
tities.
III. TTie sum of the same odd powers of two quantities is
divisible by the sum of tJie quantities.
"^^^ = a* — a«6 + a«i» — a P+ 5*,
a -{- b ' ' '
^^ = a« — a* J + a*i^ — a« J^ + a*6* — a^ + 5^,
and so on.
52 ELEMENTARY ALGEBRA.
1. Find the factors of oc^ — y^.
OPERATION.
(x» — 7/) --- (x -^ ij) = x^ + of'y + c^y^ -{- xf + y*
By I. of this article, the difTerence of the same powers of two
quantities is divisible by the difference of the quantities; therefore
X — 7/ must be a factor of x^ — y^; and dividing x^ — 1/ by x — y
gives the other factor x^ -^ x^ y -\- x^ y^ -\- x 1/ -\- y*.
2. Find two factors of c° — d^.
OPERATION.
(c^ — d') -i- (c + d) =zc'-^c^d + (^d^ — c^d^ + cd^-^d^
By 11. the difference of the same even powers of two quantities
is divisible by the sum of the quantities ; therefore c -\- d must be a
factor of c® — r/"; and dividing c® — r/* by c -\- d gives the other
factor c' — c*d-}- c' d^ — c-'^ d' -^ c d' — d\
3. Find the factors of m^ -\- n^.
OPERATION.
{m^ -|- n^) -T- {m -\- n) =:^'m^ — m^ n -\- m^'nP' — mn^ -\- n^
By III. the sum of the same odd powers of two quantities is
divisible by the sum of the quantities ; therefore m-\- n must be a
factor of m'^ -j- n^ ; and dividing irv' -\- n^ by m -\- n gives the other
factor 771* 77l' 71 -|- 771^ f? — 771 71^ -|- 7i*.
4. Find the factors of c^ — 7?.
Ans. (a — x) {c? -\- a X •\- :ji?) ,
6. Find the factors of a^ -\- x^.
Note. — In Example 2, the factors of c* — rf' there obtained are
not the only factors; for by I. c' — d^ is divisible by c — c/; and
dividing c* — d^ by c — d gives another factor,
c^ -\. c' d ^ (^ d^ ■\- (? d' ■\- c d^ ■\- d'-,
or by Art. 70,
c« — d!« « (c« + rf») (c» — d»).
FACTORING. 53
But c* — c*rf-[-c»6/«--c»(i»-t-crf* — rf»,
c^-^c*d-\-c'd'-{-c'd'-{-cd*-^d\
are not prime quantities ; for the first can be divided by c — c?, and
the quotient thus arising can be divided by c* ± c J -j- t/' ; the second
can be divided by c -j- </, and the quotient thus arising will be the
same iis after the division of the first quantity by c — </, and can
be divided by c* ± c </ -j- </'; the third can be divided by c -f- d, and
the fourth by c — d. Performing these divisions, by each method
we shall find the prime factors of c* — </' to be
c 4- J, c — (/, c' + c (/ -f- ^/», and c« — c (/ + d\
In finding the prime factors, it is better to apply first the princi-
ple of Art 70 as far as possible.
6. Find the prime factors of x^^ — y^^.
Ans. {x+y)(x—y) (x^ — 7^y-{-x'y'^ — xf
+ y') (x-^ + x'^y + x^f + ^y + y%
7. Find the prime factors of «® — 1.
Ans. (a+1) (a — 1) (a^ + a + l) (o» — a + 1).
8. Find the prime factors of a^ — 2 a- x^ -\- x^.
Ans. (a -{- x) (a + ^) i'^ — ^) (" — ^)-
9. Find the prime factors of x« + 2 a:* y* + /.
Ans. (x+y) {x + y) {j?^xy + f^) {pt^-xy + f).
10. Find the prime factors of 1 — a*.
Ans. (1 +a) (1 — a) (1 + a^).
11. Find the prime factors of 8 — c*.
Ans. (2--C) (4 + 2c + c^.
i>4 ELEMENTARY ALGEBRA.
SECTION X.
GREATEST COMMON DIVISOR.*
72. A Common Divisor of two or more quantities is any
quantity that will divide each of them without remainder.
73. The Greatest Common Divisor of two or more
quantities is the greatest quantity that will divide each
of them without remainder.
74. To deduce a rule for finding the greatest common
divisor of two or more quantities, we demonstrate the
two following theorems : —
Theorem I. A common divisor of two quantities is also
a common divisor of the sum or the difference of any
multiples of each.
Let A and B be two quantities, and let d be their common di-
visor ; d is also a common divisor of m ^ dz n B.
Suppose A -^ d == p'^ i. e. A = dp, and mA = dmp,
and B -^ d = q; I. e. B = d q, and n B == d n q\
then mA±nB = d7np±dnq = d (mp ± n q).
That is, d is contained in m A -\- n B, m p -\- n q times, and in
m A — n B, mp — n q times ; i. e. cZ is a common divisor of the
sum or the difference of any multiples of A and B.
Theorem II, The greatest common divisor of two quan-
tities is also the greatest common divisor of the less and
the remainder after dividing the greater by the less.
Let A and B be two quantities, and A"^ B;
and let the process of dividing be as appears in B) A (q
the margin. Then, as the dividend is equal to qB
the product of the divisor by the quotient plus ~
the remainder,
A = r-^qB. (1)
* See Prefia«e.
GREATEST COMMON DIVISOR. 55
And, as the remainder is equal to the dividend minus the product
of the divisor hy the quotient,
r= A—qB. (2)
Therefore, according to the preceding theorem, from (1) any divisor
of r and B must be a divisor of A ; and from (2) any divisor of A
and jS, a divisor of r ; i. e. the divisors of A and B and B and r
are identical, and therefore the greatest common divisor of A and
B must also be tlie greatest common divisor of B and r.
In the same way the greatest common divisor of B and r is the
greatest common divisor of r and the remainder after dividing B
by r.
Hence, to find the greatest common divisor of any two
quantities,
RULE.
Divide Oie greater by the less, and the less by the remain-
der, and so continue till the remainder is zero; the last dir
visor is the divisor sought.
Note 1. — The division by each divisor should be continued until
the remainder will contain it no longer.
Note 2. — If the greatest common divisor of more than two quan-
tities is required, find the greatest common divisor of two of them,
then of this divisor and a thirds and so on ; the last divisor will
be the divisor sought.
Note 3. — The common divisor of x i/ and z r is a: ; x is also the
common divisor of x and x z, or of ax y and xz\ i. e. the common
divUor of two quantities is not changed hy rejecting or introducing
into either any factor which contains no factor of the other.
Note 4. — It is evident that the greatest common divisor of two
quantities contains all the factors common to the quantities.
CASE I.
75» To find the greatest common divisor of monomials.
1. Find the greatest common divisor of S a"^ li^ c d,
WaH'c\ and 2S aH* c.
The greatest common divisor of the coefficients found by the gen-
eral rule is 4 ; it is evident that no higher power of a than a*, of
66 ELEMENTARY ALGEBRA.
h than 5', of c than itself, will divide the quantities ; and that d will
not divide them ; therefore, the divisor sought is 4 a^ b^ c. Hence,
RULE.
Annex to the greatest common divisor of the coefficients
those letters which are common to all the quantities, giving to
each letter the least exponent it has in any of the quantities.
2. Find the greatest common divisor of 63 a^ b'' c^ d^,
27 a^ b' c^ and 45 a^ b"" c« d. Ans. 9 a^ h' c\
3. Find the greatest common divisor of ^bx'y^z^ and
l2babxUfz\
4. Find the greatest common divisor of 99 a S^c^rf^a:^^
and 22 a" b^ c^ d^ xK Ans. II a b'^c'^d^ a^.
5. Find the greatest common divisor of 11 x^y"^, l^si^y^,
and 2l2bx'y^^.
CASE II.
76. To find the greatest common divisor of polynomials.
1. Find the greatest common divisor of a;^ — y^ and
x"^ — 2xy -\- y"^.
x'-^f)x^ — 2xy-{- f(l
x^ — y"^
^2xy + 2y^
Kejecting the factor 2 y
X— y)x'^ — y^(x + y
x^ — xy
xy — y^
xy — y^ Ans. x — y.
RULE.
Arrange the terms of both quantities in the order of the
powers of some letter^ and (lien proceed according to the
general rule in Art. 74.
Note 1. — If the leading term of the dividend is not divisible by
the leading term of the divisor, it can be made so by introducing
GREATEST COMMON DIVISOB. 67
in the dividend a factor which contains no factor of the divisor ; or
either quantity may be simplified by rejecting any factor which
contains no factor of the other. (Art. 74, Note 3.)
Note 2. — Since any quantity which will divide a will divide — a,
and vice versa, and any quantity divisible by a is divisible by — a,
and vice versa, therefore all the signs of either divisor or dividend,
or of both, may be changed from -|- to — , or — to -|-» "without
changing the common divisor.
Note 3. — When one of the quantities is a monomial, and the other
a polynomial, either of the given rules can be applied, although gen-
erally the greatest common divisor will be at once apparent.
2. Find the greatest common divisor of ax* — a^ar* — 8 a^x^
and 2car* — 2aca:" + 4a=^car* — 6a^cx — 20a*c.
ax* — o'z* — Bc^x*
DlTidlng by a Z*
ar* — a z* — 8 a*
2a«x» — 3a»x — 2a*
DlTidlng by a"
2z» — 3ax— 2a»
2 car* — 2ac3^-\-4a*ca^ — Sc^cx — 20a*c
Dividing by 2 C
x* — ax^-\-2a'x' — 3a*x—l0a* (1
a;* — gg' — 8a*
2a'z' — 30*3: — 2a* ntBem.
3^— ax*— 8a*
Multiplying by 2
2 a;* — 2 a x» — 16 a* (a:"
2 a:* — 3 ga:* — 2 0*0:*
a 3* -\- 2 a* x"— 16 a*
Multiplying by 2
2ax»-f4a»x'— 32a* (aa;
2a3* — 3a*x* — 2a*z
7a*x»+ 2a»z — 32a*
Multiplying by 2
14 a« a:* -f 4 o* a: — 64 a* (7 a«
14a'a:*— 21a'a:— 14 a*
25 a" ar — 50a* 25a'x — 50a* sdBem.
DlTlding by 25 O*
X— 2a)2a:' — 3ax— 2a'(2z-|-a
2a:'— .4ax
ax— 2a"
ax — 2a* Ans. x — 2a.
58 ELEMENTARY ALGEBRA.
3. Find the greatest common divisor of a* — x* and
a^ -\- a^x — ax^ — x^. Ans. a^ — x^.
4. Find the greatest common divisor of a* — x* and
a® — a^ x^. Ans. c^ — x^.
5. Find the greatest common divisor o£2ax^ — a^x — a^
and 2 x^ -\- S a X -\- a^.
6. Find the greatest common divisor of 6 a x — S a
and Q ax^ -\- ax^ — 12 ax. Ans. Sax — 4a.
v. Find the greatest common divisor of x^ — y^ and
8. Find the greatest common divisor of 3 ic^ — 24 aj — 9
and 2x^—16x — Q.
9. Find the greatest common divisor of x^ — y^ and
x^ — y^. Ans. x — y.
10. Find the greatest common divisor of 10 x* — 20x'^y
4-30/ and x^ + 2x^y -\- 2xy^ + yK Ans. x + y.
11. Find the greatest common divisor of a* + "^ H~ ^^
+ a — 4 and a^ + 2 a« + 3 a^ + 4 a — 10.
Ans. a — 1.
12. Find the greatest common divisor of 1 ax^ -\- 21ax^
-f- 14 a and x^ -{- x^ -{- x^ — x. Ans. x^ I.
13. Find the greatest common divisor of 21 a^y^ — 8 a^y
and 3/ — 2 a/ 4- 3 aV — 2 a^y\
Ans. By"^ — 2 ay.
14. Find the greatest common divisor of a^ -{- a — 10
and a'^ — 16. Ans. a — 2.
Note 5. — The greatest common divisor of polynomials can also be
found by factoring the polynomials, and finding the product of the
factors common to the polynomials, taking each factor the least num-
ber of times it occurs in any of the quaptities. (Art. 74, Note 4.)
15. Find the greatest common divisor of3a^^ — 4aar-f-
Saxy — 4: ay and a^ x — x -\- a^y — y.
Sax^ — 4:ax -{- Saxy — 4zay = a(x -\- y) (Sx — 4)
a^x — X -{- a^y — yz=:(x-\-y)(a — I) (a^ -\- a -\- 1)
Ans. so -\- y.
LEAST COMMON MULTIPLE. 60
SECTION XI.
LEAST COMMON MULTIPLE.
77. A Multiple of any quantity is a quantity that can
be divided by it without remainder.
78. A Common Multiple of two or more quantities is
any quantity that can be divided by each of them with-
out remainder.
79. The Least Common Multiple of two or more quan-
tities is the least quantity that can be divided by each
of them without remainder.
80. It is evident that a multiple of any quantity must
contain the factors of that quantity ; and, vice versa, any
quantity that contains the factors of another quantity is
a multiple of it : and a common multiple of two or more
quantities must contain the factors of these quantities ;
and the least common multiple of two or more quantities
must contain only the factors of these quantities.
CASE I.
To find the least common multiple of monomials.
1 . Find the least common multiple of 6 a^b^c, Sc^lr^c^d,
and 12 a* b ex.
The least common multiple of the coefficients, found by inspection
or the rule in Arithmetic, is 24 ; it is evident that no quantity which
contains a power of a less than a*, of b leas than fc*, of c less than
c\ and which does not contain d and a:, can be divided by each of
these quantities ; therefore the multiple sought is 24 a* l^ c^ d x.
Hence, iu the case of monomials,
60 ELEMENTARY ALGEBRA.
RULE.
Annex. (0 the least common multiple of the coefficients all
the letters which appear in the several quantities, giving to
each letter the greatest exponent it has in any of the quan-
tities.
2. Find the least common multiple of 3 a* h^ c^ QaH*e d\
and IQahcx^. Ans. 30 a'' i^c^^^^^
3. Find the least common multiple of IQahx, 80 a Z»^a:^
and ZbaHx\ Ans. 560a^Z»^ar^
4. Find the least common multiple of Qa^b^, Iba'^bx^,
and l^axy'^. Ans. ^Q a'^b^x^y^.
6. Find the least common multiple of l^a^bc^x,
2^ab^cx^y, and Z^aH'^xz.
6. Find the least common multiple of Ki^xyz, 4.5 a be,
and 25 m n.
1. Find the least common multiple of 10 a^ by"^, 13 a^ b"^ c,
and llan^c^
8. Find the least common multiple of 14 a^ b'^ c*, 20 a^ b c*,
25a«6c», and 28 abed.
CASE II.
81 1 To find the least common multiple of any two
quantities.
Since the greatest common divisor of two quantities contains all
the factors common to these quantities (Art. 74, Note 4) ; and since
the least common multiple of two quantities must contain only the
factors of these quantities (Art. 80) ; if the product of two quanti-
ties is divided by their greatest common divisor, the quotient will
be their least common multiple.
Hence, to find the least common multiple of any two
q^uautities^
LEAST COMMON MULTIPLE. Oft
RULE.
Divide one of the quantities by their greatest common di-
xrisor, and multiply this quotient by the other quantity, and
the product unit be tJieir least common multiple.
Note 1. — If the least common multiple of more than two quanti-
ties is required, find the least common multiple of two of them,
then of this common multiple and a third, and so on ; the last com-
mon multiple will be the multiple sought.
Note 2. — In case the least common multiple of several monomials
and polynomials is required, it may be better to find the least com-
mon multiple of the monomials by the Rule in Case I., and of the
polynomials by the Rule in Case II., and then the least common
multiple of these two multiples by the latter Rule.
1. Find the least common multiple of x^ — y^ and
x' — 2xy + y\
OPERATION. Their greatest common
X — y) x^ — 2xy-[-y divisor is x — y, with
which we divide one of
""^ y the quantities ; and mul-
(^ ir) (^ — y)> Ans. tiplying the other quan-
tity by this quotient, we
have the least common multiple (x* — ^ (x — y).
2. Find the least common multiple of 2 a* a:", ^ar^y,
a* — X*, and a* — a' oc^.
The least common multiple of the monomials is 4 a' x* y ; and
the least common multiple of the polynomials is a' (a* — x*).
The greatest common divisor of these two multiples is a' ; and
dividing one of these multiples by a\ and multiplying the quotient
by the other, we have 4 a? x* y (^a* — x*) as the least common mul-
tiple.
3. Find the least common multiple of Sd^l^, 6a^by,
a' — 8, and a^ — 4 a + 4.
Ans. 6 a^ b^y («» — 8) (a — 2).
62 ELEMENTARY ALGEBRA.
4. Find the least common multiple of 3 a;^ — 24 a; — 9
and 2x^—16x — 6.
(See 8th Example, Art. 76.)
5. Find the least common multiple of a* — x^ and
6. Find the least common multiple of a:* — 1, x^-\-2x-{-l,
and (x — 1)^. Ans. x^ — x'^ — x^ -\- 1.
t. Find the least common multiple of a:^ — y^ and x^ -j" y^*
8. Find the least common multiple of a^ -j~ ^ — ^^ ^^^
a* — 16.
Note 3. — The least common multiple of any quantities can also
be found by factoring the quantities, and finding the product of all
the factors of the quantities, taking each factor the greatest number
of times it occurs in any of the quantities. (Art. 80.)
9. Find the least common multiple of x^ — 2xy-{-y'^,
x^ — y^, and [x -f- yy.
x^—2xy-\-y'^={x — y){x — y)
x'-y^= (x' + f) (X + y){x- y)
(x + yy ={x + y) (x + y)
Hence L. C. M = (a; — ?/) (x — y) (a;^ _(- y'^) (x -\-y) (x + y)
= x^ — x^y^ — x'^,y^ + y^.
10. Find the least common multiple of Bax^ — 4aa?-(-
S axy — 4: ay and a^ x — x -\- a^y — y.
(See 15th Example, Art. 76.)
Ans. a{x -\- y) (S x — 4.) (a'' + a + 1) (a — l)= Sa'x'' —
4:a^X -\- Ba^xy — 4a'*?/ — Sax'^ -\- 4,ax — Baxy + 4a?/.
FRACTIONS. - 63
SECTION XII.
FRACTIONS.
82. When division is expressed by writing the dividend
over the divisor with a line between, the expression is
called a Fraction. As a fraction, the dividend is called
the numerator, and the divisor the denominator.
Hence, the value of a fraction is the quotient arising
from dividing the numerator by tlie denominator.
X V
Thus, - is a fraction whose numerator is z y and denominator y,
and whose value is x.
83. The principles upon which the operations in frac-
tions are carried on are included in the following
THEOREM.
Any multiplication or division of the numerator causes a
like change in the value of (lie fraction, and any multiplica-
tion or division of the denominator causes an opposite change
in tlw value of (lie fraction.
Let — ^ be any fraction ; its value == a: y.
Ist. Changing the numerator.
Multiplying the numerator by y,
which is y times the value of the given fraction.
Dividing the numerator by y,
— *>
y
which is — of the . value of the given fraction.
64 ELEMENTARY ALGEBRA.
2d. Changing the denominator.
Multiplying the denominator by y,
which is — of the value of the given fraction.
Dividing the denominator by y,
which is y times the value of the given fraction.
Corollary. — Multiplying or dividing both numerator and
denominator by the same quantity does not change the value
of the fraction.
For if any quantity is both multiplied and divided by the same
quantity its value is not changed.
^^' y ~ cy ~ I ~^'
84i Every fraction has three signs r one for the numer-
ator, one for the denominator, and one for the fraction
as a whole.
Thus, +^-
If an even number of these signs is changed from -\- to
— , or — to -\-, the value of the fraction is not changed ;
but if an odd number is changed, the value of the fraction
is changed from -\- to — , or — to -\-.
Thus, changing an even number,
— xy_ -\-xy_
-\ry —y
taking
1 +^y_- . jp
+^^=+^;
FRACTIONS. 6^r
and changing an odd number,
__ + ^?/ _ 1 — _5_y __, I +^y -_ __ — ^y = _ x
4-y "^ H-y — y — y
The various operations in fractions are presented under
the following cases.
CASE I.
85t To reduce a fraction to its lowest terms.
Note. — A fraction is in its lowest terms when its terms are mu-
tually prime.
1. Reduce „:— ,— ^^ to its lowest terras.
24 a' z y"
operation. Since dividing both terms
16 a' :r y 4xy 2 °^ * ^"^"'^'°" ^^^ ^^« '^°^«
24^T^ — 67y — ry quantity does not change
its value (Art. 83, Cor.), we
divide both terms by any factor common to them, as 4 a' ; and both
terms of the resulting fraction by any factor common to them, as
2xy\ or we can divide both terms of the given fraction by their
2
greatest common divisor* 8 a' x y ; the resulting traction — is the
fraction sought Hence,
RULE.
Divide both terms of the fraction by any factor common
to them ; then divide these quotients by any factor common to
them ; and so proceed till the terms are mutually prime. Or,
Divide both terms by their ffi'eatest common divisor.
2. Reduce -~i to its lowest terms. Ans.
x«y» xy
972 a' x*?/* 2x
3. Reduce T^^.— ^i to its lowest terms. Ans. ;— , —
408 a* ar y* 3 a' y
24 X 1/ z 2z
4. Reduce — - — ^— to its lowest terms. Ans. —
12a x y a
5. Reduce ^^-^,^'^ to its lowest terms.
51 crbxy
66 ELEMENTARY ALGEBRA.
6. Reduce 7^77— 7^-r^ to its lowest terms.
1. Reduce „ , , — ^—. — ^ to its lowest terms.
x^-^2xy -{-f
Ans. ^
a'—2ab-\-b'
8. Reduce -5 ^ ^ ,, , ,, to its lowest terms.
9. Reduce ^-— ^ ^ — to its lowest
2 ex* — 2acx^ -f- Aarcx' — 4 a^ c x
terms. . aa^
' ' J(2^'~^fT^'
10. Reduce — -^ — ^- — . to its lowest terms.
gr — x*
CASE II.
86. To reduce fractions to equivalent fractions having
a common denominator.
o c
1. Reduce j~ and j- to equivalent fractions having a
common denominator.
OPERATION. y^Q multiply the numerator and
a g bx denominator of each fraction by the
bij tt^xy denominator of the other (Art. 83,
T ^ ,, Cor.). This must reduce them to
c c y /
j~ — ^r^ equivalent fractions having a common
denominator, as the new denominator
of each fraction is the product of the same factors.
ORj In the second operation we find the
a a X least common multiple, b x y, of the
by hxy denominators by and fta:; as each de-
c
_ cy
nominator is contained in this multi-
Wx hxy pl®' 6^ch fraction can be reduced to
a fraction with this multiple as a de-
nominator, by multiplying its numerator and denominator by the
quotient arising from dividing this multiple by its denominator.
Hence,
FRACTIONS. 6T
RULE.
Multiply all the denominators together for a common de-
nominator, and multiply each numerator into the continued
product of all the denominators, except its own, for new
numerators. Or,
Find the least common multiple of Oie denominators for
the least common denominator. For new numerators, mul-
tiply each numerator by tJie quotient arising from dividing
this midtiple by its denominator.
2. Reduce — » ~. and —r- to equivalent fractions
xy ab aby ^
having the least common denominator.
. ahm nxy , '^
Ans. -T — » ^ > and
a b xy ab xy a bxy
3. Reduce -— :. rr-r^* and — 7 to equivalent frac-
15 6 10 6 c 25 act/ ^
tions having the least common denominator.
. 80 a' erf 45 ad xy , 126 a;
^^^' TbOabcd* IbO^bVd' ^ 150abcd'
4. Reduce — » » and z—, to equivalent fractions
m nxy b a ^
having the least common denominator.
5. Reduce , . and —^t to equivalent fractions hav-
ing the least common denominator.
. a" — 2a6-f-6* , a*m -\- ahm
6. Reduce _ and — ^— r to equivalent fractions
having the least common denominator.
7. Reduce -, v — i — » and to equivalent frac-
3^—y- x-\-y x — y ^
tions having the least common denominator.
;
ELEMENTARY
ALGEBRA.
CASE
III.
81
U To add fractions.
1.
Find the sum of - and
OPERATION.
c
X
If anvthiner
If anything is divided into equal
5 c h -\- c parts, a number of these parts rep-
"^ I ^ ^ resented by 6, added to a number
represented by c, gives h -\- c of
these parts. In the example given, a unit is divided into x equal
parts, and it is required to find the sum of h and c of these parts ; i. e.
h ^, c h -\- c
X ~^ X X
It is evident, therefore, that fractions that have a common denom-
inator can be added by adding their numerators. But fractions that
do not have a common denominator can be reduced to equivalent
fractions having a common denominator. Hence,
RULE.
Reduce the fractions, if necessary, to equivalent fractions
having a common denominator; then write the sum of the
numerators over the common denominator.
r. A jj m X J a . bmy 4-bnx -\- any
2. Add -, -, and -r- Ans. ' , '
n y ony
3. Add -, -J, and ^.
4. Add ^^l m. and ^^.
^xy b ab 8 ab xy
30 a^ b^ - f l(^a?y^-\- 35 m
40 a b xy
Ans.
5. Add -—;-, -— -y, and
3a^ ^c d 21 a c
6. Add -— 1 — and - — — • Ans
Y. Add
l-|-ti I — a I —a^
l_|_a ,1— a . 2+2 a"
— ^ — and ■-— i — Ans. -— ^ ^•
1 — a \ -\-a 1 — a^
FRACTIONS.
8. Add i--t^ and ^.
9. Add ^(-;!^^-^ and ^^.
10. Add ^-S', 5^^^ and '-^ + " . AnB. 1.
11. Add -^^ and ^""^
a:* — y* a^ — 2/*
Ans. ' + y
7 X
12. Add 7nx and -— -•
lo a
Note. — Consider m a: = — , and then proceed as before.
. 18 a ma: 4- 7x
A°«- — w^-
7 X
13. Add a? 4- y and — t-t-
14. Add a;2 + 2xy + y*and^-^.
Ans. ^ + ^y-^y'-y' + i.
% — y
CASE IV.
88. To subtract one fraction from another.
c b
1. Subtract - from -•
X X
OPERATION. If anj-thing is divided into x
h c b — c equal parts, a number of these
X X X pdrts represented by c, subtracted
from a number represented by 6,
leaves b — c of these parts ; i. e. = • Hence,
*^ ' XX X
RULE.
Reduce the fractions, if necessary, to equivalent fractions
having a common denominator; then subtract the numerator
of the subtrahend from that of the minuend, and write the
result over the common denominator.
70 ELEMENTARY ALGEBRA.
2. Subtract -r- from r — Ans.
4 8 c "'8 c
3. Subtract ;r- from ;r— •
7 X 6 ax
4. Subtract -z-ftzs from -r —
19 x^ Ida
_ CI 1 ^ ^ 29 ac „ 39 a; . 273 ar* — 116 acy
5. Subtract — -— ^ from ^; Ans. -;— ^ ^•
14 a^ 8xy 56 xr y
6. Subtract ^ from :; — j — -' Ans. -s -•
1 — a 1 -\-a a^ — 1
T. Subtract *" , from — j— ,•
X — 1 a; -}- 1
oot^x , ah 4-b c n ah — he
8. Subtract „,^^jrj from ^^— j,^-
9. Subtract r from — i-^- Ans.
a — h ^^ a-\-h 6^
10. Subtract -, r from „ ' • Ans. —
11. Subtract 16 from
a;* _ 1 ""^" a;2 — 1 ar» -f- 1
a;3_7
1 4-a;
1 fi
Note. — Consider 16 = — , and then proceed as before.
. oi?-^lQx — 2Z
Ans. --j
1 -\-x
yZ g
12. Subtract , from xy.
a — h ^
13. Subtract x -{- b from , ; , ♦ Ans. , , , •
CASE V.
89i To reduce a mixed quantity to an improper fraction.
1. Keduce x -\- - to an improper fraction.
o
OPERATION. As eight eighths make
, a 8x ^^a 8a:-|-« 3- unit, there will be in
•"S 8'8 8 X units eight times x
. , ,, . 8a; j8a:,a 8a;-4-a ^t
eighths; i. e. a: = — ; and y -f - = — -^ — . Hence,
FRACTIONS. 71
RULE.
Multiply the integral part by the denominator of the frac-
tion ; to the product add the numerator if tlie sign of the
fraction is plus, and subtract it if the sign is minus, and
under the result write the denominator.
Note. — By a change of the language, Examples 12-14 in
Art. 87, and 11-13 in Art. 88, become examples under this case.
Thus, Example 12, Art. 87, might be expressed as follows: Reduce
tnx-f- _- to an improper fraction.
7
2. Reduce ar* -f- 4 to an improper fraction.
AnB. tl+^l^.
y
a \ X
8. Reduce 25 a — 25 a: -I -I— to an improper frac-
tion.
4. Reduce a — 1 -f- . T ^^ ^^ improper fraction.
. a" — a
Ans. — j— -.
6. Reduce y -| ^—^ — to an improper fraction.
6. Reduce — ^ (a -j- 6) to an improper fraction.
. a* — ah
Ans. —J
7. Reduce x — 1 J— to an improper fraction.
Note. — It must be remembered that the sign before the dividing
line belongs to the fraction as a whole.
. , z»4-l a:"— i_x« — 1 —2 2
X — 1 -, = r— ; = — J—:' or r— -' Ans.
x-\-l ^+1 ^+1 ^-f-l
8. Reduce x -(- 1 ^— to an improper fraction.
o
Ans
1 — X
72 ELEMENTARY ALGEBRA.
9. Reduce x^ — 2ax -\- c^ — - — to an improper
fraction.
Note. — According to the same* principle an integral quantity-
can be reduced to a fraction having any given denominator, by
multiplying the quantity by the proposed denominator, and under the
product writing the denominator.
10. Reduce x -\- \ to a fraction whose denominator is
X — 1 . * ^ — 1
• Ans. -•
X — 1
11. Reduce x — 1 to a fraction whose denominator is
a — h.
12. Reduce 4 a a: to a fraction whose denominator is
a^ — z.
CASE VI.
90. To reduce an improper fraction to an integral or
mixed quantity.
Cu -* -■ ^ (2 cc I 5 fl
1. Reduce ^ to an integral or mixed quan-
X — z a
tity.
OPERATION.
* a'
X — 2 a) x^ — 4: a X -{- 5 a^ (x — 2 a -\ ^r—
x^ — 2 ax
— 2ax -\- ^ c?
— 2 a ar -(- 4 a2
As the value of a fraction is the quotient arising from dividing the
numerator by the denominator (Art. 82), we perform the indicated
division. Hence,
RULE.
Dimde the numerator by the denominator ; if there is any
remainder, place it over the divisor, and annex the fraction
so formed with its proper sign to the quotient.
FRACTIONS. 78
2. Reduce to an integral or mixed quantity.
Ans. a — 4 6.
8. Reduce ""^^ "^"^^ to au integral or mixed
quantity.
4. Reduce g^~^^ to an integral or mixed quantity.
^- Reduce ^^^~e^"^"^^ to an integral or mixed
quantity. 7
Ans. 4y-2a + ^.
6. Reduce "^^ *o ^ integral or mixed quan-
tity.
>T P«^.,^^ Sax — 10 bx — 5rx
^- deduce ^- ±- to an integral or mixed
quantity.
^- ^^^^^® 2a--2b to an integral or mixed
quantity. a ^ ^ , a
Ans. 2 a — 2 b
a — 6
9. Reduce ^--^ to an integral or mixed quantity.
10. Reduce ^-— to an integral or mixed quantity
CASE VII.
91. To multiply a fraction by an integral quantity.
1. Multiply ^ by c.
OPKRAnoN. According to the theorem
a?4-y w cx-f-cy ^° ^^ 83, multiplying the
ab ^ ab numerator by c multiplies
the value of the fraction c
times.
74 ELEMENTARY ALGEBRA.
2. Multiply ^-i^ by a.
OPERATION. According to the theorem
, I in Art. 83, dividing the de-
— — — — X 05 = — ^ — nominator by a multiplies
the value of the fraction a
times. Hence,
RULE.
Divide the denominator by the integral quantity when it
can be done without remainder; otherwise, multiply the nu-
merator by the integral quantity.
« T»r li- 1 Sax -\- 4: xy , ,
3. Multiply -^P^.~ by m + n.
Ans. ^Zn '
4. Multiply ^-^^j by ab.
Note. — Any factor common to the denominator and multiplier
may be cancelled from both before multiplying.
^ — a ..o ^* — «v^ ^*y — ^y A
36 + 3c X ^2^ = 6~+ c X ^ = -h^' ^^«-
'' ^'^'''^'y ^^x ^^ ^ -
1. Multiply y^±^ by U (x^ ^ ^).
Ans. 2{x^ — f) (a + x).
8. Multiply ^^^hyx—y.
Note. — When a fraction is multiplied by a quantity equal to its
denominator, the product is the numerator.
Ans.
FRACTIONS. 76
.9. Multiply ^y^^J^ by (x - a)«.
10. Multiply ^^-i^ by 2r» — 2 ary + y^.
X y
Ans. (a + h) {x — y).
CASE VIII.
92. To multiply an integral quantity by a fraction.
1. Multiply ar* + 2xy + y^ by ^-.
OPERATION.
4(ar« + 2xy + y«)-(x + y) = 4(x + y)
We first multiply the multiplicand by the numerator 4; but the
multiplier is 4 -^ (x -|- if) \ and therefore this product is a: -j- y times
too great, and this product divided by x -j- .V must be the product
sought.
It is evident that the result would be the same if the division were
performed first, and the multiplication afterward. Hence,
RULE.
Divide the integral quantity by the denominator when it
can be done without remainder, and muUiply the quotient
by the numerator. Otheruxise, multiply the integral quantity
by the numerator, and divide live product by the denom-
inator.
2. Multiply a» - 3 a» i + 3 a 4« - i^ by ^._//ft^y -
Ans. Tx(a — b).
3. Multiply a* — x* by ^
a'-f X*
g
4. Multiply 7 a* — 4xy by . «
. 21a»— 12xy
5. Multiply 17(ar' — y') by ^i^-
76 ELEMENTARY ALGEBRA.
Note. — Since the product is the same, whichever quantity is con-
sidered as the multiplier, by considering the integral quantity as the
multiplier, Case VIII. becomes the same as Case VII.
CASE IX.
93. To divide a fraction by an integral quantity.
According to the theorem in Art.
83, dividing the numerator by a de-
creases the value of the fraction a
times.
According to the theorem in Art.
83, multiplying the denominator by c
decreases the value of the fraction c
times. Hence,
RULE.
Divide the numerator by the integral quantity when it loill
divide it without remainder; otherwise, multiply the denom-
inator by the integral quantity.
3. Divide 7-^— by a. Ans. ri— •
46c "^ 46c
4. Divide -y by Uf- Ans. ^yj-
5. Divide -r- — by Q abc.
6. Divide l^byQaJ^r^. Ans ^"^
1.
Divide - by a.
OPERATION.
a 1
6-^« = 6
2.
Divide 7- by c.
OPERATION.
a a
3267 "■' -*• 32 6's'
T. Divide ilg^J ^y 2 (« + ^) (X + y).
Ans.
13 {X + yy
FRACTIONS. 77
CASE X.
91 • To divide an integral quantity by a fraction.
1. Divide x by y
OPERATiox. ^ -7- o = - ; but the divisor is not a,
X but a-^h. Dividing by a, therefore,
"^ a is dividing by a divisor h times too
X hx great, and the quotient will be h times
a^ 'a ^^^ small; therefore the quotient sought
. X ^^ , hx --
18 - X = — . Hence,
a a
RULE.
Divide the integral quantity by the numerator, and mul-
tiply the quotient by the denominator.
« T^- .J . u 3x . 16a
2. Divide 4 a a: by — • Ans. -g-»
3. Divide 7x2 by ^_'. Ans. '^-^'
^ abc 3?
4. Divide a + 6 by -. Ans. ''^"^^^ >
5. Divide a2 + 2ax-fa:» by ^^.
6. Divide x* — hx^ by -•
7. Divide 2x' + 3y by ?^i^-
8. Divide 1 by -• Ans. -•
Note. — Hence, the reciprocal of a fraction is the fraction inverted,
CASE XI.
95. To multiply a fraction by a fraction.
1. Multiply T by —
78 ELEMENTARY ALGEBRA.
OPERATION. We first multiply ^ by a: ; but the
a ax multiplier is not x, but x ~- y^ there-
h h fore the product is y times too great ;
d X Q, X
ax ax and -^ -^ y = r- (Art. 93) must be
-J- -^ y -=2 — b y ^ '
"y the product sought. Hence,
RULE.
Multiply the numerators together for a new numerator,
and the denominators for a new denominator.
Note 1. — Common factors in the numerators and denominators
may be cancelled before multiplication.
Note 2. — Cases VIL and VIIL «an be included in this by writ-
ing the integral quantity as the numerator of a fraction, with a unit
as the denominator.
Note 3. — Mixed quantities may be reduced to improper fractions
before multiplying. ■
2. Multiply J- by ^^. ^^^--j^d^-
3. Multiply ^4>y 1"^^.
4. Multiply —r- by
^ '' aoc '' mx
5. Multiply ^Sy'-=^. Ans.^^
6. Multiply ""-f^ by -^^^.
1. Multiply -,-^"t-^ by ^.
8. Multiply ^^by ^'_f-;.
9. Multiply - — by ^ „ o * • Ans. ^-,~y.-^'
^ "^ 2Xy •'2a* — 8 a* 3 -f • «
FRACTIONS. 79
10. Multiply ,^^_, by ,,^^^._,,^^ '
11. Multiply YT-^-'iTy ^y T?' ^°«- ^"i^*
12. Multiply ^by^.
13. Multiply y + -^ by y "•'
a-l-3/
Ads. -^ ^
a«-i^
14. Multiply together ^^' 7^' ^^^ V^'
Ads. 1.
15. Multiply together ~^» '2_/s' *°^ T" *
16. Multiply together a + - , 5 + - ' a°d ^ — u'
Ans. a6v4-i^ — i 5*
if ^ by y^
CASE XII.
96i To divide a fractioD by a fraction.
1. Divide - by -7 •
y '' h
OPERATION. X X
- -f- a = — (Art. 93) ; but the
- -4- a = — ... a
y ay divisor is not a, but - ; we have used
. . b
_^ y, I __ ^ a divisor h times too great, and there-
^^ ^ fore the quotient — is 6 times too
I. °^
X ox
small, and the quotient sought is — X ^ ■=■ (Art. 91). It will
be noticed that the denominator of the dividend is multiplied by the
numerator of the divisor, and the numerator of the dividend by the
denominator of the divisor. Hence,
RULE.
Invert the divutor, and then proceed as in multiplication of
a fraction by a fraction.
80 ELEMENTARY ALGEBRA.
Note 1. — All cases in division of fractions can be brought under
this rule, by writing integral quantities as fractions with a unit for
the denominator.
Note 2. — After the divisor is inverted, common factors can be
cancelled, as in multiplication of fractions.
Note 3. — Mixed quantities should be reduced to improper frac-
tions before division.
2. Divide - by — • Ans.
c •' n cm
x^ X* 4
3. Divide - by -• Ans. -^— •
4. Divide 3^ by ^^.
. ^. ., 20^4- 2c , 3 a . 10 a«c 4- 10 c»
6. Divide ^,3 ', — by ^-. Ans. — — ,.. ^ , ■
6. Divide i^^±# by -1-
m — n
7. Divide ^ by ^+^?. Ans. ;|^
8. Divide — ^—j- by —
a — h *' 4
3 x* ar 3 T
9. Divide 3 . , by ■ , • Ans.
10. Divide ^"^7/ by ^-=1^^.
x-}- 1 '^ 4
11. Dmde ^ by =t
12. Divide /"'•':-!"' . by '"' + '"" + "' ■
Ans. 3 (m« + n«).
13. Divide 1 + "^-^ by -4 1.
Ans ^ <^^ + y) + ^^ +'^)' .
c {c — x — y)
14. Divide x + y—^ by -4-a;-|-y.
FRACTIONS. 81
15. Divide ^-^ + ^^ ^y r+^-
Ans ^IL±^)*
A^«- (i_x«j-«-
Note. — The division of fractions is sometimes expressed by writ-
a
ing the divisor under the dividend. Thus, — . Such an expression
y
is called a Complex Fraction. A Complex Fraction can be
reduced to a simple one by performing the division indicated.
16. Reduce y to a simple fraction. Ans. z— •
5
17. Reduce ^ to a simple fraction.
c Ans. ' — i— •
lex — bx
x+l
X 1
18. Keduce . to a simple fraction.
19. Reduce — , . to a simple fraction.
\—x
^--^
20. Reduce ^ . ? to a simple fraction.
Ans. — i —
21. Reduce — ^— ^ to a simple fraction.
a 4- 6 Ans. (x + y) (a + h).
Note. — A Complex Fraction can also be reduced by multiplying
its numerator and denominator by the least common multiple of
the denominators of the fractional parts. Thus, if both terms of
the fraction in Ex. 16 be multiplied by 5 a:, or both in Ex. 17 by
ex, the result will be the same as above.
4* F
82 ELEMENTARY ALGEBRA.
SECTION XIII.
EQUATIONS
OF THE FIRST DEGREE CONTAINING BUT ONE UNKNOWN
QUANTITY.
97i An Equation is an expression of equality between
two quantities (Art. 9). Tliat portion of the equation
which precedes the sign = is called the first member, and
that which follows, the second member.
98. The Degree of an equation containing but one un-
known quantity is denoted by the exponent of the highest
power of the unknown quantity in the equation.
An equation of the first degree, or a simple equation, is
one that contains only the first power of the unknown
quantity. For example,
2 a; — a a; = 27.
An equation of the second degree, or a quadratic equation,
is one in which the highest power of the unknown quantity
is the second power. For example,
x^ — ax-=zh -\- c, or ax^ — 5 = IT.
An equation of the third degree, or a cubic equation, is one
in which the highest power of the unknown quantity is the
third power, and so on.
99. The Reduction of an Equation consists in finding the
value of the unknown quantity, and the processes involved
depend upon the Axioms given in Art. 13. The processes
can be best understood by considering an equation as a pair
of scales which balance as long as an equal weight remains
in both sides : whenever on one side any additional weight
is put in or taken out, an equal weight must be put in or
EQUATIONS OP THE FIRST DEGREE. 88
taken out on the other side, in order that the equilibrium
may remain. So, in an equation, whatever in done to one
side muat be done to tJie oifier, in order that the equality may
remain.
1. If anything is added to one member, an equal quantity
must be added to the other.
2. If anything is subtracted from one member, an equal
quantity must be subtracted from the other.
3. If one member is multiplied by any quantity, the other
member must be multiplied by an equal quantity.
4. If one member is divided by any quantity, the other
member must be divided by an equal quantity.
6. If one member is involved or evolved, the other must
be involved or evolved to the same degree.
TRANSPOSITION.
100. Transposition is the changing of terms from one
member of an equation to the other, without destroying
the equality.
The object of transposition is to bring all the unknown
terms into one member and all the known into the other,
80 that the unknown may become known.
I. Find the value of x in the equation x-|- 16 := 24.
Subtracting 16 from the first
OPERATION. member leaves x ; but if 1 6 is sub-
^ ~r 16 = 24 tracted from the first member, it
x = 24 — 16 = 8 must also be subtracted from the
second.
2 Find the value of x in the equation x — b = a.
OPERATION. Adding b to the first member
J. ft = a gives x ; but if 6 is added to the
__ I 1 first member it must also be added
to the second.
84 ELEMENTARY ALGEBRA.
3. Find the value of x in the equation 2 x = a; -j- 16.
OPERATION.
Subtracting x from both mem-
2x = a: + 16
2a: — x= 16
bers, we have 2 a; — a; = 16, or
x= 16.
a: =16
It appears from these examples that any term which dis-
appears from one member of an equation reappears in the
other with the opposite sign. Hence,
RULE.
Any term may be transposed from one member of an
equation to the other, provided its sign is changed.
4. Find the value of x in the equation Sx — 15 = 4x4-5.
OPERATION.
Sx — 15 = 4:X-\- 6
Transposing, Sx — 4x= 5 -\~ 15
Uniting terms, 4 a; = 20
Dividing both members by 4, x := 5
6. Find the value of x in 4 x -)- 46 = 5 x -|- 23.
Note. — Reducing, we have — x = — 23. If each member of
this equation is transposed, we shall have 23 = a: ; i. e. 23 equals
ar, or x equals 23. Dividing both members by — 1 will give the
same result. Hence, the signs of all the terms of an equation may
be changed without destroying the equality.
6. Find the value of x in It ar + It = 19 x + 13.
Ans. x = 2,
1. Find the value of a: in 8 a; — 14 = 13 a; — 29.
8. Find the value of x in 5 a: + 25 = 10 a: — 25.
Ans. X =: 10.
9. Find the value of x in 24x — 17 = 11 x + 74.
10. Find the value of x in 37 x — (4 + 7) = 41 x — 23.
EQUATIONS OF THE FIRST DEGREE. 85
CLEARING OF FRACTIONS.
101. To clear an equation of fractions.
1. Find the value of a: in the equation - — 2 = -+ 1.
OPERATION. If the given equation is mul-
X 2 -4-1 tipHed by 6, the least common
3 6 ' multiple of 6 and 3, it will give
2x — 12 = a:-|-6 2x — 12 = x-|-6, an equation
X = 18 without a fractional term. Hence,
RULE.
Multiply each term of the equation by the least common
multiple of the denominators.
Note 1. — In multiplying a fractional term, divide the multiplier
by the denominator of the fraction and multiply the numerator by
the quotient.
Note 2. — An equation may be cleared of fractions by multipljnng
it first by one denominator, and the resulting equation by another, and
so on, till all the denominators disappear; but multiplying by the
least common multiple is generally the more expeditious method.
"Note 3. — Before clearing effractions it is better to unite terms
which can readily be united ; for instance, the equation in Ex. 1,
X X
by transposing — 2, can be written - = - -|- 3.
o o
Note 4. — When the sign — is before a fraction and the de-
nominator is removed, the sign of each term that was in the nu-
merator must be changed.
2. Given ^ _ ^ + 25 = 33 — ^^-
operation.
n -oc X X ^ X — 6
1 ransposing 25, - — I = ^ 5 —
Multiplying by 20, 5 x — 4 x = 160 — 10 x + 60
Transposing and uniting, 11 x = 220
Dividing by 11, x=:20
86 ELEMENTARY ALGEBRA.
Note. — The sign of the numerator of — - is -|-, and must be
o
changed to — when the denominator is removed ; for — (-}- 4 x)
== — 4 a:; and so the sign of each term of the numerator of the fraction
— must be changed when the denominator 2 is removed ; for
— (-f 10 a: — 60) = — 10 a: -|- 60.
102i To reduce an equation of the first degree contain-
ing but one unknown quantity, we deduce from the preced-
ing examples- the following
RULE.
Clear the equation of fractions, if necessary.
IVanspose the known terms to one member and the un-
known to the other, and reduce each member to its simplest
form.
Divide both members by the coefficient of tlie unknown
quantity.
Note 1. — To verify an equation, we have only to substitute in
the equation the value of the unknown quantity found by reducing
the equation. For instance, in Ex. 2, Art. 101, by substituting 20
for X, in ^ — f -f 25 = 33 — ^ "~ , we have
4 5' 2
4 5' 2
6 _ 4 _|_ 25 = 33 — 7,
26 = 26.
Note 2. — When answers are not given, the work should be veri-
fied.
103* Since the relations between quantities in Algebra
are often expressed in the form of a proportion, we intro-
duce here the necessary definitions.
EQUATIONS OF THE FIRST DEGREE. 87
104. Ratio is the relation of one quantity to another of
the same kind ; or, it is the quotient which arises from di-
viding one quantity by another of the same kind.
Ratio is indicated by writing the two quantities after
one another with two dots between, or by expressing the
division in the form of a fraction. Thus, the ratio of a to
b is written, a : b, or j ; read, a is to b, or a divided by b.
105. Proportion is an equality of ratios. Four quan-
tities are proportional when the ratio of the first to the
second is equal to the ratio of the third to the fourth.
The equality of two ratios is indicated by the sign
of equality (=) or by four dots (::).
Q C
Thus, a \ b =. c : dy ox a '. b '. : c : rf, or 7 = - ,; read, a to 6
ha
equals c to d, or a is to b s^ c is to d, or a divided by b
equals c divided by d.
The first and fourth terms of a proportion are called the
extremen, and tlie second and third the means.
106. In a proportion the product of the means is equal
to the product of the extremes.
Let a : b = c : d
a c
Clearing of fractions, ad = be
A proportion is an equation ; and making the product
of the means equal to the product of the extremes is
merely clearing the equation of fractions.
Examples.
1. Reduce 1^ + 10 = ^^ + 13. Ans. x = 30.
2. Reduce 17 « — 14 = 12 x — 4. Ans. x r= 2.
88 ELEMENTARY ALGEBRA.
3. Reduce 6 a: -— 25 + a; = 135 — 3 a: — 10.
Ans. x-==. 15.
4. Reduce 3a:+5 — a! = 38 — 2a:. Ans. x = 8^.
5. Reduce ""-^ -j- ^ =, 30 — ^-i-^- Ans. x = 12.
6. Reduce a: — 7^ = — — • Ans. a; =: lly^.
T. Reduce ^ + ^ + ^ + ^= 154. Ans. a: = 120.
2 o 4 o
8. Reduce f + |= 16 + f . Ans. a: = 24.
9. Reduce --|-a=7 \- d.
h c
OPERATION.
6 ' ^ c '
Multiplying byJc^, chx-\- ahch:=:zh ex — hhx-\-hcdh
Transposing, chx — b c x-\-hhxr=:h c d h — ahch
Factoring 1st mem., (c A — hc-\-hh)xr=hc dh — a hch
Tx-.i. 1 m ' , /. b c dk — ah ch
"^& "J ^"
WiXJ.V>iVXJ.U VA ^,
ch — hc-{-hh
10
Reduce
X -\- mx =: c.
A TTS '*» — —
1 4-m
11.
Reduce
'T «-^-
Ans. a:= ^^
12.
Reduce
1 , h
- + - = a;.
Ans. X = ' —
a c
13.
Reduce
a; ' X
Ans. X = — '—
c
14.
Reduce
«=.-^+«-
^ns. X = 9.
15.
Reduce
2 3a
= c.
XXX
Ans. a: = — ^ —
EQUATIONS OF THE FIRST DEGREE. 89
X , X , X
16. Reduce - + - + - = 39.
i i \
2 X X
17. Reduce - - — --\-c = d.
18. Reduce (a — 3) x + ^ = i
/-r -r\
19.
Reduce x — / 1 — ^\ =z 5. Anfl. ar = 6.
20. Reduce 6 — ^^^tl __ a: ._ 4.
5
21. Reduce 2 a: ^ =18 {
Ans. X = 9.
ortr»j ^17 — X „ .a: — 95
22. Reduce - =3xH --•
23. Reduce 2 a: — ?^^ = 14 — —'^- Ans. x = 5.
24. Reduce 6 a: + T^ - | = 9^ - ^/ + ^.
Note. — Before clearing of fractions, transpose 7^ and unite it
X 1 1 X
with 9^; also transpose — -, and unite it with -5-.
25. Reduce 4ar+?-±i = 5 + iliil
X
3~
26. Reduce ^—i — ^^ = 21 — -"ti. Ans. a: = 39.
Jo G
27. Reduce - 4- -r 4- - = (/. Ans. x = r — ; ; — -.'
a ^ b ^ c be -\- ac-\- ab
28. Reduce — ~^ — 6 = -7-? + 7.
ftAOj ^ — 1 /» 22 — a: 3 + x . ^
29. Reduce — ^ — = 6 ^- Ans. x =: 7.
O
90 ELEMENTARY ALGEBRA.
on r> J in I 2a: — 22 3 x — 75 . 284. — 4.x
30. Reduce 19 + —^ — = 1 ^
oi T>j 4:X-\-5 5x — 5 a- 4-1 ,
31. Reduce — -^ — = — ^ 1.
5 4 b .
Ans. X = 5.
32. Reduce 1^^ - '-^ = 4.-17+ '-^.
o 4 '6
oo r> J A ar— 12 , _ 20a: 4- 21 1
33. Reduce 4:X [- 5 = } •
o ' 4 4
34. Keduce = ~ » Ans. a; r=:
a; c m ' hm-\-cd
35. Reduce 7 — — = 1 — 3 « c.
oa x> A 5a:+3 3a:-j-15 . , 6a:4-10
36. Reduce — ^ h 6 ,' = 4 -] \
2 ' 4 '4
OK -o J o 3a — 19 _ 23— a: , 5.r— 38 , -^
31. Reduce Zx 8 =: — — - 4 \- 10.
z 4 o
Ans. X = 19»
«« _,, 13 — 3x 3a:4-2 ^ „ ,8a:— 13
38. Reduce — ,^ J— - = T — 6 :r ^
10 5 '0
on T> A 4.(x—7) 3(a:4-l) 7.t— 17 a;
39. Reduce ^ + 11 = —lo ^ 21 '
,^ _., 4.r — 6,„ 19— 4a: 5.r — 6, 7 a: + 8
40. Reduce a: 1-3=:— ^ ^—] ^.
41. Reduce ^ + ^ + ^ + ^ = m.
,^_., 7x4-5, 6a: — 30 ,,
42. Reduce - J— + ~ — = x + I.
7 ' 7 X — 7 '
Note. — Multiply by 7, transpose, and unite.
43. Reduce 2 (3 4- a:) : 6 a: — 9 = 2 : 3. Ans. x= 6.
44. Reduce l + f: ^^^J-, - H -^
45. Reduce b : c -\- d = - : n.
X. ' a:
EQUATIONS OF THE FIRST DEGREE. ftl
PROBLEMS
PRODUCING EQUATIONS OF THE FIRST DEGREE CON-
TAINING BUT ONE UNKNOWN QUANTITY.
107. The problems given in this Section must either con-
tain but one unknown quantity, or the unknown quanti-
ties must be so related to one another that if one be-
comes known the others also become known.
108. With "beginners the chief difficulty in solving a
problem is in translating the statements or conditions
of the problem from common to algebraic language ; i. e.
in preparing the data, and forming an equation in accord-
ance with the given conditions.
1. If three times a certain number is added to one half
and one third of itself, the sum is 115. What is the
number ?
SOLUTION.
Let X =. the number ;
then by the conditions of the problem.
Clearing of fractions, 18x + 3a; + 2a: = 690
Uniting terms, 23 ar = 690
Dividing by 23, ar=: 30
VERIFICATION.
3X30 + f + ^»=U5
115 = 116
In this problem there is but one unknown quantity, which we rep-
resent by X.
2. There are three numbers of which the first is 6 more
than the second, and 11 less than the third ; and their sum
is 101. What are the numbers?
92 ELEMENTARY ALGEBRA.
SOLUTION.
Let X r= the first, In this problem
then X — 6 = the second, there are three un-
and a: + 11 = the third. known quantities;
Their sum,3x+ 6 = loT 1^"^*^"^ "^" ^^ ^^
lated to one an-
other that, if any
x= 32, the first, one becomes known,
X— 6 = 26, the second, t^g other two will
a? + 11 = 43, the third. be known.
VERIFICATION.
32 + 26 + 43 = 101
101 = 101
From these examples we deduce the following
GENERAL RULE.
Let X [or some one of the latter letters of the alphabet)
represent the unknown quantity ; or, if there is more than one
unknown quantity, let x represent one, and find the others by
expressing in algebraic form their given relations to the one
represented by x.
With tJw data thus prepared form an equation in accord-
ance with the conditions given in the problem.
Solve the equation.
The three steps may be briefly expressed thus : —
1 St. Preparing the Data ;
2d. Forming the Equation ;
3d. Solving the Equation.
3. The sum of three numbers is 960 ; the first is one
half of the second and one third of the third. What are
the numbers ? Ans. 160, 320, and 480.
4. Find two numbers whose difference is 18 and whose
sum 112. Ans. 47 and Qb,
EQUATIONS OF THE FIBST DEGREE. 98
5. A man being asked how much he gave for his horse
said, that if he had given $ TO more than three times as
much as it cost, he would have given S445. How much
did his horse cost him ?
6. A man being asked how many sheep he had, replied
that if he had as many more, and two thirds as many,
and three fifths as many, he should have 8 more than
three times as many as he had. IIow many sheep had he ?
*?. Divide $675 between A and B in such a manner
that B may have two thirds as much as A.
Ans. A^s share, $ 345 ;
B^s " $230.
8. A father divided his estate among his three children
80 that the eldest had $ 1440 less than one half of the
whole, the second $500 more than one third of the
whole, and the youngest $ 250 more than one fourth
of the whole. What was the value of the estate ?
SOLUTION.
Let X •=. whole estate.
Then ^ — 1440 = share of the eldest,
1+ 500= " " " second,
1+ 250= " '* " youngest,
13 X
Their sum — 690 =: x, whole estate.
i= 690
X = 8280, whole estate.
9. A gentleman meeting five poor persons, distributed
$7.50, giving to the second twice, to the third three times,
to the fourth four times, and to the fifth five times as much
as to the first How much did he give to each ?
94 ELEMENTARY ALGEBRA.
10. Divide 195 into two such parts that the greater di-
vided by 3 shall be equal to the less divided by 2.
Note. — To avoid fractions, let 3 a: = the greater and 2 a; = the less.
Ans. 477 and 318.
11. Divide a into two such parts that the greater di-
vided by h shall be equal to the less divided by c.
SOLUTION.
Let X = the greater,
then a — x = the less.
. , ' X a — X
And T =
b c
Clearing of fractions, cx^= ah — hx
Transposing, hx -\- ex =. ab
Dividing by 5 -f- c, x-= r-^-^ the greater,
ah ac , , ,
a — x=:a — f—. — = 5— j — , the less.
b-\'C b-\-c
12. What number is that which, if multiplied by 7, and
the product increased by eleven times the number, and
this sum divided by 9, will give the quotient 6 ?
13. If to a certain number 55 is added, and the sum
divided by 9, the quotient will be 5 less than one fifth
of the number. What is the number? Ans. 125.
14. As A and B are talking of their ages, A says to B,
*' If one third, one fourth, and seven twelfths of my age
are added to my age, the sum will be 8 more than twice
my age.^' What was A's age ?
15. A farmer having bought a horse kept him six weeks
at an expense of $20, and then sold him for four fifths of
the original cost, losing thereby $ 50. How much did he
pay for the horse? Ans. $150.
16. A man left $ 18204, to be divided among his widow,
three sons, and two daughters, in such a manner that the
widow should have twice as much as a son, and each son
as much as both daughters. What was the share of each ?
EQUATIONS OF THE FIRST DEGREE. 95
IT. If a certain number is divided by 9, the sum of the
divisor, dividend, and quotient will be 89. What is the
number? Ans. 72.
18. If a certain quantity is divided by a, the sum of the
divisor, dividend, and quotient will be b. What is the
quantity ?
19. Verify the answer to the preceding problem.
20. A farmer mixed together corn, barley, and oats. In
all there were 80 bushels, and the mixture contained two
thirds as much corn as barley and one fifth as much bar-
ley as oats. How many bushels of each were there ?
21. Three men, A, B, and C, built 572 rods of fence. A
built 8 rods per day, B 7, and C 5. A worked one half
as many days as B, and B one third as many as C. How
many days did each work ?
22. What number is as much greater than 340 as its
third part is greater than 34 ? Ans. 459.
23. A man meeting some beggars gave 3 cents to each,
and had 4 cents left. If he had undertaken to give 5 cents
to each, he would have needed 6 cents to complete the dis-
tribution. How many beggars were there, and how much
money did he have ?
SOLUTION.
Let X = the number of beggars ;
then, according to the first statement,
3 X + 4 = the number of cents he had,
and, according to the second statement,
bx — 6 = the number of cents he had.
Therefore, 6a: — 6 = 3x + 4
2a:=10
ar = 5, the number of beggars,
and 8x -[- 4 = 19, the number of cents he had.
96 ELEMENTARY ALGEBRA.
24. A boy wishing to distribute all his money among
his companions gave to each 2 cents, and had 3 cents
left ; therefore, collecting it again, he began to give 3
cents to each, but found that in this case there was one
who had received none, and another who had only 2
cents. How many companions, and how much money
had he? Ans. 7 companions, and IT cents.
25. What two numbers whose difference is 35 are to
each other as 4 : 5 ?
26. A man being asked the hour, answered that three
times the number of hours before noon was equal to three
fifths of the number since midnight. What was the time
of day ?
SOLUTION
Let X = the number of hours since midnight, i. e. the time ;
then 12 — x = the number of hours before noon.
Then . 36— 3^:=?^
5
Clearing of fractions, 180 — 15 x =z^x
Whence 18x = 180
X =z 10. Ans. 10 o'clock.
2Y. A gains in trade $ 300 ; B gains one half as much
as A, plus one third as much as C ; and C gains as much
as A and B. What is the gain of B and C ?
Ans. B's, $375; C's, $675.
28. What number is to 28 increased by one third of
the number as 2 : 3 ? Ans, 24.
29. What number is that whose fifth part exceeds its
sixth b}^ 15 ?
30. Divide $3740 into two parts which shall be in the
ratio of 10 : 7.
31. Divide a into two parts which shall be in the ratio
of6:c, . ah , ac
Ans. i— 1 — and 7—-, —
h-\- c -\-c
EQUATIONS OF THE FIRST DEGREE. 97
32. What number ia that the sum of whose fourth part,
fifth part, and sixth part is 37 ?
33. What quantity is that the sum of whose third part,
fifth part, and seventh part is a ? » 105 a
-ft-ns. ^^
34. A farmer sold IT bushels of oats at a certain price,
and afterward 12 bushels at the same rate ; the second
time he received 55 shillings less than the first. What
was the price per bushel ?
35. A certain number consists of two figures whose
sum is 9 ; and if 27 is added to the number, the order of
the figures will be inverted. What is the number ?
SOLUTION.
liCt X =z the left-hand figure ;
then 9 — x = the right-hand figure.
As figures increase from right to left in a tenfold ratio,
10 x + (9 — x) = 9 a: -f- 9 = the number ;
and when the order of the figures is inverted,
10 (9 — ar) -f- a: = 90 — 9 a: := the resulting nu mber.
Therefore 9x+9 + 27 = 90— -9a:.
Or 18 a: = 54
Whence ar = 3, the left-hand figure,
and 9 — x = 6, the right-hand figure.
Ans. 36.
36. A certain number consists of three figures whose
sum is 6, and the middle figure is double the left-hand
figure ; and if 198 is added to the number, the order of
the figures will be inverted. What is the number ?
Ans. 123.
37. Two men 90 miles apart travel towards each other
till they meet. The first travels 5 miles an hour and the
second 4. IIow many miles does each travel before they
meet?
6 o
98 ELEMENTARY ALGEBRA.
38. A man hired six laborers, to the first of whom he
paid t5 cents a week more than to the second ; to the
second, 80 cents more than to the third ; to the third, 60
cents more than to the fourth ; to the fourth, 50 cents
more than to the fifth ; to the fifth, 40 cents more than
to the sixth; and to all he paid $68.15 a week. What
did he pay to each a week ?
89. What number is that to which if 20 is added two
thirds of the sum will be 80 ?
40. What number is that to which if a is added - of
c
the sum will be c?? \^„ ^^ ^
Ans. -T a.
41. A man spent one fourth of his life in Ireland, one
fifth in England, and the rest, which was 33 years, in
the United States. To what age did he live?
42. A post is one fifth in the mud, two sevenths in
the water, and 18 feet above the water. How long is
the post ?
43. What number is that whose half is as much less
than 40 as three limes the number is greater than 156?
Ans. 56.
44. Two workmen received the same sum for their la-
bor ; but if one had received $15 less and the other $15
more, one would have received just four times as much
as the other. What did each receive ?
45. Of the trees on a certain lot of land five sevenths
are oak, one fifth are chestnut, and there are 32 less wal-
nut trees than chestnut. How many trees are there ?
46. Divide 474 into two parts such that, if the greater
part is divided by 7 and the less by 3, the first quo-
tient shall be greater than the second by 12.
Ans. 357 and 117.
EQUATIONS OF THE FIRST DEGREE. 99
4Y. Two persons, A and B, have each an annual income
of S1500. A spends every year $400 more than B, and
at the end of five years the amount of their savings is
$6000. What does each spend annually?
Ans. A $1100, and B $700.
48. In a skirmish the number of men captured was 41
more, and the number killed 26 less than the number
wounded ; 45 men ran away ; and the whole number en-
gaged was four times the number wounded. How many
men belonged to the skirmishing party ? Ans. 240.
49. A and B have the same salary. A runs into debt
every year a sum equal to one sixth of his salary, while
B spends only three fourths of his ; at the end of five
years B has saved $1000 more than enough to pay A's
debt What is the salary of each ? Ans. $ 2400.
60. A man lived single one third of his life : after hav-
ing been married two years more than one eighth of his
life, he had a daughter who died ten years after him,
and whose age at her death was one year less than two
thirds the age of her father at his death. What was the
father's age at his death ?
SOLUTION.
Let X =: his age ;
then - = his age at marriage,
o
^ + f + 2 = his age at daughter's birth,
8 o
and ar — ( 1 4- ^ -|- 2 ) = her age at his death.
Then x-f-|-2+10 = '^-l
Transposing and uniting, — - = — 9
X = 72, the father's age.
100 ELEMENTARY ALGEBRA.
51. Divide $ 864 among three persons so that A shall have
as much as B and C together, and B $5 as often as C $ 11.
52. A father and sou are aged respectively 32 and 8.
How long vi^ill it be before the son will be just one half
the age of the father ?
53. A man's age was to that of his wife at the time
of their marriage as 4:3, and seven years after, their
ages were as 5:4. What was the age of each at the
time of their marriage ?
54. One fifth of a certain number minus one fourth of
a number 20 less is 2. What is the number? Ans. 60.
55. There are two numbers which are to each other as
J : ^ ; but if 9 is added to each, they will be as i : ^.
What are the numbers ? Ans. 9 and 6.
56. A person having spent $ 150 more than one third
of his income had $ 50 more than one half of it left.
What was his income ?
5T. A merchant sold from a piece of cloth a number
of yards, such that the number sold was to the number
left as 4 : 5 ; then he cut off for his own use 15 yards,
and found that the number of yards left in the piece was
to the number sold as 1:2. How many yards did the
piece originally contain ? Ans. 45.
58. Four places. A, B, C, and D, are in a straight line,
and the distance from A to D is 126 miles. The distance
from A to B is to the distance from B to C as 3 : 4, and
one third the distance from A to B added to three fourths
the distance from B to C is twice the distance from to
D. What is the distance from A to B, from B to C, and
from C to D ?
59. A laborer was hired for 40 days ; for each day he
wrought he was to receive $2.50, and for each day he
was idle he was to forfeit $1.25. At the end of the time
he received $58.75. How many days did he work?
Ans. 29.
A e.^,-
EQfUATIONS OF THE HBST DEGREE. 101
60. A cask which held 44 gallons was filled with a
mixture of brandy, wine, and water. 'T^crif) 'were '19: gal-
lons more than one half as muct ,wine as brandy, ^nd , as
much water as brandy and wJDje/'-Hpw* ulai^y*' g&>)[^V8'
were there of each ?
61. Two persons, A and B, travelling each with S80,
meet with robbers who take from A $ 5 more than twice
what tliey take from B; then B finds he has $26 more
than twice what A has. IIow much is taken from each ?
Ans. From A, $69 ; from B, $32.
62. Four persons, A, B, C, and D, entered into part-
nership with a capital of $84810; of which B put in
twice as much as A, C as much as A and B, and D as
much as A, B, and C. How much did each put in ?
63. In three cities, A, B, and C, 1188 soldiers are to
be raised. The number of enrolled men in A is to that
in B as 3 : 5 ; and the number in B to that in C as 8 : T.
How many soldiers dught each city to furnish ?
Ans. A, 288 ; B, 480 ; C, 420.
64. Divide $65 among five boys, so that the fourth
may have $2 more than the fiftli and $3 less than the
third, and the second $4 more than the third and $5
less than the first.
65. A merchant bought two pieces of cloth, one at the
rate of $ 5 for 7 yards, and the other $ 2 for 3 yards ;
the second piece contained as many times 3 yards as the
first times 4 yards. He sold each piece at the rate of
$6 for 7 yards, and gained $24 by the bargain. How
many yards were there in each piece ?
Ans. First, 84 ; second, 63.
66. A drover had the same number of cows and sheep.
Having sold 17 cows and one third of his sheep, he finds
he has three and a half times as many sheep as cows left.
How many of each did he have at first?
102 ELEMENTARY ALGEBRA.
67 A flour dealer sold one fourth of all the flour he
had aod one foilrth'. of: a barrel ; afterward he sold one
third. c>f wli^t he had left .and one thh-d of a barrel; and
ttlen. ofte Jigjf of tl^Q remainder and one half of a barrel;
and had 15 barrels left. How many had he at first ?
SOLUTION.
Let X = number at first;
3 X 1
then — — - = number after first sale,
2 /3 X 1\ 1 a; 1
g{~l ;) — 3^^^2 — 2^^ number after second sale,
and 2 §2 — 2/ — 2^^4 — 4^^ number after third sale.
Then ^ -^ ? =:: 15
4 4
Clearing of fractions, x — 3 = 60
Whence x = 63, number at first.
«
68. A merchant bought a barrel of oil for $50; at the
same rate per gallon as he paid, he sold to one man 15
gallons ; then to another at the same rate two fifths of
the remainder for $ 14. IIow many gallons did he buy
in the barrel ?
69. Two pieces of cloth of the same length but dif-
ferent prices per yard were sold, one for $5 and the
other for St. 50. If there had been 6 more yards in
each, at the same rate per yard as before, they would
have come to $ 15.41^^. How many yards were there in
each? Ans. 21.
10. A and B began trade with equal sums of money.
The first year A lost one third of his money, and B
gained $ 150. The second year A doubled what he had
at the end of the first year, and B lost $150, when the
two had again an equal sum. What did each have at
first ?
EQUATIONS OF THE FIRST DEGREE. 108
11. A man distributed among his laborers $2.50 apiece,
and had $25 left. If he had given each $3 as long as
his money lasted, three would liave received nothing.
How many laborers were there, and how much money
did he have? Ans. 68 laborers, and S195.
72. A man who owned two horses bought a saddle for
S35. When the saddle was put on one horse, their value
together was double the value of the other horse ; but
when the saddle was put on the other horse, their value
together was four fifths of the value of the first horse*
What was the value of each horse ?
T3. From a cask two thirds full 18 gallons were taken,
when it was found to be five ninths full. How many
gallons will the cask hold ?
H. A farmer had two flocks of sheep, and sold one
flock for $60. Now a sheep of the flock sold was worth
4 of those left, and the whole value of those left was $8
more than the price of 8 sheep of those sold, and the flock
left contained' 40 sheep. How many sheep did the farmer
sell, and what was the value of a sheep of each flock ?
Ans. Number sold, 15; value, $4 and $1.
75. A man has seven sons with 2 years between the
ages of any two successive ones, and the sum of all their
ages is ten times the age of the youngest. What is the
age of each ?
76. Divide 75 into two parts such that the greater in-
creased by 9 shall be to the less diminished by 4 as 3 : 1.
77. Divide a into two parts such that the greater in-
creased by b shall be to the less diminished by c as m : n.
78. What two numbers are as 3:4, while if 8 be
added to each the sums will be as 5 : 6 ?
79. Divide 127 into two parts, such that the difference
between the greater and 130 shall be equal to five times
the difference between the less and 63.
104 ELEMENTARY ALGEBRA.
SECTION XIV.
EQUATIONS
OF THE FIRST DEGREE CONTAINING TWO UNKNOWN
QUANTITIES.
109. Independent Equations are such as cannot be de-
rived from one another, or reduced to the same form.
Thus, a; + y= 10, I + 1=5, and 4a:+32^=:40— y
are not independent equations, since any one of the three
can be derived from any other one ; or they can all be
reduced to the form x -\- y = 10. But a; -j- y = 10 and
4:X = 1/ are independent equations.
110. To find the value of several unknown quantities,
there must be as many independent equations in which
the unknown quantities occur as there are unknown
quantities.
From the equation x -}- i/ = 10 we cannot determine the value
of either a: or y in known terms. If y is transposed, we have
x= 10 — y; but since y is unknown, we have not determined the
value of X. We may suppose y equal to any number whatever,
and then x would equal the remainder obtained by subtracting y
from 10. It is only required by the equation that the sum of two
numbers shall equal 1 ; but there is an infinite number of pairs
of numbers whose sum is equal to 10. But if we have also the
equation 4x = y, vre may put this value of y In the first equation,
X -{- y = 10, and obtain a: -j- 4 a: = 10, or a; = 2 ; then 4 a: = 8 = y,
and we have the value of each of the unknown quantities.
ELIMINATION.
Ill, Elimination is the method of deriving from the
given equations a new equation, or equations, containing
one less unknown quantity. The unknown quantity thus
excluded is said to be eliminated.
EQUATIONS OF THE FIRST DEGREE. 105
There are three methods of elimination : —
I. By substitution.
II. By comparison.
111. By combination.
CASE I.
112, Elimination by substitution.
1. Given |*^ + ^y = 2^l, to find a;and y.
OPERATION.
4a: + 6y = 23 (1) 6a: + 4y= 22 (2)
(3) 6x + 4(?i^)= 22 (4)
25X + 92 — 16aj = 110 (5)
?^ = 3 (7) x= 2 (6)
23 — 4 z
y = — s —
Transposing 4 x in (1) and dividing by 5, we have (8), which
gives an expression for the value of y. Substituting this value of
y in (2), we have (4), which contains but one unknown quantity ;
i. e. y has been eliminated. Reducing (4) we obtain (6), or
a: == 2. Substituting this value of x in (3), we obtain (7), or
y = 3. Hence,
RULE.
Find an expression for the value of one of the unknown
quantities in one of the equations, and substitute this value
for the same unknown quantify in the other equation.
Note. — After eliminating, the resulting equation is reduced by
the rule in Art. 102. The value of the unknown quantity thus
found must be substituted in one of the equations containing the
two unknown quantities, and this reduced by the rule in Art. 102.
Find the values of x and y in the following equations : —
2. Given 1^ + !'= in. Ans. j- = l«-
106 ELEMENTARY ALGEBRA.
- + ^=12
3. Given ^ - - L. Ans. j^=^^•
5 + ^ - . J (5^ = 15.
4. Given
5. Given
(2. -I- 3 = 0) ^^^^ (
(a;+y — 29 = 0) ^
■J 2 "^ 3 36 >• .
(2a7+ 32/ = 2 )
x= 1.
^ = 22.
6. Given ] 2~3-" ^L
(3a: — 3^ = 16)
CASE II.
113« Elimination by comparison.
1. Given {2rJy = 2?}' *^ ^^^ ^ ^^^ y-
OPERATION.
a: — 2y = 6 (1) 2a: — y = 27 (2)
a: = 6 + 2s^ (3) ^_ 27 + y (^^
«l/ —
" . 2
6 + 2y=^' + ^
(5)
12 + i!, = 21+y
(6)
P= 6
0)
a;= 6 + 10 = 16
(8)
Finding an expression for the value of x from both (1) and (2),
we have (3) and (4). Placing these two values of x equal to
each other (Art. 13, Ax. 8), we form (5), which contains but one
unknown quantity. Reducing (5) we obtain (7), or* ?/ = 5. Sub-
stituting this value of y in (3), we have (8), or a: = 16. Hence,
EQUATIONS OF THE FIRST DEGREE.
RULE.
107
Find an expression for ihe value of the same unknovm
quantity from each equation, and put these expressions equal
to each other.
By this method of elimination find the values of x and y
in the following equations : —
{. Given -l^^""! — ^[..
3. Given
+ y = 12
= 8
4. Given |3a: + 5y = 2|
5. Given
(3(x-y)-9 = 0)
7. Given i6*-5y=17|.
( 2a; — v = 13>
8. Given ■
:-!=«
Ana.
( V = 3.
Ans
( V =
= 24.
Ans.
Ans.
(a: = 2.
108
ELEMENTARY ALGEBRA.
CASE III.
114i Elimination by combination.
1. Given ■< ^ ^ ?- , to find x and y.
OPERATIOS
.
Zx — 2y— 1
(1)
2a; — 3y = 3
(2)
Qx — iy— 14
(3)
6 a; — 9.y= 9
(4)
by= 5
(5)
2a; — 3 =3
0)
y= 1
(6)
a;=:3
(8)
If we multiply (1) by 2, and (2) by 3, we have (3) and (4),
fn which the coefficients of x are equal; subtracting (4) from (3),
we have (5), which contains but one unknown quantity. Redu-
cing (5), we have (6), or y = 1 ; substituting this value of y in
(2), we obtain (7), which reduced gives (8), or a; = 3.
2. Given -
-X y __
2 4
13 "^ 2
6
12
- , to find X and y.
•
DPI
SRATION.
!-!= ' w
.-1=12
(2)
(3)
9-1= 6 (6)
'i-^^
(4)
y =
12 (1)
a: = 18
(5)
If we multiply (1) by 2, we have (3), an equation in which y has
the same coefficient as in (2) ; since the signs of y are different in
(2) and (3), if we add these two equations together, we have
(4), which contains but one unknown quantity. Reducing (4), we
have (5), or a: = 18. Substituting this value of x in (1), we have
(6), which reduced gives (7), or 3/ =-12. Hence,
EQUATIONS OF THE FIRST DEGREE.
RULE.
109
Multiply or divide the equations so that the coefficients of
the quantity to he eliminated shall become equal ; then, if the
signs of this quantity are alike in both, subtract one equa-
tion from the other ; if unlike, add the two equations to-
getlier.
Note. — The least multiplier for each equation will be that which
will make the coefficient of the quantity to be eliminated the least
common multiple of the two coefficients of this quantity in the
given equations.' It is always best to eliminate that quantity whose
coefficients can most easily be made equal.
By this method of elimination find the values of x and y
in the following equations : —
3. Given |^- + 3y = 33> ^^ jx
x=3.
4.
4. Given
f 8a: + 6t/
(lOx — 3y
}■
Ans
6. Given ■
19 • 6
y __
12
5
Ans. 1^ = 27.
\y = 12.
6. Given
fx-f y
-^— 1 =0
1. Given
2
T X
110 ' ELEMENTARY ALGEBRA.
115* Find the values of x and y in the following
Examples.
Note. — Which of the three methods of elimination should be
used depends upon the relations of the coefficients to each other.
That one which will ehminate the quantity desired with the least
work is the best.
1. Given P- + 32/ = 25|.
\'!x + 2t/ = 2Si
(5x— y= 0>
2. Given
3. Given
4. Given
2
x—2y
3
— y=6
(x—l , 17
( 2x— .4y==17
Ans. y '
Ans
Ans.
= 3.
15.
fx = ll.
ly= 1.
(. = 9.
5. Given
2a;-{-3y 1 a;-|-22/ + 3
Z "~ 3 ~ 2
^__2.-2y^3
Ans,
(a; = ll.
(2^= 5.
6. Given
Y. Given
7+3 -^
L 8 + 16 — "^^r
—3 3— - ^^
^ — y 3: + y ___ J.
EQUATIONS OF THE FIRST DEGREE.
Ill
8. Given
2x_, 5y 8^
T ' T 15
3 7
Ans
9. Given
10. Given
5x +
54
+ 5 y = 102
ar + y I y^ '^y — ^
4 •"3'" 2
11. Given \ 4 + 3 "= ^ [ •
( 4 a; — 3 V = 25 ;
Ans
( v =
= 10.
20.
m
\ ^-l-:
= 48.
28.
(.y = — 3.
12. Given
^ — .V _o
Ans
1 —
13. Given
5 J
Adb,
10.
20.
14. Given
^-±y = y^2
4y — 4
2
15. Given
112
16. Given
ELEMENTARY ALGEBRA.
3 — ^ 3J
IT. Given
r ^+y _^ 1
"X" — ^~"3
18. Given ^
2y
3a:
a: + 3
4
7 — y
3
6
14
19. Given <
4a;— 7y
5
3rr + y
X — 4
= y + 3
PROBLEMS
PRODUCING EQUATIONS OF THE FIRST DEGREE CON-
TAINING TWO UNKNOWN QUANTITIES.
116t Many of the problems given in Section XIII. con-
tain two or more unknown quantities ; but in every case
these are so related to each other that, if one becomes
known, the others become known also ; and therefore
the problems can be solved by the use of a single let-
ter. But many problems, on account of the complicated
conditions, cannot be performed by the use of a single
letter. No problem can be solved unless the conditions
given are suflScient to form as many independent equa-
tions as there are unknown quantities.
1. A grocer sold to one man Y apples and 5 pears for
41 cents ; to another at the same rate 11 apples and 3
pears for 45 cents. What was the price of each ?
EQUATIONS OF THE FIRST DEGREE. 113
SOLUTION.
Let X = the price of an apple,
and y =z " " "a pear.
Then, by the conditions.
'Jx + 5y = 41
(1)
and Hx-f- 3y= 45
(2;
55x + 15ff = 225
(3)
21a;+15y=123
(4)
2l + 5yr=41
(•?)
34x=102
(5)
y= 4
(8)
x= 3
(6)
We multiply (2) by 6 and (1) by 8, and obtain (3) and. (4);
subtracting (4) from (3) we have (5), which reduced gives (6), or
X = 3. Substituting this value of x in (1), we have (7), which re-
duced gives (8), or y = 4.
2. There is a fraction such that if 2 is added to the
numerator the fraction will be equal to ^ ; but if 3 is
added to the denominator the fraction will be equal to ^.
What is the fraction ?
SOLUTION.
Let - = the fraction.
y
Then, by the conditions,
'-±1 = \ (1) and^,=i (2)
Zx = y-\-^ (3)
2x + 4=y (4)
X — 4 = 3 (5)
x=1 (6)
14 + 4 = 18=y a) ^ = f, (8)
Clearing (1) and (2) of fractions, we obtain (3) and (4) ; sub-
tracting (4) from (3), we obtain (5), which reduced gives (G), or
a; = 7. Substituting this value of x in (4), we have (7), ot y= 18.
Hence, - - ~
114 ELEMENTARY ALGEBRA.
3. There are two numbers whose sum is 28, and one
fourth of the first is 3 less than one fourth of the second.
What are the numbers ? Ans. 8 and 20.
4. The ages of two persons, A and B, are such that 5
years ago B's age was three times A's ; but 15 years hence
B's age will be double A's. What is the age of each ?
Ans. A's, 25; B's, 65.
6. There are two numbers such that one third of the
first added to one eighth of the second gives 39 ; and
four times the first minus five times the second is zero.
What are the numbers ?
6. Find a fraction such that if 6 is added to the nu-
'nerator its value will be ^, but if 3 be added to the de-
nominator its value will be -^ ? Ans. /j.
t. What are the two numbers whose difference is to
their sum as 1:2, and whose sum is to their product
as 4 : 3 ?
SOLUTION.
Let X = the greater and y = the less.
Thenx — i/:x + y=l:2 (1) x-]-y : xy = 4.: S (2)
2x — 2y = x + y (3) Sx + Sy = 4:xy (4)
x = Sy (5) 9y + Sy=l2f (6)
x = 3 (1) l=:y (8)
Having written (1) and (2) in accordance with the statement in
the problem, we form from them (3) and (4) by Art. 106. Re-
ducing (3), we obtain (5) ; substituting this value of x in (4), we
have (6), which, though an equation of the second degree, can be
at once reduced to an equation of the first degree by dividing each
term by y ; performing this division and reducing, we obtain (8) or
y = 1 ; substituting this value of y in (5) we obtain (7), or ar = 3.
EQUATIONS OF THE FIRST DEGREE. 115
8. What are the two numbers whose difference is to
their sum as 3 : 20, and three times the greater minus twice
the less is 35 ?
9. There is a niimb^r consisting of two figures, which
is seven times the sum of its figures ; and if 36 is sub-
tracted from it, the order of the figures will be inverted.
What is the number ? Ans. 81.
10. There is a number consisting of two figures, the
first of which is the greater ; and if it is divided by the
sum of its figures, the quotient is 6 ; and if the order of
the figures is inverted, and the resulting number divided
by the difference of its figures plus 4, the quotient will
be 9. What is the number? Ans. 54.
11. As John and James were talking of their money,
John said to James, " Give me 15 cents, and I shall have
four times as much as you will have left." James skid
to John, "Give me 7^ cents, and I shall have as much
as you will have left/' How many cents did each
have ? Ans. John, 45 cents ; James, 30 cents.
12. The height of two trees is such that one third of
the height of the shorter added to three times that of
the taller is 360 feet ; and if three times the height of
the shorter is subtracted from four times that of the taller,
and the remainder divided by 10, the quotient is 17. Re-
quired the height of each tree.
Ans. 90 and 110 feet.
13. A farmer who had $41 in his purse gave to each
man among his laborers $2.50, to each boy SI, and had
$15 left. If he had given each man S4 and then each
boy $3 as long as his money lasted, 3 boys would have
received nothing. How many men and how many boys
did he hire ?
116 ELEMENTAKY ALGEBRA.
14. A man worked 10 days and his son 6, and they
received $ 31 ; at another time he worked 9 days and
Ms son 1, and they received $29.50. What were the
wages of each ?
m
15. A said to B, "Lend me one fourth of your money,
and I can pay my debts." B replied, "Lend me $100
less than one half of yours, and I can pay mine." Now
A owed $1200 and B $1900. IIow much money did
each have in his possession ?
Ans. A, $800 ; B, $1600.
16. If a is added to the difference of two quantities,
the sum is b ; and if the greater is divided by the less,
the quotient will be c. What are the quantities ?
. he — ac , h — a
Ans. ■ - and -•
c — 1 c — 1
lY. A man owns two pieces of land. Tliree fourths
of the area of the first piece minus two fifths of the area
Df the second is 12 acres ; and five eighths of the area
of the first is equal to four ninths of the area of the
second. How many acres are there in each ?
Ans. 1st, 64 acres ; 2d, 90 acres.
18. A and B begin business with different sums of
money; A gains the first year $350, and B loses $500,
and then A's stock is to B's as 9 : 10. If A had lost
$500 and B gained $350, A's stock would have been to
B's as 1:3. With what sum did each begin ?
Ans. A, $1450; B, $2500.
19. If a certain rectangular field were 4 feet longer
and 6 feet broader, it would contain 168 square feet more ;
but if it were 6 feet longer and 4 feet broader, it would
contain 160 square feet more. Required its length and
breadth.
EQUATIONS OF THE FIRST DEGREE. 117
20. A market-man bought eggs, some at 3 for Y cents
and some at 2 for 5 cents, and paid for the whole $2.62 ;
he afterward sold them at 36 cents a dozen, clearing
$0.62. How many of each kind did he buy?
21. A and B can perform a piece of work together in
12 days. They work together 7 days, and then A fin-
ishes the woRk alone in 15 days. How long would it
take each to do the work ? Ans. A 36 and B 18 days.
22. "I was ten times as old as you 12 years ago,''
said a father to his son ; " but 3 years hence I shall be
only two and one half times as old as you.'' What
was the age of each ?
23. If 3 is added to the numerator of a certain frac-
tion, its value will be f ; and if 4 is subtracted from the
denominator, its value will be j-. What is the fraction?
24. A farmer sold to one man Y bushels of oats and 5
bushels of corn for $12.76, and to another, at the same
rate, 5 bushels of oats and 7 bushels of corn for $13.40.
What was the price of each ?
25. Find two quantities such that one third of the first
minus one half the second shall equal one sixth of a ;
and one fourth of the first plus one fifth of the second
shall equal one half of a. . 34 a , 15 a
Ans. — and -23^.
26. A person had a certain quantity of wine in two
casks. In order to obtain an equal quantity in each, he
poured from the first into the second as much .as the
second already contained; then he poured from the sec-
ond into the first as much as the first then contained ;
and, lastly, he poured from the first into the second as
much as the second still contained ; and then he had 16
gallons in each cask. How many gallons did each origi-
nally contain? Ans. 1st, 22 ; 2d, 10 gallons.
118
ELEMENTARY ALGEBRA.
SECTION XV.
EQUATIONS
OF THE FIRST DEGREE CONTAINING MORE THAN TWO
UNKNOWN QUANTITIES.
117. The methods of elimination given for solving equa-
tions containing two unknown quantities apply equally
well to those containing more than two unknown quantities.
( ^+ y— ^= 4
1. Given -l^x -\- ^y -{- Az
izx — 2y\-bz
= A.
= n k to
= 5)
find X, y, and z.
OPERATION.
a: + y — 2 = 4 (1) 2x + 3y + 4.z=l7 (2)
3x
— 2r/+ 5z= 5 (3)
2x + 2y — 2z= 8 (4)
3x
+ 3y— 32; =12 (5,^
y + 62= 9 (6)
by— 82= 7 (7{
5y + 302=45 (8j
x + 3 — 1=4 (13) y + 6 = 9 (11)
382 = 38 (9)
x = 2 ' (14) y = S (12)
z= 1(10)
Multiplying equation (1) by 2 gives equation (4), which we sub-
tract from (2), and obtain (6) ; multiplying (1) by 3 gives (5), and
subtracting (5) from (3) gives (7). We have now obtained two
equations, (6) and (7), containing but two unknown quantities. Mul-
tiplying (6) by 5, we obtain (8), and subtracting (7) from (8), we
obtain (9), which reduced gives 2=1. Substituting this value of
z in (6), and reducing, we obtain y == 3. Substituting these values
of y and z in (1), and reducing, we obtain x = 2.
2. Given -
X + y =26'
y +z=29
z -|- t^> = 66
w -\-u =Sl
M + « = 4:6
■ , to find u, w, X, y, and z.
EQUATIONS OF THE FIRST DEGREE. 119
OPERATION.
z + y«26 (1) y + « = 29 (2) *-Hw-56 (8) ir + « =« 81 (4) « + x»-46 (5|
y + z = 28 «— z— 3 to+z=53 «— z = 28
X— z- 8 (6) i»+z=53 (7) « — z=28 (8) 2z = 18 (9)
y = 17 (11) z = 12 (12) ir = 44 (13) u = 37 (14) z= 9(10)
Here we subtract (1) fipom (2), and obtain (6) ; then (6) from
(3), and obtain (7) ; then (7) from (4), and obtain (8) ; then (8)
from (5), and obtain (9), which reduced gives (10), or x = 9. Sub-
stituting this value of x in (1), (6), (7), and (8), and reducing, we
obtain (11), (12), (13), and (14), or y = 17, z = 12, w = 44, and
w = 37.
Hence, for solving equations containing any number
of unknown quantities,
RULE.
Fr(mi the given equations deduce equations one less in
number, containing one less unknown quantify; and con-
tinue thus to eliminate one unknown quantity after an-
other , until one equation is obtained containing but one
unknown quantity. Reduce this last equation so as to find
the value of this unknown quantity ; then substitute this value
in an equation containing this and but one other unknown
quantity, and reducing the resulting equation, find the value
of this second unknown quantity ; substitute again these values
in an equation containing no more than these two and one
otJier unknown quantity, and reduce as before ; and so con-
tinue, till the value of each unknown quantity is found.
Note. — The process can often be very much abridged by the
exercise of judgment in selecting the quantity to be eliminated, the
equations from which the other equations are to be deduced, the
method of elimination which shall be used, and the simplest equa-
tions in which to substitute the values of the quantities which have
been found.
120
ELEMENTARY ALGEBRA.
Find the values of the unknown quantities in the fol-
lowing equations : —
3. Given
X -{- f/ -\- z -{- w = 16"
x-{-i/-\-w-\-u= 14
X-\-1/-{-Z-\- It z=l[>^
Note. — If these equations are added together and the sum di-^
vided by 4, we shall have x -\- y -{- z -{- w -\- u = 20 ; and if
from this the given equations are successively subtracted, the values
of the unknown quantities become known. r ^ g
U = 3.
Ans.<i z =z. Q,
6.
4. Given
x=2.
Ans. -J y = 4:.
(z=:6.
6. Given -
^2x + 33/ + 4^ = 6T-]
4
2 "> 2
2y + ^
'ZbJ
Ans
rx= 1.
iz = n.
6. Given
a;— y — 2;=: 1
X-{-2l/— 10z=: 1
2x — 4y-f- 3s= 1
rx =z 5.
Ans. -|y r= 3.
U = i.
T. Given
\^+y+\^=22
1 ,1,1
i^ + iy + 2^
1 , 1 . , 1
24
10
Ans.
E
20.
12.
32.
EQUATIONS OF THE FIBST DEGREE.
121
8. Given ^
1,1 5
x'^y— 6
y^2 12
1 , 1 ___ 3
Note. — The best method for this example is that used in Ex-
ample 3, without clearing of fractions.
9. Given
( ^+ y+ ^= 6j fxz=
-] 2a: + 3y + 4^ = 20 [■ . Ans. U =
xz= 1.
2.
3.
10. Given
11. Given
rx + iy = 3U
]y + ^z = ll[
iz +ix = 29)
x + y = a
11
rx-^y = a^ /a? = ^ (a — ft -j- c)
■\y + ^ = f>y- Ans. }y = i{a + b—c)
(a: + z = c) iz=^{b-\-c — a)
M
Ans. < f/z=-
abx-\-abyz=:a-\- b
raox-^a()y=za-\- b ^
Given <acx-{'acz=za-\-c>
\bcy-\-bcz:=b +c)
13. Given
I
x+ 21 =^y+28
9x = 2y
\
122 ELEMENTARY ALGEBBA.
PROBLEMS
PRODUCING EQUATIONS OF THE FIRST DEGREE CON-
TAINING MORE THAN TWO UNKNOWN QUANTITIES.
118. 1. A merchant has three kinds of flour. He can
sell 1 bbl. of the first, 2 of the second, and 3 of the third
for $85; 2 of the first, 1 of the second, and ^ bbl. of
the third for $45.50 ; and 1 of each kind for $41. What
is the price per bbl. of each ?
Ans. 1st, $ 12 ; 2d, $ 14 ; 3d, $ 15.
2. Three boj'^s. A, B, and C, divided a sum of money
among themselves in such a manner that A and B re-
ceived 18 cents, B and C 14 cents, and A and C 16. How
much did each receive? Ans. A, 10 ; B, 8 ; C, 6 cents.
3. As three persons, A, B, and C, were talking of their
ages, it was found that the sum of one half of A's age, one
third of B's, and one fourth of C's was 33 ; that the sum
of A's and B's was 13 more than C's age ; while the sum
of B's and C's was 3 less than twice A'a age. What
was the age of each? Ans. A's, 32 ; B's, 21 ; C's, 40.
4. As three drovers were talking of their sheep, says
A to B, "If you will give me 10 of yours, and C one
fourth of his, I shall have 6 more than C now has."
Says B to C, *'If you will give me 25 of yours, and A
one fifth of his, I shall have 8 more than both of you
will have left." Saj^s C to A and B, "If one of you
will give me 10, and the other 9, I shall have just as
many as both of you will have left." How many did
each have?
5. Divide 32 into four such parts that if the first part
is increased by 3, the second diminished by 3, the third
multiplied by 3, and the fourth divided by 3, the sum,
difierence, product, and quotient shall all be equal.
Ans. 3, 9, 2, and 18.
EQUATIONS OF THE FIRST DEGREE. 123
6. If A and B can perform a piece of work together
in 8|2j. days, B and C in 9^r^ days, and A and C in 8J
days, in how many days can each do it alone?
Ans. A in 15, B in 18, and C in 21 days.
7. Find three numbers such that one half of the fijst,
one third of the second, and one fourth of the third shall
together be 56 ; one third of the first, one fourth of the
second, and one fifth of the third, 43 ; one fourth of the
first, one fifth of the second, and one sixth of the third, 35.
8. The sum of the three figures of a certain number is
12 ; the sum of the last two figures is double the first ;
and if 297 is added to the number, the order of its fig-
ures will be inverted. What is the number?
Ans. 417.
9. A man sold his horse, carriage, and harness for
$450. For the horse he received $25 less than five
times what he received for the harness ; and one third
of what he received for the horse was equal to what he
received for the harness plus one seventh of what he
received for the carriage. What did he receive for each?
Ans. Horse, $225; carriage, $175; harness, $50.
10. A man owned three horses, and a saddle which
was worth $45. If the saddle is put on the first horse,
the value of both will be $30 less than the value of the
'second ; if the saddle is put on the second horse, the value
of both will be $55 less than the value of the third ; and
if the saddle is put on the third horse, the value of both
will be equal to twice the value of the second minus $ 10
more than one fifth of the value of the first. What is
the value of each horse ?
Ans. Ist, $100; 2d, $175; 3d, $275.
11. The sum of the numerators of two fractions is 7,
and the sum of their denominators 16 ; moreover the sum
of the numerator and denominator of the first is equal
124 ELEMENTARY ALGEBRA.
to the denominator of the second ; and the denominator
of the second, minus twice the numerator of the first, is
equal to the numerator of the second. What are the
fractions ? Ans. f and f .
l/Q. A man bought a horse, a wagon, and a harness,
for $ 180. The horse and harness cost three times as
much as the wagon, and the wagon and harness one half
as much as the horse. What was the cost of each ?
13. A gentleman gives $600 to be divided among three
classes in such a way that each one of the best class is
to receive $10, and the remainder to be divided equally
among those of the other two classes. If the first class
proves to be the best, each one of the other two classes
will receive $5 ; if the second class proves to be the best.
each one of the other two classes will receive $4f ; but
if the third class proves to be the best, each one of the
other two classes will receive $2, What is the number
in each class?
14. A cistern has 3 pipes opening into it. If the first
should be closed, the cistern would be filled in 20 min-
utes ; if the second, in 25 minutes ; and if the third, in
80 minutes. How long would it take each pipe alone
to fill the cistern, and how long would it take the three
together ?
Ans. 1st, 85f minutes ; 2d, 46-^7j- minutes ; 3d, 35^^^
minutes. The three together, IQ-^r minutes.
15. Three men. A, B, and C, had together $24. Now
if A gives to B and C as much as they already have
and then B gives to A and C as much as they have after
the first distribution, and again C gives to A and B as
much as they have after the second distribution, they will
all have the same sum. How much did each have at
first? Ans. A, $13 ; B, $1, and C, $4.
EQUATIONS OF THE FIRST DEGREE. 125
SECTION XYI.
POWERS AND ROOTS.
119. A Power of any quantity is the product obtained
by taking that quantity any number of times as a factor;
and the exponent shows how many times the quantity is
taken (Art. 24). Thus,
a=ia^ is the first power of a ;
a a=Ma^ " . second power, or square, of a ;
a a a:=: a^ " third power, or cube, of a ;
a a a az:^ a^ " fourth power of a ;
and so on.
Though not strictly within the definition, the quantity
itself is called the first power, and the first root of itself.
120. In order to explain the use of negative indices,
we form, by the rules of division, the following series: —
' 8 4 3 2 1^1 111
^' «' «' «' «' ^' a' ^' ^' a<' d'^
a\ a\ a», a^ a\ a^ a-\ a"*, a"*, a'^ a-\
We form the first series as follows: a* divided by a gives a*;
a* by a, gives a*; a' by a, gives a'; a' by a, gives a; a by a, gives
1 ; 1 by a, gives -; - by a, gives ,; , by a, gives -j, and so on.
The second series is formed in the same way from a' to a; but if
wc follow the same rule of division from a toward the right as from
rt* to a, viz. subtracting the index of the divvsor from that of the div-
l^nd^ a divided by a, gives cfi\ cfi by a, gives a-^ ; read a, with the
negative index one; a~* by a, gives o""; a~* by a, gives a~*\ and
Bo on.
From this we learn,
Ist. That the power of every quantity is 1 ;
2d. That cr^, a"*, ar^, &c., are only different ways of
•^ 111^
wrxtinq -, -,, — ,» (Jcc.
^ a or or
126 ELEMENTARY ALGEBRA.
Any two quantities at equal distances on opposite sides
of a°, or 1, are reciprocals of each other.
121. The rules given for the multiplication and divis-
ion of powers of the same quantity (Arts. 50 and 54)
apply equally well whether the exponents are positive
or negative. For
1 a'
a^ X ar^ = a^ X -, = ~, = ci^
a« -T- a-^ = a^ -f- - = a^ X ci' = c^
2_^
a'
a' a"
The following examples in multiplication are to be done
according to the rules for the multiplication of powers of
the same quantity by each other, given in Art. 50 ; and
those in division, by the rule for the division of powers
of the same quantity by each other, given in Art. 5JL.
1. Multiply x' by x'^. Ans. x^.
2. Multiply a^ by a'^.
3. Multiply x^ by x'^. Ans. a;^ or 1.
4. Multiply y-'^ by y*.
5. Multiply a-^x^y^ by a~^x~^y'^.
Ans. ar^x^y^, or ar^y''.
6. Multiply Ix-^y-^z by 3xV^*-
1. Multiply llx'^fz-^ by A^x-^y-^^^,
8. Multiply ^ll|-^' by 5 a^ b-"- c\
9. Divide x^ by x~^. Ans. x^^.
10. Divide x^ by x''^.
11. Divide a:~" by x~^. Ans. x-*.
EQUATIONS OP THE FIBST DEGREE. 127
12. Divide y-^ by f.
13. Divide y' by y*. Ans. y'^.
14. Divide a~*6c^ by a^h~* c'^. Ans. a~'^U*c*.
15. Divide 16x'y-*2 by 4a;-y-*2:«.
16. Divide 4ar»y-»z by 2aar-2y-«2*.
IT. Divide *la^hx-^y^ by 10 a i*^ a:* y-«2«.
18. Divide 144<7«6c-^x*y-'2 by 16 «''6-^c-='ar^/.
122t It follows from the preceding article that a factor
may he transferred from the numerator of a fraction to its
denominator, or vice versa, provided the sign of the expo-
nent of the factor is changed from -\- to — , or — to +. For
^ = a« X ^4 = a« X a:-* = a'x-*
a a 1 v> _^ ^
-,= ^=a^-, = aX^ = a2/
y ""y ^ ""y * ar'^y * "^ "~x-^y
— = 1 X - = —
X (/ ^ X dx
1. Transfer the denominator of ^-^ — r to the numerator.
bc'y-'^
Ans. cU Ir^ c'"^ x^ y .
2. Transfer the numerator of -j- , - , to the denommator.
3. Transfer the denominator of /, . to the numerator.
a <r
4. Transfer the numerator of ° . ^ ^ to the denominator.
128 ELEMENTARY ALGEBRA.
6. Free from negative exponents ,_^ —4.
Ans. ■= — I — ^-
lac*xy
6. Free from negative exponents
1. Free from negative exponents
8. Free from negative exponents . ::_a /"^ x •
3.— n yn
9. Free from negative exponents -zitj:-
INVOLUTION.
123. Involution is the process of raising a quantity to
a power.
124. A quantity is involved by taking it as a factor as
many times as there are units in the index of the re-
quired power.
125. According to Art. 48,
(+«) X (+a) X (+a) = (+a^) X {+a)= + a\
and so on ;
and ( — a ) X ( — a)= + fl^^
(-a) X (-«) X (-«) = (+a') X {-a) = -a\
(—a) X (— «) X i—a) X (— «) = (—a') X {—a)= + a\
and so on.
Hence, for the signs we have the following
RULE.
Of a positive quantity all the powers are positive.
Of a negative quantity the even powers are positive, and
the odd povjers negative.
INVOLUTION. 129
INVOLUTION OF MONOMIALS.
126. To raise a monomial to any required power.
1. Find the third power of 2 a* 6.
OPERATION.
(2 a» J)» = 2 an X 2 a« i X 2 a« 6 (1)
= 2. 2. 2. a'^a^'a^bbb (2)
= 8a»6» (3)
According to Art 124, to raise 2 0*6 to the third power we take
it as a factor three times (1) ; and as it makes no difference in the
product in what order the factors are taken, we arrange them as
in (2) ; performing the multiplication (Art. 60) expressed in (2),
we have (3). Hence,
RULE.
Multiply the exponent of each letter by the index of the
required power, and prefix the required power of the nu-
merical coefficient, remembering that the odd powers of a
negative quantity are negative, while all other powers are
positive.
Note. — It follows that the power of the product is equal to the
product of the powers.
2. Find the square of 2 x. Ans. 4 x*.
3. Find the cube of 3 x^. Ans. 21 a:«.
4. Find the fourth power of a* 6*. Ans. a" ft*.
6. Find the third power of 4 a'' a:. Ans. 64 a* a:*.
6. Find the square of 2 x'^. Ans. 4 x'^,
7. Find the cube of 3ar-*y^ Ans. 27a:-»y.
8. Find the mth power of a b. Ans. a*" ft'\
9. Find the third power of — 3 a'* b. Ans. — 27 a' ft^
130 ELEMENTARY ALGEBRA.
10. Expand (—2a^xy. Ans. 16a^^x\
11. Expand (3 aH")"*. Ans. S^a^^If'\
12. Expand (2x'yy.
13. Expand {—la'^ot^y,
14. Expand (--3a;»y)8.
15. Expand (— a-y. Ans. — a-\
16. Expand (x-^y^.
11. Expand (— 4a;-8 3/)2. Ans. i^.
18. Expand (3a'»a;2)4.
19. Expand (— 2 a;-^ 3/-«)».
20. Expand (— 3a;-2» y'»)6.
21. Expand (—9 a-2i-'»a;V)'-
INVOLUTION OF FRACTIONS.
127t To involve a fraction.
a*
1. Find the cube of ^r-r'
2
V2&7 ""26
OPERATION. According to Art. 124,
sy ^^ ^ ^ ^ *^ fi"^^ *^® ^^^ of ^'^y
^ 2^ '^ 26 86» quantity we must take it
three times as a factor:
taking — =- three times as a factor, and performing the multiplica-
a'
tion by Art. 95, we have 77^5. Hence,
BULE.
Involve both numerator and denominator to the required
power.
INVOLUTION. 131
2. Find the square of r— 3-
3. Find the cube of — ir-r-,- Ans. —
4. Find the fourth power of
26c«
3xr"
6. Find the fifth power of — ^^'.
2 a z~*
6. Find the third power of
7. Find the mth power of
2 a'ar**
8. Find the fourth power of , _, ^ «
3 a* 6-* c-'
9. Find the third power of — h'^^fr^'
10. Find the fourth power of — -—_,-:_,-■
■^ m ■ n^ p
11. Find the fifth power of — ~i'lr*'<*'
INVOLUTION OF BINOMIALS.
128. A BINOMIAL can be raised to any power by suc-
cessive multiplications. But when a high power is re-
quired, the operation is long and tedious. The Binomial
Theorem, first developed by Sir Isaac Newton, enables us
to expand a binomial to any power by a short and speedy
process.
129t In order to investigate the law which governs the
expansion of a binomial we will expand a-\-h and a — h
to the fifth power by multiplication.
132 ELEMENTARY ALGEBRA.
a -\-h
a +h •
a^-i- ab
ah 4-y
a^ -\- 2 a b -\- h^ 2d power.
a +b
an-\- ^ab^ -\-b^
a^j^Zan+ 3 a 5^ + 6« . . . .3d power.
a -\-b
an 4- 3 g^ ^2 _[_ 3 a 53 _[_ ^4
a*-(-4an-|- 6a2^2_|_ 4^j3 _|_54 ^ ^ 4th power.
g -f&
a«_|-4:an+ 6an2+ 4a2 6«4- a6*
a^-l- 4a«62 4- 6g2^«-j-4aZ^^ + ^«
a5_j_6an 4- 10^6^+ l^ aH^ -\- b a ¥ -\- b^ 6th power.
a — h •
a — h
o^ ■— a 6
— g 6 + 6^
g2 — 2 a 6 + 62 . . , , . .2d power.
a — h
— g26-f- 2 g62 — 58
aS _ 3 ^2 5 _j_ 3 gj 52 _ j8
a — b
ai^San+ 3g262_ ^Ti^
— g^ft-i- 3g2&2_ 3a6 « + &^
a4_4aH-f ^an^— 40^^^ - . 4th power.
a — b
a5 — 4an+ 6gH2_ 4,aH^+ a¥
•— g^-f 4an2_ 6a2&«-f 4a&^-~-&s
o^ — 5 a^ 4- 10 a" ^' -- 10 a2 6^ + 5 a 6^ — 6^ 6th powen
3d power.
INVOLUTION. 13B
By examining the diflferent powers of a -\- h and a — b
in these Examples, we shall find the following invariable
laws governing the expansion : —
1st. The leading quantity (i. e. the first quantity of th^
binomial) begins in the first term of the power with an ex-
ponent equal to the index of the power, and its exponent
decreases regularly by one in each successive term till it dis-
appears ; the following quantity (i. e. the second quantity
of the binomial) begins in the second term of the power ivitJ.
the exponent one, and. its exponent increases regularly hj
one till in the last term it becomes the same as the index of
the power.
Thus, in the fifth power the
Exponents of a are 5, 4, 3, 2, 1.
Exponents of 6 are * 1, 2, 3, 4, 6.
It will be noticed that the sum of the exponents of the
letters in any term is equal to the index of the power.
2d. The coefficient of the first term is one ; of the second,
the same as the index of the power ; and universally, the co-
efficient of any term, multiplied by the exponent of the lead-
ing quantity, and this product, divided by the exponent of
Die following quantity increased by one, will give the co-
efficient of the succeeding term.
Thus, in the fifth power, 5, the coefficient of the second
term, multiplied by 4, a's exponent, and divided by
1 plus 1, 6's exponent plus 1, =: — ^ = 10, the coeffi-
cient of the third term.
The coefficients are repeated in the inverse order after
passing the middle term or terms, so that more than half
of the coefficients can be written without calculation. The
number of terms is always one more than the index of
134 ELEMENTARY ALGEBRA.
the power ; i. e. the second power has three terms ; the
third power, four terras ; and so on. When the number
of terms is even, i. e. when the index of the power is
odd, the two central terms have the same coefficient.
3d. When both terms of the binomial are positive, all the
terms of the power are positive; but when the second term
is negative, those terms which contain odd powers of the
following quantity are negative, and all the others positive;
or every alternate term, beginning with the second, is negor
tive, and the others positive.
1. Expand {x -{- yy.
OPERATION.
According to the law, the first term will be
and the second term
The coefficient of the third term will be
and the third term
A
+ 8x^y.
4
^X 7
2. '
+ 28^3^.
2
28 X^
X '
■\-^^^f-
14
^X5
The coefficient of the fourth term will be
and the fourth term
The coefficient of the fifth term will be
and the fifth term 70 a;* 2^*.
Having found the preceding coefficients and the coefficient of thi
middle term, we can write the others at once. Hence,
(X + y)8 = z8 + 8x7y + 28x«y2 ^ 56z5y3 ^. 70x*y4 + 56x3y5 ^ 28x2y6 + ^xyi + yS.
2. Expand {a — hy.
Ans. a«— 6a5&+15a*62~20a353+15a2i*— 6a65+5«.
3. Expand (m + '*)^.
4. Expand {b — yy.
6. Expand (a ■— a:)^^
INVOLUTION. 185
6. Expand {b + cy\
T. Expand (x + 1)».
Note. — Since all the powers of 1 are 1, 1 is not written when
it appears as a factor; but its exponent must be used in obtaining
the cocrticients.
Ans. X* 4- 5 X* + 10 ar' + 10 x2 + 5 a: 4- 1.
8. Expand (I — y)^
Ans. l — 6y + 15y2 — 203/^ + 153/* — 6/ + /.
9. Expand (a -- l)\
130. When the terras of the binomial have coefficients
or exponents other than 1, the theorem can be made to
apply by treating each term as a single literal quantity.
In the expansion, each factor should be enclosed in a
parenthesis, and after the expansion of the binomial by
the binomial theorem, the work should be completed by
the expansion of the enclosed factors, according to the
rule for the expansion of monomials.
1. Expand (2x — fy.
OPERATION.
(2 xy - 4 (2 x)» (y«) + 6 (2 x)' (/)» - 4 (2 x) (f)' + (f)*
Expanding each factor as indicated, we have
16 ar* — 32 a:«y« + 24 ar^^ — 8 x/ + y«
2. Expand (3x« — 2y)^
(8x«)s - 6(8x«)4 (2y) + 10(8z9)» (2yy> - 10(8a»r« (2y)» + 6(3z«) (2y)4 — (2y)5.
Ans. 243x" — 810z»y-j-1080z*y>— 720x*y»-}-240x*y*— 32/.
Note. — Any letters, as a and b, might be substituted for 3 a:"
and 2 y, and the expansion of (a — by written out, and then the
Talues of a and b substituted.
3. Expand (a' — Sby.
Ans. a'—UaH + 54ca*b^—lOSaH^ + Slb*.
4. Expand {x^^y^y.
136 ELEMENTARY ALGEBRA.
5. Expand (2 a + iy.
Ans. 8 a3+ 84 a'' + 294 a + 343.
6. Expand (2ac — xy.
Ans. 16a*c^ — 32a3c3a; + 24a2c2a:2 — 8aca;«+a:4.
7. Expand (a^x--2yy.
8. Expand (^ + xY.
9. Expand (I- ly.
^^«-il + I + ¥ + 2^« + x*.
Ans ^__i^ f i^_5a^ 5aa;* a:*
■ 32 48 "f" 36 54 »" "162 243
10. Expand — 1 V.
11. Expand (^^^_L.)'.
27 "^ ^^ 8 64
12. Expand (i + iy.
13. Expand /ac— iV"
14. Expand /^a: + -Y,
15. Expand (l — -V.
16. Expand /'2 a^ — IV.
n. Expand (l+lY.
18. Expand (i-iy.
EVOLUTION. 137
131* The Binomial Theorem can be applied to the ex-
pansion of a polynomial. Thus, in a -\- b — c, a -\- b
can be treated as a single term, and the quantity can be
written (a -J- i) — c. In like manner, a -\- b -\- x — y can
be written (a -]- b) -{- (x — y). In such cases it is easier
to substitute a single letter for the enclosed terms, and
after the expansion to substitute the proper values.
1 . Expand (a + ^ — <?)*.
OPERATION.
Put a -\- b = X
(x — c)» = aH» — Sar'c + Sxc^ — c»
Substituting for x, its value, a -f- 6 ;
(a + 6 — c)' = o8 + 8o«6 + 3a6a + &3 — 3a5c — 6a6 c — 36« c + 3ac« + 86c« — c»
2. Expand (2a — b — c — dy,
Note. — For 2a — b — c — d write (2 a — 6) — (c -f d).
Ans. 4a«— 4a6-|-ft* — iac—4ad-\-2bc-\-2bd-\-c'-\-2cd-\-cP.
3. Expand (3a:— ^y — a + by.
4. Expand (i x — a + by.
EVOLUTION.
132. Evolution is the process of extracting a root of
a quantity. It is the reverse of Involution.
133* A ROOT of any quantity is a quantity' which taken
as a factor a given number of times will produce the
given quantity.
The number of times the root is to be taken as a factor
depends upon the name of the root. Thus, the second
or square root of a quantity is a quantity which taken
twice as a factor will produce the given quantity ; the
third or cube root is a quantity which taken three times
as a factor will produce the given quantity ; and so on.
138 ELEMENTARY ALGEBRA.
A Root is indicated by the radical sign m^ ^ or by a
fractional exponent. Thus,
\/x, or x^ indicates the square root of x.
3/— i
s/ X, or x^
tt
(t
cube
y/x, or x'«
tt
ft
mth
ft H {I
it it It
134. A root and a power may be indicated at the same
time. Thus, \^ x^, or x^ , indicates the cube root of the
fourth power of x, or the fourth power of the cube root
of X ] for a power of a 7'oot of a quantify is equal to the
same 7V0t of the same power of the quantity. /^8^ or 8^
is the square of the cube root of 8, or the cube root of
the square of 8, i. e. 4.
135. A perfect power is a quantity whose root can be
found. A perfect square is one whose square root can
be found ; a perfect cube is one whose cube root can be
found ; and so on.
136. Since Evolution is the reverse of Involution, the
rules for Evolution are derived at once from those of
Involution. And therefore, as according to Art. 125 an
odd power of any quantity has the same sign as the
quantity itself, and an even power is always positive,
we have for the signs in evolution the following
RULE.
An odd root of a quantity has the same sign as the quan-
tity itself
An even root of a positive quantity is either positive or
negative.
An even root of a negative quantity is impossible, or im-
aginary.
EVOLUTION. 139
SQUARE ROOT OF NUMBERS.
137. The Square Root of a number is a number which,
taken twice as a factor, will produce the given number.
]38« The square of a number has twice as many fibres
08 the rooty oi' one less than twice as many. Thus,
Roots, 1, 10, 100, 1000.
Squares, 1, 100, 10000, 1000000.
The square of any number less than 10 must be less than 100;
but any number less than 10 is expressed by one figure, and any
number less than 100 by less than three figures; i. e. the square of
a number consisting of one figure is a number of either one or two
figures. The stjuare of any number between 10 and 100 must be
between 100 and 10000; i. e. must contain more than two figures
and less than five. And the square of any number between 100
and 1000 must contain more than four figures and less than seven.
Hence, to ascertain the number of figures in the square
root of a given number.
Beginning at units, point off the number into periods of
two figures each ; there will be as many figures in (lie root
(W there are periods, and for the incomplete period at the
left, if any, one more.
139t To extract the square root of a number.
1. Find the square root of 6329.
From the preceding explanation, it is evident that the square
root of 5329 is a number of two figures, and that the tens figure
of the root is the square root of the greatest perfect square in 53 ;
i. e. ^Ad, or 7. Now, if we represent the tens of the root by a
and the units by h, a -\- b will represent the root ; and the given
number will be
(a -f 5)' = a« -f 2 a ft -|- 6».
Now a' = 70» = 4900 ;
therefore, 2 a 6 -f 6» = 5329 — 4900 = 429.
But . 2a6-f 6»=(2a-f 6)6;
140 ELEMENTARY ALGEBRA.
If therefore 429 is divided hy 2 a -\- b, it will give h the units of
the root. But b is unknown, and is small compared with 2 a ;
we can therefore use 2 a = 140 as a trial divisor. 429 -^ 140,
or 42 -|- 14 = 3, a number that cannot be too small but may be
too great, because we have divided by 2 a instead of 2 a -j- &.
Then b = 3, and 2 a -f 6 = 140 + 3 = 143, the true divisor ; and
(2 a + 5) & = 143 X 3 == 429 ; and therefore 3 is the unit figure of
the root, and 73 is the required root. The work will appear as
follows : —
OPERATION.
5 3 2 9 (7 3
49
1 4 3) 4 2 9
429
Hence, to extract the square root of a number,
RULE.
Separate the given number into periods of two figures each,
by placing a dot over units, hundreds, dbc.
Find the greatest square in the left-hand period, and place
its root at the right.
Subtract the square of this root figure from the left-hand
period, and to the remainder annex the next period for a
dividend.
Double the root already found for a trial divisor, and,
omitting the right-hand figure of the dividend, divide, and
place the quotient as the next figure of the root, and also at
the right of the trial divisor for the true divisor.
Multiply the true divisor by this new root figure, subtract
the product from tlie dividend, and to the remainder annex
the next period, for a new dividend.
Double the part of the root already found for a trial di-
visor, and proceed as before, until all the periods have been
employed.
EVOLUTION. 141
Note 1. — When a root figure is 0, annex also to the trial di-
visor, and bring down the next period to complete the new dividend.
Note 2. — If there is a remainder, after using all the periods in
the given example, the operation may be continued at pleasure by-
annexing successive periods of ciphers as decimals.
Note 3. — In extracting the root of any number, integral or deci-
mal, place the first point over unit's place ; and in extracting the
square root, over every second figure from this. If the last period in
the decimal periods is not full, annex 0.
2. Find the square root of 46225.
We suppose at first that a rep-
OPKRATlON. j^gj^^ ^jjg hundreds of the root,
46225 (2 15 and b the tens ; proceeding as in
4 Ex. 1, we have 21 in the root
Then letting a represent the
hundreds' and tens together, i. e.
21 tens, and b the units, we have
4 2 5) 2 I 2 5 2 a, the 2d trial divisor, = 42
2 12 5 tens ; and therefore 6 = 5; and
2 a -j- 6 = 425 ; and 215 is the
required root.
8. Find the square root of 5013,4.
OPERATION.
5013.46(7 0.805+
49
140.8)113.40
112.64
41)62
41
141.605) .T60000
.7 8025
4. Find the square root of 288369. Ans. 537.
5. Find the square root of 42849. Ans. 207.
6. Find the square root of 173.261. Ans. 13.16+.
7. Find the square root of .9. Ans. .948+.
142 ELEMENTARY ALGEBRA.
8. Find the square root of 2. Ans. 1.4142-f-.
9. Find the square root of 484.
10. Find the square root of 48.4.
11. Find the square root of .064.
12. Find the square root of .00016.
Note. — As a fraction is involved by involving both numerator and
denominator (Art. 127), the square root of a fraction is the square root
of the numerator divided hy the square root of the denominator.
13. What is the square root of | ? Ans. §.
14. What is the square root of ^| ?
15. What is the square root of -^jj ? -^^ = Z^. Ans. f .
Note. — If both terms of the fraction are not perfect squares,
and cannot be made so, reduce the fraction to a decimal, and then
find the square root of the decimal. A mixed number must be re-
duced to an improper fraction, or the fractional part to a decimal,
before its root can be found.
16. What is the square root of | ? Ans. .53-|-.
1 1 . What is the square root of 2-fT ^
18. What is the square root of -^^ ?
19. What is the square root of Tf ?
CUBE ROOT OF NUMBERS.
140. The Cube Root of a number is a number which,
taken three times as a factor, will produce the given
number.
141. The cube of a number consists of three times as
many figures as the root, or of one or two less than three
times as many.
Roots, 1, 10, 100, 1000.
Cubes, 1, 1000, 1000000, 1000000000.
The cube of any number less than 10 must be less than 1000;
but any number less than 10 is expressed by one figure, and any
EVOLUTION. 143
number leas than 1000 by less than four figures; i. e. the cube of
a number consisting of one figure is a number of less than four
figures. The cube of any number between 10 and 100 must be be-
tween 1000 and 1000000; i. e. must contain more than three figures
and less than seven. And in the same way -we see that the cube
of any number between 100 and 1000 must contain more than six
figures and less than ten.
Hence, to ascertain the number of figures in the cube
root of a given number,
Becfinning at units, point off the number into periods of
three figures each ; there tvill be as many figures in the root
as there are periods, and for tJie incomplete period at the
left, if any, one more.
142* To extract the cube root of a number.
1. Find the cube root of 42875.
From the preceding explanation, it is evident that the cube root
of 42873 is a number of two figures, and that the tens figure of
the root is the cube root of the greatest perfect cube in 42 ; i. e.
^ 27, oi*8. Now, if we represent the tens of the root by a and the
units by 6, a -\- b will represent the root, and the given number
will be
(a -I- 6)» = a» + 3 a« & + 3 a &« + 6».
Now a* = 30' = 27000;
therefore, 3 a* 6 + 3 a 6« + 6* = 42875 — 27000 == 15875.
But 3a>&-|-8aft«-f 6» = (3a*-f3a&-l-i»)6.
If therefore 15875 is divided by 3 a' + 3 a 6 -f &» it will give 6,
tlie units of the root. But 6, and therefore 3 a 6 -j- &*, a part of
the divisor, is unknown, and we must use 3a^ = 2700 as a trial
divisor. 15875 -^ 2700, or 158 -^ 27 = 5, a number that cannot
be too small but may be too great, because we have divided by 3 a*
instead of the true divisor, 3 a^ -{- 3 a b -\- l^. Then b = 5, and
3a'-|- 3aft-f^= 2700 -f- 450 -f- 25 = 3175, the true divisor;
and (3 a' 4- 3 a 6 -f- />») 6 = 31 75 X 5 = 15875, and therefore 5 is
the unit's figure of the root, and 35 is the required root The work
will appear as follows : —
144 ELEMENTARY ALGEBRA.
OPERATION.
4 2 8 T 5 (35 Root.
21
Trial divisor, 3 a^ = 2 Y '
3ab=z 450
b^=z 2 5
True divisor, B a^ + 3ab + b^ = 3 1 75
1 6 8 1 5 Dividend.
158T5
Hence, to extract the cube root of a number,
RULE.
Separate the number into periods of three figures each, by
placing a dot over units, thousands, Sc.
Find the greatest cube in the left-hand period, and place
its root at the right.
Subtract this cube from the left-hand period, and to the re-
mainder annex the next period for a dividend.
Square the root figure, annex two ciphers, and multiply
this result by three for a trial divisor ; divide the dividend
by the trial divisor, and place the quotient as the next figure
of the root.
Multiply this root figure by the part of the root previously
obtained, annex one cipher and multiply this result by three ;
add the last product and the square of the last root figure to
the trial divisor, and the sum will be the true divisor.
Multiply the true divisor by the last root figure, subtract
the product from the dividend, and to the remainder annex
the next period for a dividend.
Find a new trial divisor, and proceed as before, until all
the periods have been employed.
Note 1. — The notes under the rule in square root (Art. 139)
apply also to the extraction of the cube root, except that 00 must
be annexed to the trial divisor when the root figure is 0, and after
placing the first point over units the point must be placed over
every third figure from this.
Note 2. — As the trial divisor may be much less than the true
EVOLUTION.
145
divisor, the quotient is frequently too great, and a less number
must be placed in the root.
2. Find the cube root of 1819144T.
OPERATION.
l8t Trial Divisor, 8a* =1200]
Sab =- 360
6« = 36
l8t True Divisor, 3a'-|-3a& + 6»
2d Trial Divisor,
18191447 (2 63
_8
10191 1st Dividend.
1596J
8c^ =202800
Sab = 2340
6^= 9
9576
2d True Divisor, 3a«-f3a6-f-6»=205149
61544 7 2d Div.
615447
We suppose at first that a represents the hundreds of the root
and 6 the tens: proceeding as in Ex. 1, we have 26 in the root
Then letting a represent the hundreds and tens together, i. e. 26
tens, and b the units, we have 3 a', the 2d trial divisor, = 202800 ;
and therefore 6 = 3; and 3 a^ -\- S ab -\- b*, the 2d true divisor,
= 205149; and 263 is the required root.
Note. — Though the 1st trial divisor is contained more than 8
times in the dividend, yet the root figure is only 6.
3. Find the cube root of 68116.47.
OPERATION,
6 8116.4 7 0(4 0.9 5+
64
48 0.0
10 8.0
.8 1
4 7 1 6.4 7
49 8.8 1
50 18.4 30
6.13 50
.0025
50 24.56 7 5
4417.9 2 9
2 9 8.5410 00
251.228375
4 7.3126 25^
146 ELEMENTARY ALGEBRA.
4. Find the cube root of 292420Y. Ans. 143.
5. Find the cube root of 8120601. Ans. 201.
6. Find the cube root of 36926037.
7. Find the cube root of 67911.312.
8. Find the cube root of 46417.8.
9. Find the cube root of .8. Ans. .928+.
10. Find the cube root of .17164.
11. Find the cube root of .0064.
12. Find the cube root of 25.0001Y.
13. Find the cube root of 2.7.
Note. — As a fraction is involved by Involving both numerator and
denominator (Art. 127), the cube root of a fraction is the cube root
of the numerator divided by the cube root of the denominator.
14. What is the cube root 2 r ? Ans. f.
15. What is the cube root of //^ ?
16. What is the cube root of f||? |-f| == |f|.
Ans. f.
Note. — If both terms of the fraction are not perfect cubes, and
cannot be made so, reduce the fraction to a decimal, and then find
the cube root of the decimal. A mixed number must be reduced
to an improper fraction, or the fractional part to a decimal, be-
fore its root can be found.
17. What is the cube root of ■^\? Ans. .899+.
18. What is the cube root of ^j?
19. What is the cube root of 3^ ?
20. What is the cube root of 117??
EVOLUTION. 14T
EVOLUTION OP MONOMIALS.
1I3* As Evolution is the reverse of Involution, and
since to involve a monomial (Art. 126) we multiply the
exponent of each letter by the index of the required
power, and prefix the required power of the numerical
coefficient,
Hence, to find the root of a monomial,
RULE.
Divide the exponent of each letter by the index of the re-
quired root, and prefix the required root of the numerical
cvefficient.
Note 1. — The rule for the signs is given in Art. 136. As an
even root of a positive quantity may be either positive or negative,
we prefix to such a root the sign ± ; read, plus or minus.
Note 2. — It follows from this rule that the root of the product
of several factors is equal to the vroduct of the roots. Thus,
V/"36 = v'l V^'O = 6.
1. Find the cube root of So:*/. Ans. 2xy^.
2. Find the square root of 4 x^. Ans. ±2x. .
3. Find the third root of — 125 a«ar.
Ans. — 5 a^x^.
4. Find the fourth root of 81a"* b.
Ans. ± 3 a-* b^.
5. Find the fifth root of 32 a^^ b\ Ans. 2 a* bK
6. Find the cube root of — 729 arV-
Ans. — 9x1^.
7. Find the fourth root of 256 xV-
8. Find the cube root of — 512 a-H'.
9. Find the fifth root of 243 a:* y^.
148 ELEMENTARY ALGEBRA.
Note. — As a fraction is involved by involving both numerator
and denominator (Art. 127), a fraction must be evolved by evolving
both numerator and denominator.
4 q2 2 a
10. Find the square root of r— r* Ans. ± ^r^,-
Perform the operations indicated in the following ex-
pressions : —
11. \/—129aH^c\
12. (ida^x^f)^,
13.
/ 9 a? X*
Y 36Z>V
14. it^a'^x"
15. {25Qa''x^'y^Y.
,4^10^,16\-J
16. /^Slan\
11. \^a'"62"*c"»".
SQUARE ROOT OF POLYNOMIALS.
144. In order to discover a method for extracting the
square root of a polynomial, we will consider the rela-
tion of a 4" ^ ^^ ^ts square, a^ ~\- 2 a b -{- h^. The first
term of the square contains the square of the first term
of the root ; therefore the square root of the first term of
the square will be the first term of the root. The second
term of the square contains twice the product of the two
terms of the root ; therefore, if the second term of the
square, 2 a b, is divided by twice the first term of the
root, 2 a, we shall have the second term of the root b.
Now, 2 a b -\- b^ = (2 a -\- b) b', therefore, if to the trial
divisor 2 a we add b, when it has been found, and then
EVOLUTION. 149
multiply the corrected divisor by h, the product will be
equal to the remaining terras of the power after a^ has
been subtracted.
The process will appear as follows : —
OPERATION. Having written a, the square
a^-\-2ab-\-lt^{a-\-b root of a", in the root, we sub-
a^ tract its square (a') from the
2a-|-i)2a64-^ g»ven polynomial, and have
\ , \ VI 2ah 4- h" left. Dividing the
' first term ol this remamder,
2 a h, by 2 a, which is double the term of the root already found,
we obtain ft, the second term of the root, which we add both to
the root and to the divisor. If the product of this corrected divisor
and the last term of the root is subtracted from 2 a ft -j- 6', nothing
remains.
145* Since a polynomial can always be written and
involved like a binomial, as shown in Art. 131, we can
apply the process explained in the preceding Article to
finding the root, when this root consists of any number
of terms.
1. Findthe8quarerootofa2+2a5+ft=* — 2ac— 2ftc + (r».
OPERATION.
a^^2ab + P—2ac — 2bc + c^(a + b^c
2a + b)2ab + lP
2ab + if'
2a + 2b — c)—2ac — 2bc + c*
-^2ac — 2bc-\-(^
Proceeding as before, we find the first two terms of the root a-\-b.
Considering a -\- h zr 2^ single quantity, we divide the remainder
— lac — 2ftc-)-c* by twice this root, and obtain — c, which we
write both in the root and in the divisor. If this competed divisor
is multiplied by — c, and the product subtracted from the dividend,
nothing remains.
150 ELEMENTARY ALGEBRA.
Hence, to extract the square root of a polynomial,
RULE.
Arrange the terms according to the powers of some letter.
Find the square root of the first term, and write it as the
first term of the root, and subtract its square from the given
polynomial.
Divide the remainder by double the root already found,
and annex the result both to the root and to the divisor.
Multiply the corrected divisor by this last term of the root,
and subtract the product from the last remainder. Proceed
as before with the remainder, if there is any.
2. Find the square root of 4 .<;^ — 4 a:^^ -[" y*.
Ans. 2x — y^.
3. Find the square root of a^ -\-2ab-\-P-\-4:ac
+ 4 5 c + 4 c2. Ans. a + h + 2c.
4. Find the square root of 9 x^ — 12 x^ -{- 4=x^ -{- 6 ax^
— 4:ax-\-a^. Ans. Sx'^ — 2x-\-a.
5. Find the square root of4a^ -\- Sab — 4a -|~ ^^^
— 46+1 Ans. 2a + 2b—l.
6. Find the square root of 25 x^ — lOx^ -{- Gx"^ — x
-\- {. Ans. 5^2 — oc -\- ^.
1. Find the square root of a?^ + 2 x^ — x^ — 2x^ -{- x^.
8. Find the square root of 4«^ — 4:ab -\- b^ — 4ac
^4.ad-[-2bc-\-2bd-\-c''-^2cd-[-d\
Ans. 2 a — b — c — d.
9. Find the square root of x^ — 4a?^ -[" ^^^ — ^^*
J^bx'' — 2x-{- 1.
10. Find the square root of 4a4 -|- Sa^b — SaH^
^I2ab' + 9b\
F.VOLUTION. 161
Note 1. — According to the principles of Art 136, the signs of
the answers given above may all be changed, and still be correct
Note 2. — No binomial can be a perfect square. For the square
of a monomial is a monomial, and the scjuare of the polynomial with
the least number of terms, that is, of a binomial, is a trinomial.
Note 3. — A trinomial is a perfect square when two of its terms
are perfect squares and the remaining term is equal to twice the
product of their square roots. For,
(a-|-6)» = a'-f 2a6 + 6»
(a — 6/ = a' — 2a& + 6»
Therefore the square root of a' ± 2 a 6 -[- i' is a ± 6. Hence, to
obtain the square root of a trinomial which is a perfect square,
Omitting the term that is equal to twice the product of the square
roots of the other two^ connect the square roots of the other two by the
tign of the term omitted.
1 1 . Find the square root of / + — *
4 A 4
Ans. 5-~
12. Find the square root of x' -f- 2x + 1.
Ans. X -\-\.
13. Find the square root of 4a:^ — 8xy -|- 4y^.
14. Find the square root of ^- — 2ab -\- ^h^.
15. Find the square root of 16y^ + 40yr* + 25 2*.
Note. — By the rule for extracting the square root, any root whose
inde.x is any power of 2 can be obtained by successive extractions
of the square root Thus, the fourth root is the sqtiare root of the
square root ; the eighth root is the square root of the square root of
the square root; and so on.
16. Find the fourth root of a« — I2a«6 + 54a*6»
— 108o^» + 8l6^ Ans. a" — Zb,
152 ELEMENTARY ALGEBRA.
IT. Find the fourth root of -, + -1 + -^. 4- — 4- -
Ans. -A
X
18. Find the fourth root of x^ — 4.x' + 10a;« — - 16a:«
J^lQx^—Ux^-\-\{)x^ — 4.x-\-l. Ans. t^^x+I.
146t To find any root of a polynomial.
Since, according to the Binomial Theorem, when the terms of a
power are arranged according to the power of some letter begin-
ning with its highest power, the first term contains the first term
of the root raised to the given power, therefore, if we take the re-
quired root of the first term, we shall have the first term of the root.
And since the second term of the power contains the second term of
the root multiplied by the ne^jt inferior power of the first term of the
root with a coefficient equal to the index of the root, therefore if we
divide the second term of the power by the first term of the root raised
to the next inferior power with a coefficient equal to the index of the
root, we shall have the second term of the root. In accordance with
these principles, to find any root of a polynomial we have the following
RULE.
Arrange the terms according to the powers of some letter.
Find the required root of the first term, and write it as
the first term of the root.
Divide the second term of the polynomial by the first term
of the root raised to the next inferior power and multiplied
by the index of the root.
Involve the whole of the root thus found to the given power,
and subtract it from the polynomial.
If there is any remainder, divide its first term by the di-
visor first found, and the quotient will be the third term of
the root.
Proceed in this manner till the power obtained by involv-
ing the root is equal to the given polynomial.
EVOLUTION. 153
Note 1. — This rule verifies itself. For the root, whenever a new
term is added to it, is involved to the given power, and whenever the
root thus involved is equal to the given polynomial, it is evident that
the required root is found.
Note 2. — As powers and roots are correlative words, we have
used the phrase given power, meaning the power whose index is equal
to the index of the required root, and the phrase next inferior power
meaning that power whose index is one less than the index of the
required root.
1. Find the cube root of a« — 3 a'^ + 5 a* — 3 a — 1.
OPERATION.
Constant divisor, 3 a*) a' — 3 a^ -|- 5 a* — 3 a — 1 (a^ — a — 1
gg _ 3 flS -I- 3 q^ _ gg
— 3 a*, Ist term of remainder.
a«_.3a» + 6a« — 3a— 1
The first term of the root is a*, the cube root of a*, a* raised
to the next inferior power, i. e. to the second power, with the co-
efficient 3, the index of the root, gives 3 a*, which is the constant
divisor. — 3 a*, the second term of the polynomial, divided by
3 o*, gives — a, the second terra of the root, (a* — a)' = a* — 3 a*
4" 3 a* — a* ; and subtracting this from the polynomial, we have — 3 a*
as the first term of the remainder. — 3 a* divided by 3 a* gives
— 1, the third term of the root, (a' — a — 1)'= the given poly-
nomial, and therefore the correct root has been found.
2. Find the fourth root of 16 x* — 32 ar« y* + 24 ar* y*
— 8ar/ + y".
OPERATION.
4 X ( 2 x) • =* 3 2 a:") 1 6 a:* — 3 2 z" y» -f- 24 x« y* — 8 a; / -f y» ( 2 z — y»
16 a;* — 82a:»y»-|-24a:'y* — 8xy'-}-y'
3. Find the cube root of a» + 3 a" 6 + 3 a ft» + 6^ — 3 a^ c
_>.6a6c — 36='c + 3ac« + 3ir»— c«.
4. Find the fourth root of 16 a* c* — 32 a» c» x + 24 a« (r* ar»
— 8acx^ + x*.
7*
154 ELEMENTARY ALGEBRA.
SECTION XVII.
RADICALS.
147. A Radical is the indicated root of any quantity,
as \/ X, d^ , \/2, 3^, &c.
148. In distinction from radicals, other quantities are
called rational quantities.
149. The factor standing before the radical is the co-
efficient of the radical. Thus, 2 is the coefficient of \/2
in the expression 2 \/ 2.
150. Similar Radicals are those which have the same
quantity under the same radical sign. Thus, \/a, 2 \/a,
and X h/a are similar radicals ; but 2 \/a and 2 \^b, or
2 x^ and 2 x* are dissimilar radicals.
151. A Surd is a quantity whose indicated root cannot
be found. Thus, \/2 is a surd.
The various operations in radicals are presented under
the following cases.
CASE I.
152. To reduce a radical to its simplest form. ^
Note. — A radical is in its simplest form when it contains no
factor whose indicated root can be found.
1. Reduce \/^5a^b to its simplest form.
OPERATION.
^15aH = A/25a^ X Sb=z^25a^xV^h = 5a\/Sb
We first resolve 75 a' b into two factors, one of which, 25 a', is
the greatest perfect square which it contains ; then, as the root of
RADICALS. 155
the product is equal to the product of the roota (Art 143, Note 2)»
we extract the square root of the perfect square 25 a", and annex
to this root the factor remaining under the radical. Hence,
RULE.
Hesolve the quantity under the radical sign into two fac-
tors, one of which is the greatest perfect power of the same
name as the root. Extract the root of the perfect poicer,
multiply it by the coefficient of the radical, if it has any, and
annex to the result the other factor, with the radical sign
between them.
Reduce the following expressions to their simplest
form : —
2.
3.
^/12a:.
9a«
v/'
3c
v/1
Ans. 2\/3x.
Ans. 7 x' i^~x.
4.
^72a»6».
Ans. 2a^96».
6.
b^64:ab\
Ans. 10 6/^^4 a.
6.
1.
SA^U1aH\
25 \/ 56 a:.
Ans. 21 a 6=^3.
Ans. 60^7x.
8.
9.
4x/128r«y.
-C/343x».
10.
/ 27a«c
Y l2S2*y
11.
12.
1 1l€?C
V 128x*y-
Y 64 2
>v/16r^y» — 32
Vl6x«y«.
— 32:
rV
Gx^y^Vl — 2ar'y«
= 4a:y \/l — 2x2yS Ans.
156 ELEMENTARY ALGEBRA.
13. 4-^81a«c + 27a«. Ans. 12a4^3c + l.
14. (a + h) \/3a^ — Qab-\- Sb\ Ans. (a" — b^) ^"3.
15. 1 ^250x^f—12ox^y\
16. (a: — y) (a^ar — a^y)*.
18. \/^^^16.
-v/— 16 = \/16 V— 1=4:\/— 1, Ans.
19. 4/ —1250.
20. Vl9«' — 4^2
153. When a fraction is under the radical sign, it can
be transformed so as to have only an integral quantity
under the radical sign, by multiplying both terms of the
fraction by that quantity which will make its denominator a
perfect power of the same name as the root, and then re-
moving a factor according to the Bule in Art. 152.
1. Reduce iy to its simplest form.
OPERATION.
<j1 = ^li = slh^~^=\'^^
Transform each of the following expressions so as to
have only an integral quantity under the radical sign.
2. iy/|- Ans. i v'^e.
3. 4^|. Ans. *^^9.
4. ?y/]. Ans. 1^^343.
5- Isjm^- Ans. i>/347.
RADICALS. 16T
le- Ana. 1^30.
10. .vi51
*' 1876
11. (a + 6)y/^. Ans. V^-6».
CASE II.
151* To reduce a rational quantity to the form of a
radical.
1. Reduce 3 a:* to the form of the cube root.
OPERATION. Since 3 a:* is to be placed
3 2^ __ 3^ 27 a:* under the form of the cube
root without changing its
value, we cube it and then place the radical sign, ^, over it It is
evident that ^ 21 z? = 3 a:". Hence,
RULE.
Involve (he quantity to the power denoted by the index of
the root required, and ph/ce the corresponding radical sign
over the power thus produced.
2. Reduce 4 a* 6 to the form of the square root.
Ans. A^lQa^i^.
3. Reduce 2ab'^c~^ to the form of the fifth root.
Ans. 4/32a'b^''c'^.
4. Reduce ^a'c^ to the form of the cube root.
9
168 ELEMENTARY ALGEBRA.
5. Reduce -z — r to the form of the fourth root.
Bxyt
6. Reduce x — 2i/ to the form of the square root.
Ans. \/ x^ — 41x1/ -\- 4: y^.
155% On the same principle the rational coefiScient of a
radical can be placed under the radical sign, by involv-
ing the coefficient to a power of the same name as the root
indicated by the radical sign, multiplying it by the radical
quantity, and placing the given radical sign over the product.
1. Place the coefficient of hi^ly under the radical
sign.
OPERATION.
6 A^ 2y = >^ 125 >ij/ 2^^ = ^ 250y
In the following examples, place the coefficient under
the radical sign.
2. 3>^4a;8y. Ans. ><y324x'y.
3. 2xyf^2x'y. Ans. ^Ua^y^.
4. f x/i.
5. ^\/l4.
6. {a — h) ^^^' Ans. ^a^ —■2aH + ab"",
T. 4.xy\^l — 2x^f.
CASE III.
156. To reduce radicals having different indices to
equivalent ones having a common index.
1. Reduce \/« and ^ b to equivalent radicals having
a common index.
RADICALS. 169
OPERATION. In thirf case we write the radicals
4 I B/ — ii witli their fractional indices : and
a^ =: a^ =: /^ a . .
then, as the denominator is the in-
b^ = b^ =z A^ h^ dex of the root, in order that the
two radicals may have the same
root-index, we reduce the fractional indices to equivalent ones hav-
ing a common denominator. It is evident that we have not changed
the values of the given radicals by the process. Hence,
RULE.
Reduce the fractional indices to equivalent ones having a
common denominator ; involve each quantity to the power de-
noted by the numerator of the reduced index ^ and indicate
tlie root denoted by the denominator,
2. Reduce \^2 and ^^3 to equivalent radicals having
a common index.
3* = 3^^ = -^3^ == ^27 )
Ans.
3. Reduce \/ | and v^ ^ to equivalent radicals having
a common index. Ans. ^j^^ and /^ r^.
4. Reduce 4/ - and 1/ | to equivalent radicals having
a common index.
5. Reduce ^ a, ^ a — b, and \/ a -{- b to equivalent
radicals having a common index.
Ans. v' < v' (^i^^y, and iy (a~+by.
6. Reduce \/2, \/4, and /^ 3 to equivalent radicals
having a common index.
7. Reduce \/ x and v^y to equivalent radicals having
a common index. Ans. \/x^ and \/y".
160 ELEMENTARY ALGEBRA.
CASE IV.
157. To add radical quantities.
1. Add \/ic and Vy. Ans. \/« + Vy^
It is evident that the addition can only be expressed.
2. Add Zf>/x and b\fx. Ans. ^ s/x.
It is evident that 3 times the ^ x and 6 times the ^~x make 8
times the y/a;.
3. Add V'S and \/ 50 together.
OPERATION. In this case we make the radi-
f^~~^ = 2 \/^ ^^^ ipa-rts similar by reducing them
.-rx c /"H- *o t^®"* simplest form (Art. 152),
^ "^^ — — — and then add their coefficients as
Sum = T \/ 2 in Example 2. Hence,
RULE.
Make the radical parts similar when they are not, and
prefix the sum of the coefficients to the common radical. If
the radical parts are not and cannot he made similar, con-
nect the quantities with their proper signs.
4. Add 2\/60«a; and 3\/98aa;. Ans. 31\/2aic.
6. Add 4>^24ar8 and x^^l. Ans. 110:^^3.
6. Add \/2T and \/^363. Ans. 14\/'3.
Y. Add -^512a:* and .^162y*.
Ans. {4.x -{-Zy) 4^2.
8. Add ^1) and ^/\
VT=\/^\/5 = i\/5; V5 + i\/5 = f\/"5, Ana.
9. Add ^^ and ^1|^. Ans. ^ ^l2 .
RADICALS. 161
10. Add Vl, 10^7f» and 6/v/20. Ana. 13 V^.
11. Add Vl^ and a/2^.
CASE V.
158i To subtract one radical from another.
1. From \/T5 take \/'2T.
OPERATION. We make the radical parts sim-
.-=T r /-q ilar by reducing them to their sim-
~~ plest form (Art. 152). And 3 y/l
taken from 5 y^ 3 evidently leaves
2 v' 3. Hence,
V27 = 3^/3
2v^3
RULE.
Make the radical parts similar when they are not, sub-
tract the coefficient of the subtrahend from that of the min-
uend, and prefix the difference to the common radical. If
the radical parts are not and cannot be made similar, indi-
cate the svbtracHon by connecting them with the proper sign.
__ _ •
2. From ^^ take >^ 3. Ans. 2-^3.
3. From 9 s/ a^xy^ take 3 a \/ x^*.
Au8. ^ayjs/x.
4. From T \/20x take 4\/45a:. Ans. 2\/5ar.
6. From /^ 500 take ^T08. Ans. 2 ^1^,
6. Prom 2^^^ take \/^. Ans. ttV^-
7. From >v/ | take \/^. Ans. ;^ V^O.
8. From 2^n6a:» take .^y891x».
9. From a 4/1^ take 7 *i/a^3^.
10. From >C^ 1174 take ^1892.
162 ELEMENTARY ALGEBRA.
CASE VI.
159. To multiply radicals.
1. Multiply 3 V « by 5 f^l.
OPERATION.
3 s/~a X 5 \/T = 3 X 5 X V « X V^ = 15 s/~ab
As it makes no diflference in what order the factors are taken,
we unite in one product the numerical coefficients ; and >^ ay^^h
= \/~ab (Art. 143, Note 2).
2. Multiply 4\/2a6 by 5 /v^3«y.
We reduce the radical
OPERATION. . , . ,
parts to equivalent radicals
4v2a6= 4Y^ 8a o having a common index
(Art.
ply as
ample.
5^Sax= 6^ 9a^x^ (Art. 156), and then multi-
Product = 20 ^W^W ^^y ^' ^^ *^' preceding ex-
3. Multiply ^a by \/ a.
OPERATION.
a^ X a2 zz= >^a'^ X a^ = \/a\ or a*
Multiplying as in the preceding examples, we have ^ a*, or a^ ;
but I = ^ -|- ^ ; i. e. the index of the product is the sum of the
indices of the factors.
From these examples we deduce the following
RULE.
I. Reduce the radical paints, if necessary, to equivalent
radicals having a common index, and to the product of the
radical parts placed under the common radical sign prefix
the product of their coefficients.
II. Boots of the same quantify are multiplied together by
adding their fractional indices.
RADICALS.
163
4.
Multiply 3VT0by 4V'5.
. Ans. 60 V 2.
6.
Multiply 4 -C^ a ar* by a Vx.
Ana. ^axA^/d^x.
6.
Multiply oa/c by b^/c.
Ads. a&c.
1.
Multiply \/ x" by ^ x.
Ans. x«.
8.
Multiply ^fhyV^.
Ans. 1*, or ^16807.
9.
Multiply \^x by v^x.
Ans. V^""^".
10. Multiply 2 /v/a + 6 by 6x^a + 6.
Ans. 12 x-^ (« + *)*.
11. Multiply /k/x, *yx, and /^x together. Ans. x^*.
12. Multiply 3 a{/^ by 2 V 8. Ans. 6 >C/2.
13. Multiply a\/x-^y by h mJ xy. Ans. aJy.
14. Multiply (a + ft)i by (a — 5)^ Ans. (a^ — J^jl
16. Multiply i \/T by 3 \/ 1
16. Multiply 2 V'S by 4 V"8.
CASjE VII.
160* To divide radicals.
1. Divide 60\/15x by 4\/6x.
OPERATION. As division ia finding
60 \/T5x -i- 4 i^~hx= 15 V 3" * quotient which, multi-
plied by the divisor, will
produce the dividend, the coefficient of the quotient must be a
number which, multiplied by 4, will give 60, the coefficient of the
dividend, i. e. 15; and the radical part of the quotient must be a
quantity which, multiplied by y^Sx, will give v^lSa:, i.e. v^3; the
quotient required, therefore, is 15 v' 3.
164 ELEMENTARY ALGEBRA.
2. Divide 6 M^Ty by 2 .^ 2y.
OPERATION.
We reduce the radical parts to equivalent radicals having a com-
mon index (Art. 155), and then divide as in the preceding example.
3. Divide \/a by /^ a.
OPERATION.
^ ^ „i = ^a^^Zr^2 _ ^- or J
Dividing as in the preceding examples, we have ^~a, or a^. But
^ = i — i ; i' e. the index of the quotient is the index of the divi-
dend minus the index of the divisor.
From these examples we deduce the following
RULE.
I. Reduce the radical parts, if necessary, to equivalent
radicals having a common index, and to the quotient of the
radical parts placed under the common radical sign prefix
the quotient of their coeficients.
II. Roots of the same quantity are divided by subtracting
the fractional index of the divisor from that of the dividend.
4. Divide 16 a/ ax by S\/a^x. Ans. 2\/«~^.
6. Divide 4:A^d' — b"^ by 2 \/« — *•
Ans. 2A/a -\- b.
6. Divide Q\/'21 hj Sa/S. Ans. 6.
T. Divide *>/ x by \/x. Ans. v^"*""-
8. Divide \/a by Xl/b. Ans, i/r-
9. Divide 3 by \^~3. Ans. -v/'S.
RADICALS. 165
10. Divide x by ^x. Ans. ^x^.
11. Divide 4a«\/x by 2cr^\/~y, Ans. ^^^Jy
12. Divide V 6" by ^5^
13. Divide 4^1 by </T.
14. Divide /^a by f^ a.
15. Divide I ^1 by I ^f.
CASE VIII.
161 1 To involve radicals.
1. Find the cube oi Z^x.
OPERATION.
(3V^)»==3\/x X Zs/xX 3\/7
In accordance with the definition of involution, we take the quan-
tity three times as a factor. By Art. 159 the product is 27 y/ a:*.
2. Find the square of 2 .^ a.
OPERATION. ^° ^^^ ^^® ^^ ^^^^ "^^^
_ 4 1 *^® fractional exponent, and
(2 aJ/ a)* = (2 a*)* = 4 a* found the square of the given
quantity by multiplying its
exponent by the index of the required power, according to Art. 1 26.
Hence,
RULE.
I. Involve the radical as if it were rational, and placing
it under its proper radical sign, prefix the required power
of its coefficient.
II. A radical can be involved by multiplying its fractional
exponent by the index of the required power.
166 ELEMENTARY ALGEBRA.
Note. — Dividing the index of the root is the same as multiply-
ing the fractional exponent. Thus the square of ^ a is i/~a. For
(a^f = a^, or y' a.
3. Find the cube of 3x\/ a.
Ans. 21 a x^ a/ a, or 21 a^ x.
X 2
4. Find the square of 4 a^. Ans. 16 a^, or 16 ^a^.
5. Find the fourth power of 3 V^. Ans. 81 x^.
6. Find the nth power of a/^/x. Ans. dl't^l^.
1. Find the fourth power of 6 a/^, Ans. 25.
8. Find the cube of B \/~T. Ans. 189 VT.
9. Find the fourth power of r.
10. Find the cube of 2 a/Tx.
CASE IX.
162. To evolve radicals.
1. Find the cube root of Sa^-^o^.
OPERATION. As the root of the product is
' * ^ (Art. 143, Note 2), we prefix to
the cube root of the radical part the cube root of the rational part.
The cube root of the radical part must be a quantity which, taken
three times as a factor, will produce &a^3?\ i. e. ^ax.
2. Find the fourth root of ^ x.
— — f \\\ i_ the fractional exponent, and
V (^a: = (a;*) = x^, or '^ x found the fourth root by di-
viding the exponent of the
given quantity by the index of the required root, according to
Art 143. Hence,
RADICALS. 167
RULE.
I. Evolve the radical as if it were rational, and, placing
it under its proper radical sign, prefix the required root of
its coefficient.
II. A radical can he evolved by dividing its fractional
exponent by the index of the required root.
Note. — Multiplying the index of the root is the same as divid-
ing the fractional exponent Thus, the square root of ^ a is ^ a.
For (a^)^ = ai, or ^ a.
3. Find the square root of ba/^A^x.
(5 a A^Ti)^ = (^SOOa'ar)^ =>^500a«x, Ans.
4. Find the cube root of x'^ ^ a^ b. Ans. 1/ -W"*
6. Find the fifth root of ar^\/^. Ans. \/x.
6. Find the fourth root of i VT- Ans. ^'^.
1. Find the cube root of 1 a,/S. Ans. ^TlT.
8. Find the square root of 12 /^b.
POLYNOMIALS HAVING RADICAL TERMS.
163. It appears from the principles already established,
that the laws which apply to calculations with quantities
which have exponents, apply equally well whether the
exponents are positive or negative, integral or fractional.
The following examples, therefore, can be done by rules
already given.
1. Add 4a — 3 \/y and 3 a + 2 \/y. Ans. la — Vy.
2. Add 3 X + >^ 135 and 1 x — ^ 1080. _
Ans. lOar — 3>^6.
168 ELEMENTARY ALGEBRA.
3. Add 2 V 28 — V 27 and 2 V 63 + \/i8.
4. Subtract 15 a; — \/50a from 13 a: — s/^a.
Ans. 3\/^ — 2ar.
5. Subtract f^ as? — \/4 6 from ^J ax — \/16 6.
Ans. A^ ax — Xfs/a — 2\/6.
6. Subtract .i^"32 — ^"242 from — 3 f^l — T \/"3.
T. Multiply /v/a — \/^ by \/a — /\/a;.
OPERATION.
\/a — fn/ X
a — ^ ah — /^ ax -\- /^ bx
8. Multiply xy -\- \^ ah by 4 — s/ ah,
Ans. ^xy -\- (4 — a; 3/) /y/o^ — ah.
9. Multiply T + \/T0 by 6 — \/l0.
Ans. 32 — \/T0.
10. Multiply \/a -j- V^ by \/« — V^- -^i^s. a — J.
11. Multiply V'5 — 4-^"3 by ^45 + /^"O.
12. Multiply ^ VI + T \/"3 by i Vl^ — T V 3.
13. YiWi^Q s/ ax -\- M^ ay -\-x-\- s/ xy \y^ s/ a-\' s/ X.
OPERATION.
\/a -j- \/a;) is/ ax -\- ^ ay -\- x -\- s/ xy (\/ar -f- \/y
^/ ax -f- a:
V^ + V^
s/ ay +V^
RADICALS. Ig9
14. Divide V^ — V^ — V^ + \/^ by v^ — V^.
Ans. s/~ot. — \/^.
16. Divide <^ x-\-^x-\-^y^-\-^y^ bya:+y*
16. Divide a: — y by V^ — Vy. Ans. V^ + V.y.
17. Divide4xy + 4>v/^--ary>v/^ — a*by4 — V"^.
18. Expand (V^ + \/~yy. Ans. x + 2 V^ + y.
19. Expand (a^ — IT^y. Ans. a — 2 4/?^+ 1.
20. Expand (^ — ^1)*.
Ans. o^ — 4a*6i + 6a6 — 40*6' 4- ja
21. Expand (4 — V^)». Ans. 100 — 61^3.
22. Expand (a-i — ar-i)».
Ans. a-l — 3 a-^ x-^ + 3 a-4a:-« — x-*.
?3. Expand (l-^ly.
Ans. l_-4. + i. ?_ I i.
24. Expand (^|--^|y.
Ans. f?-i4j^+:.y_l^ , ^
25. Find the square root of a — 2 a^ i^ -j- i^.
Ans. t^~a — /^~W,
26. Find the cube root of x* — 3 x^y^ + 3 ary^ — y.
Ans. X — ys.
27. Find the fourth root of 16 a — 32 a^y* + ^4 ar^yl
-8xV + y^. Ans. 2 a^~yf
170 ELEMENTARY ALGEBRA.
SECTION XVIII.
PURE EQUATIONS
WHICH REQUIRE IN THEIR REDUCTION EITHER INVO-
LUTION OR EVOLUTION.
164. A Pure Equation is one that coDtains but one
power of the unknown quantity ; as,
A^x -\- ac = h, 4 aj2 -|- 3 = T, or 14 a;" = a 5.
165. A Pure Quadratic Equation is one that contains
only the second power of the unknown quantity ; as,
6x'^'-14:a=5lb, ay^=lScd, ov acz'^ 14.
166. Radical Equations, i. e. equations containing the
unknown quantity under the radical sign, require Invo-
lution in their reduction.
167. To reduce radical equations.
1. Reduce \/ a; — 3»=^.
operation.
V"^ — 3 = 8
Transposing, a/ x =^ 11
Squaring, x = 121
2. Reduce /^x — 4 + 7 = 10.
operation.
^ 1^4 _|_ 7 == 10
Transposing, ^x — 4 = 3
/Cubing, as -^ 4 == 2t
Transposing, *c =? 31
RADICAL EQUATIONS. * 171
3. Reduce ^^^'±^~^
y/a
= v<..
OPERATIOM.
Clearing of fractions,
V/df2 + Va:= a
Squaring",
d^ + A^x= a"
Transposing,
^x=z a^ — d^
Squaring,
a: = (a^ — d'f
Hence, to reduce radical equations, we deduce from
these examples the following general
RULE.
Transpose the terms so that a radical part shall stand by
itself; then involve each member of the equation to a power
of the same name as the root ; if the unknown quantity is
still under the radical sign, transpose and involve as before ;
finally reduce as usual.
4. Reduce 4 + J + 3 \/lc = ^J-. Ans. x=l6.
5. Reduce - 4/ - = - . Ans. x = 2.
6. Reduce (>v/7+ 4)* = 2. Ans. x = 144.
T. Reduce V 11 + arm \/x + 1. Ans. a; = 25.
8 Reduce ^x — 7 == a^ x -f 18 — VS.
Ans. ar = 27.
9. Reduce ■ ^^ = -^ . Ans. x =
X ex yj X 1 <?
10. Reduce ^- ~^ = .V^^nJL . Ans. x = 9.
V^z-f-10 v/x-f28
172 ELEMENTARY ALGEBRA.
11. Reduce s^ x -\- a^ x — a= .
SJ X — a
12. Reduce Vx — 30 + \^ x -j- 21 = \/ x — 19.
13. Reduce \/9^+ 13 = 3\/^+ 1. Ans. a; r= 4.
14. Reduce V — =^ ^, ' Ans. a? = 5.
y/Sx-j- 1 v^5x-j- 2
15. Reduce \/^"^ — 32 = ic — ^V 32.
168i Equations containing the unknown quantity in-
volved to any power require Evolution in their reduction.
169. To reduce pure equations containing the unknown
quantity involved to any power.
1. Reduce --_-=-.
OPERATION.
4a^ 3 97 Clearing the given equation of
5 7 35 fractions, transposing, and divid-
28a;'^ — 15= 97 ing, we have x^ == 4; extract-
28 ic^ = 112 ing the square root of each mem-
^2__ ^ ber of this equation, we have
x=±2 x=±2. (Art. 136.)
2. Reduce 7 a;« — 89 = 100.
OPERATION.
7 a:» — 89 = 100
7a:«=189
ar«= 27
Transposing and dividing, we
have 3? = 27; extracting the
cube root of each member of
this equation, we have a: = 3
X := 3 Hence,
RULE.
Reduce the equation so as to have as one member the un-
known quantity involved to any degree, and then extract that
root of each member which is of the same name as the
power of the unknown quantity.
PURE EQUATIONS ABOVE THE FIRST DEGREE. 173
Note. — It appears from the solution of Example 1 that every
pure quadratic equation has two roots numerically the samej but with op-
posite signs,
3. Reduce-5ar»+ T==?a:^ + 3. Ans. x = ± 6.
X^ x^
4. Reduce a = i — ^ •
c a
. , bed — acd
Ans. X =
■^V c-d
5.
Reduce = 10.
Ans. x= ± 4.
6.
Reduce 3x« + ^ = i.
Ans. 0: = ^-.
7.
Reduce ^^ + 60 = 1.
Ans. a: = — T.
8.
^ , X — 4 2r— 1
Reduce ^^_^,= ^_^^'
Ans.
x=±V-5.
9.
Reduce 4a:' — 4a:» = 0.
10. Reduce 6ar»— 3a: =3x2 — 32: + 60.
1 7
1 1 . Reduce x 4- - = :5 1.
12. Reduce 2x + 2= (x+ If.
13. Reduce I -\- Ux-^=2 ^ 2x-\
14. Reduce Sar^ — 6ar« = 2x-2 + Sx"^ — J.
16. Reduce (c + x)» — 6 c2x= (c — x)»+ 16c*.
iz>r>j 85 8z — 3 8x4-3
16. Reduce ,^ — - — r— = - — ^.
42 3x-}- 3 3x — 3
IT T> J a:«-f-x4-8 , x» 4- x — 8 „
17. Reduce -^^-f- + -J—^- = 2.
174 ELEMENTARY ALGEBRA.
170i Equations containing radical quantities may re-
quire in their reduction both Involution and Evolution;
and in this case the rule in Art. 167, as well as that in
Art. 169, must be applied. Which rule is first to be ap-
plied depends upon whether the expression containing the
unknown quantity is evolved or involved.
1. Reduce 17 — \/x« — 2 = 12.
OPERATION.
IT — Va:^ — 2 = 12
Transposing, &c., s/ x^ — 2 = 5
Squaring, a:^ — 2 = 25
Transposing and uniting, ^ = 27
Extracting the cube root, a: = 3
2. Reduce (V^c' — 4 + 3)^ = 125. Ans. a; = 2.
3. Reduce i/ — =z a^ x. Ans. x-=z ± -
y 2a; .2
a-\-h
4. Reduce \f x -\- a =.
\ X — a
Ans. x= ± A^2a^-{-2ab + b\
5. Reduce f- ^3 (.^ + ii)^^
6. deduce Ai^2x^ + Sx^ + 24.x^+S2x=zx + 2. .
7. Reduce 4^9 {x' + 19) + 100 — ^ = ^^ •
171 1 Equations which contain two or more unknown
quantities may require for their reduction involution, or
evolution, or both. In these equations the elimination is
effected by the same principles as in simple equations.
(Arts. 112-114.)
3^
1. Given -^5 4 "^^ [>-, to find x and y.
2x^ + ,
^ — 14 ?
y = 54)
PURE EQUATIONS .
ABOVE THE FIRST DEGREE.
OPERATION.
^-H^*
(1)
2x' + y= 54
(2)
r ^-^«
(3)
r=iio
(4)
15-?=14
C)
a^= 25
(5)
y= 4
(8)
X =±5
(6)
176
Subtracting four times (1) from (2), we obtain (4), which re-
duced gives (6), or X = ± 5 ; substituting this value of x in (1),
we obtain (7), which reduced gives (8), or y = 4.
Find the value of the unknown quantities in the follow-
ing equations: —
2. Given \'^=''-^l. Ans.l^=±^-
3. Given |'"-'^='^|. Ans. \-=±^-
Ma:«2 = 20 \ ra:== ± 1.
4. Given ^ 2 x ar = 10 >• . Ans. -j y = ± 3.
(3y.z = 45) (2:= ±5.
6. Given K^-^^=n. Ans. |- = 2J-
6. Given |x« + y = 97 >
(x — y = 1/ — 2x>
7. Given
(x*--2y«=14 )
176 ELEMENTARY ALGEBRA.
PROBLEMS
PRODUCING PURE EQUATIONS ABOVE THE FIRST
DEGREE.
172. Though the numerical negative values obtained in
solving the following Problems satisfy the equations
formed in accordance v^ith the given conditions, they are
practically inadmissible, and are therefore not given in
the answers.
1. A gentleman being asked how many dollars he had
in his purse, replied, " If you add 21 to the number and
subtract 4 from the square root of the sum, the remainder
will be 6.^' How many had he?
SOLUTION.
Let X = number of dollars.
Then, ^/x + 2i—4.= 6
Transposing, /\/ x -{- 21 =. 10
Squaring, a; + 21 = 100
Transposing, x= 19, number of dollars.
2. Divide 20 into two parts whose cubes shall be in
the proportion of 27 to 8. Ans. 12 and 8.
3. What two numbers are those whose sum is to the
less as 8 : 3, and the sum of whose squares is 136?
Ans. 10 and 6.
4. What number is that whose half multiplied by its
third gives 54 ?
6. What number is that whose fourth and seventh
multiplied together gives 46f ? Ans. 36.
6. There is a rectangular field containing 4 acres whose
length is to its breadth as 8:5. What is its length and
breadth ?
PURE EQUATIONS ABOVE THE FIRST DEGREE. 177
V. There are two numbers whose sum is IT, and the
less divided by the greater is to the greater divided by
the less as 64 : 81. What are the numbers?
Aus. 8 and 9.
8. The sum of the squares of two numbers is 65, and
the difl'erence of their squares 33. What are the numbers ?
9. The sum of the squares of two quantities is a, and
the difference of their squares b. What are the quantities ?
Ans. ± VTTM^) a"d ± Vi la~^h).
10. A gentleman sold two fields which together con-
tained 240 acres. For each he received as many dollars
an acre as there were acres in the field, and what he
received for the larger was to what he received for the
smaller as 49 : 25. What are the contents of each ?
Ans. Larger, 140; smaller, 100 acres.
11. What are the two quantities whose product js a
and quotient hi . . — - , la
Ans. ± V a ^ and ±1/7-
12. What two numbers are as m : n, the sum of whose
squares is a? ^ mi/a , . nd'a
Ans. ± -, ^ and ± -; ^
Y/(m«-}-n«) V(m'-}-n«)
13. What two numbers are as m : n, the difference of
whose squares is a? , - ,-
Ans. ± ^ and ± -, —
14. Several gentlemen made an excursion, each taking
$484. Each had as many servants as there were gentle-
men, and the number of dollars which each had was four
times the number of all the servants. How many gen-
tlemen were there? Ans. 11.
16. Find three numbers such that the product of the
first and second is 12 ; of the second and third, 20 ; and
the sum of the squares of the first and third, 34.
8» L
178 ELEMENTARY ALGEBRA.
SECTION XIX.
AFFECTED QUADRATIC EQUATIONS.
173f An Affected Quadratic Equation is one that con-
tains both the first and second powers of the unknown
quantity ; as,
Sx^ — 4: x=: 16; or ax — bx^ = c.
174. Every affected quadratic equation can be reduced
to the form
x"^ -]- bx = c,
in which h and c represent any quantities whatever, posi-
tive or negative, integral or fractional.
For all the terms containing a^ can be collected into one term
whose coefficient we will represent by a ; all the terms containing x
can be collected into one term whose coefficient we will represent
by d; and all the other terms can be united, whose aggregate we
will represent by e. Therefore every affected quadratic equation
can be reduced to the form
Dividing (1) by a,
d e
Letting - =b, and - = c, we have
175. The first member of the equation x^ -\- bx = c
cannot be a perfect square. (Art. 145, Note 2.) But
we know that the square of a binomial is the square of
the first term plus or minus twice the product of the two
terms plus the square of the last term; and if we can
find the third term which will make x"^ -{- bx a perfect
aa^-\-dx = e
(1)
'a a
(2)
3^-{-bx = c
(3)
EQUATIONS OP THE SECOND DEGREE. 179
square of a binomial, we can then reduce the equation
x^ -\- bx = c.
Since b x has in it as a factor the 8qi\are root of a:*, a:* can be
the first term of the square of a binomial, and b x the second terra
of the same square; and since the second term of the square is
twice the product of the two terms of the binomial, the last term of
the binomial must be the quotient arising from dividing the second
term of the square by twice the square root of the first term of the
square of the bi no-
inial ; i. e. the last
OPERATION. ^^ ^^ ^i^g ^j„^
x2-f-5ar = c (1) . , . bx b
«" + ^^ + 4 = 4 + « (2) ^^^ therefore the
J rn — third term of the
x-\- -^=- ±. kI T '\~ ^ (^) square must be
2 Y * b\ .
— to each mem-
4
ber, we have (2), an equation whose first member is a perfect
square. Extracting the square root of each member of (2), and
transposing, we obtain (4), or x ^ — 9 "*■ v/ T "^ ^' ^^'^^ ** *
general expression for the value of x in any equation in the form
of z* -|" ^ ^ = <^*
Hence, as every affected quadratic equation can be re-
duced to the form x^ -\- hx^=. c, in which b and c repre-
sent any quantities whatever, positive or negative, integral
or fractional, every affected quadratic equation can be re-
duced by the following
RULE.
Beduce the equation to the form x^ -\- bx=z c, and add
to each member the square of half the coefficient of x.
Extract the square root of each member, and then reduce
as in simple equations.
180 ELEMENTARY ALGEBRA.
1. Reduce Ta;2 — 28 a; + 14 = 238.
OPERATION.
*lf — 2Sx-{- 14 = 238
Transposing, T a;^ — 28 a: = 224
Dividing by Y, x"^ — 4a;= 32
Completing the square, x^ — 4 x -j- 4 = 36
Evolving, X — 2 = ±6
Transposing, a;:=:2±6 = 8, or — 4
Note. — Since in reducing the general equation x^ -{- bx = c
vre find x = — ^±i/j -f-c, every affected quadratic equation
must have two roots ; one obtained by considering the expression
2 ~\~ ^ positive, the other by considering this expression nega-
/ b^
tive. Whenever 4 / - -j- c = these two roots will be equal.
o -D J ^ rr , 13 x^ , X
2. Reduce ^--- + -^ = - + -.
operation.
Clearing of fractions, 4a:^ — 2a: + 13 = lOa:^ -|- 5x
Transposing, — Car* — 1x = — 13
Dividing by — 6, x^ -\ = —
Completing the square, ^' + i) +^^ = ^, + ~ =~
Evolving, a; + ^ = ± ^
Transposing, ^ = — — ± j^ ^\, or— 2^
EQUATIONS OF THE SECOND DEGREE. 181
Note. — In completing the square, as the second term disappears
when the root is extracted, we have written ( ) in place of it.
3. Reduce 3x^ — 25 + 6x = 80.
Ans. x=z 5, or — t.
24 X
4. Reduce x = 3. Ans. x=z6, or — 4.
X
5. Reduce2x + ^-±4=7. Ans. x = 2.
' X 1
Note. — In this example both roots are 2.
x* 4- 4
6. Reduce 1x ^^ =zbx — 1.
X — 4
Ans. X = 8, or — 1.
t. Reduce 17 - ^-^-=^ == ^£^ + 10.
Ans. X = 7, or — 27.
8. Reduce T , + 3 = ^* A^^s. x = 10, or — 1 J.
x+5
9. Reduce j -^ = 4.
,/v T> J 16 100 — 9x «
10. Reduce —^ — = 3.
x 4z"
176. Whenever an equation has been reduced to the
form x^ -\- bx = c, its roots can be written at once; for
this equation reduced (Art. 175) gives x = — - ± t /- + c.
Hence,
TJie roots of an equation reduced to the form x^ -\- bx = c
are equal to one half the coefficient of x with the opposite
sign, plus or minus the square root of the sum of the square
of one half this coefficient and the second member of the
equation.
182 ELEMENTARY ALGEBRA.
In accordance with this, find the roots of x in the fol-
lowing equations : —
1. Reduce a:^ -|- 8 a: = 65.
a; = — 4 ± /s/16 + 65 = 6, or — 13, Ans.
2. Reduce a:2~ 10a; = — 24.
x = b ± /v/25 — 24 = 6, or 4, Ans.
3. Reduce x'^ — Qx=z — ^, Ans. a; = 5, or 1. •
4. Reduce a:^ _|_ T a; = ITO.
7 /^9
^ = — 2 ± \/ X + ^"^^ = ^^' ^^~ ^'^' ^"«-
6. Reduce a;^ + -x = --
^ = -i±\/^ + ^ = i or -1, Ans.
1 9
6. Reduce x^ + -a: =-• Ans. a: = 1, or — 1^.
Y. Reduce a;2 — -zx^^ — — • Ans. a; = -, or -•
5 25 5 5
8. Reduce - = | -j- 5^. Ans. a: = 7, or — 5^.
SECOND METHOD OF COMPLETING THE SQUARE.
177. The method already given for completing the
square can be used in all cases ; but it often leads to in-
convenient fractions. The more difficult fractions are in-
troduced by dividing the equation by the coefficient of
x^y to reduce it to the form a;^-|-ia: = c. To present a
method of completing the square without introducing these
fractions, we will reduce equation (1) in Art. 174.
EQUATIONS OF THE SECOND DEGREE. 183
1. Reduce ax^ -\- dx = e.
OPERATION.
ax' + dx = e (1)
a^a^-^-adx^ae (2)
a'^x^ + adx + j=j + ae (3)
ax
X
Multiplying (1) by a, the coefficient of a^, we obtain (2), in which
the first term must be a perfect square. Since adx, the second
terra, has in it as a factor the square root of a* r", a* 2* can be the
first tenn of the square of a binomial, and adx the second term ;
and since the second term of the square is twice the product of the
two terms of the binomial, the last term of the binomial must be the
second term of the square divided by twice the square root of the first
term of the square of the binomial, or „ — = - ; and therefore the
2 ax 2
d*
term required to complete the square is — , which is the square of
d*
one half of the coefficient of x in (1). Adding - to both members
of (2), we obtain (3), whose first member is the square of a binomial.
Extracting the square root of (3) and reducing, we obtain (5), or
l{-Ht+-}
Hence, to reduce an aflfected quadratic equation, we
have this second
RULE.
Reduce the equation to the form ax^ -\- dx = e; then mul-
tiply tlie equation by the coefficient of ar*, and add to each
member the square of half the coefficient of x.
Extract the square root of each member, and then reduce
as in simple equations.
184 ELE]VIENTARY ALGEBRA.
Note 1. — This method does not introduce fractions into the equa-
tion when the numerical part of the coefficient of x is even. When
the coefficient of x* is unity, this method becomes the same as the
first method.
Note 2. — If the coefficient of 3? is already a perfect square the
square can be completed without multiplying the equation, by add-
ing to both members the square of the quotient arising from dividing
the second term by twice the square root of the frst. This method
also becomes the same as the first method when the coefficient of
3^ is unity.
Note 3. — As an even root of a negative quantity is impossible
or imaginary, the sign of the first term, if it is not positive, must
be made so by changing the signs of all the terms of the equation.
2. Reduce 3 a:^ + 8 a; = 28.
OPERATION.
3x2_|_8a: = 28
Completing the square, 9 a;^ + ( ) -f 16 = 16 + 84 = 100
Extracting square root, 3a;-[-4=± 10
Whence, 3a; = — 4 ±10 = 6, or — 14
And x = 2, or — 4f
3. Reduce 25ar» — 10 a; = 195.
operation.
25a;2_-i0a:=195
Completing sq. by Note 2, 25a;2— () + 1 = 1 + 195 = 196
Extracting square root, 5x — 1 = ± 14
Whence, 5a:=l ± 14 = 15, or— 13
And x= 3, or — 2f
4. Reduce 5 a;^ — 20 a; = — 15. Ans. a; = 3, or 1.
5. Reduce Ta:2 — 8 a; =12f. Ans. a; = f ± f V 26.
6. Reduce Y a;'' — 4:ax=-—-' Ans. a; =: — , or — -•
Y. Reducer- ^"7 =x — 3. Ans. x= 10, or 31.
14 — z S •*
EQUATIONS OF THE SECOND DEGREE. 185
8. Reduce ar^ + l^ = — i^ — 4.
' 4 4
9. Reduce — = g — = — ^
THIRD METHOD OF COMPLETING THE SQUARE.
178i The method of the preceding Article introduces
fractions whenever the numerical coefficient of x is not
even. To present a method of completing the square
without introducing any fraction, we will again reduce
equation (1) in Art. 174.
1. Reduce ax^ -{- dx = e.
OPERATION.
aar^-j- dx = e
(1)
' a a
(2)
^■+"a'*'"*"4a«""4a«"+"a
(3)
a»a:» + 4arfx + rf2==ef- + 4ae
W
2ax + d= ± Vrf^ + 4«e
(5)
_ — d± v/d»4-4atf
(6)
Dividing (1) by a, the coefficient of 2*, we have (2); then com-
pleting the square according to the Rule in Art. 1 75, we have (3) ;
and if we multiply (3) hy 4 a\ it will give (4), an equation free
from fractions (unless o, d, or c in (1) are themselves fractions),
and one whose first member is the square of a binomial. To pro-
duce this equation directly from (1), we have only to multiply (1)
by 4a; i. e. by four times the coefficient of z*, and add to both
members rf* ; i. e. the square of the coefficient of x. Reducing we
have (S), which is a general expression for the value of x in any
equation in the form o{ ax^ -{- dx = e.
186 ELEMENTARY ALGEBRA.
Hence, to reduce an affected quadratic equation, we
have this third
RULE.
JReduce the equation to the form ax^ -\- dx=z e; then mul-
tiply the equation by four times the coefficient of x^ and add
to each member the square of the coefficient of x.
Extract the square root of each member, and then reduce as
in simple equations.
Note. — The third Note under the Rule in Art 177 is applica-
ble in all cases.
2. Reduce 5x'^ — 1x = 24:,
OPERATION.
5 a:2 — 7 r = 24
Multiplying by 5 X 4 and
adding 7^ to each member, 100.r — { ) + 49 = 49 + 480 = 529
Extracting the square root, lOx — 7 = ±23
Transposing, 10 x = 7 ± 23 = 30, or — 16
Whence, a: = 3, or — 1.6
Note. — The multiplication of the coefficient of x^ need only be
expressed. Its coefficient after evolving is double its original coef-
ficient.
3. Reduce ^^"^"^^^ = 293.
OPERATION.
44 x^ — 15 a:
= 293
7
Clearing of fractions, 44 a;^ — 15 a: = 2051
Completing square, 176 X 44ar2 — { ) + 225 = 225 + 360976 = 361201
Evolving, 88 a: — 15 = ± 601
Transposing, 88 a: = 15 ± 601 = 616, or — 586
293
Whence, a: = 7, or -—
44
4. Reduce 1 x^ — 15 a; = — 2. Ans. a: = 2, or |.
EQUATIONS OF THE SECOND DEGREE. 187
6. Reduce —5 ^ — rir = ^ — 3.
XT O X -f- V
6. Reduce — 1—^ + - == 5.
7. Reduce — ^ + ? = 3.
o „j 4X-I-4 3x— 3
8. Reduce ' -z
X 2x — 1
1 3
9. Reduce
7 — 2x ' 2x-f-4
Ans. x =
1, or-
-28.
Ans. X =
3, or —
■u-
Ans. X =
= 2, or -
-i
lOx-flO
3x
13
10*
10. Reduce ^x^ — a^ = x — b.
h , lAcf — V
Ans. a: = -±y/-3^^.
11. Reduce 5 — 3x-^ = llOar-*.
Note. — Multiply by x*.
12. Reduce s/l^ + V^* = 6 \/^.
Note. — Divide by y^x!
179. The rules which have been given for the solution
of affected quadratic equations apply equally well to any
equation containing but two powers of the unknown quan-
tity whenever the index of one power is exactly twice that
of the other. By the same reasoning as in Art. 1T4, it can
be shown that all such equations can be reduced to the
fonn
aa:^" -|" ^^ = ^>
or
It will be seen that the first member is composed of
two terms so related that they may be the first two terms
of a binomial square, and we can supply the third by one
of the rules already given for completing the square.
OPERATION.
x^ — 2x^ = 4.8
(1)
r6_.2x«+ 1 = 1 + 48 = 49
(2)
x^—l=±1
(3)
x^ = S,or — 6
(4)
a: = 2, or ^— 6
(5)
188 ELEMENTARY ALGEBRA.
1. Reduce ;r^ — 2a:^==48.
Since the squu*e root
of x^ isa^, it is evident
that the second term
contains as one of its
factors the square root
of the first term ; i. e.
the first member of the
equation is composed
of two terms so related that they may be the first two terms of the
square of a binomial. Completing the square, we have (2) ; extract-
ing the square root of each member of (2), we obtain (3) ; transpos-
ing we have (4), and extracting the cube root of (4) we have x = 2,
or ^^^
2. Reduce Sx^ — 4.x^ = 160.
OPERATION.
3a;l_4^l— 160 (1)
36;k^— + 16 = 16+ 1920 = 1936 (2)
6a;^ — 4=±44 (3)
6 a:^ = 48, or — 40 (4)
x^ = 8, or — 5f- (5)
. x^'= 2, or ^ — -s^o (6)
a:= 16, or (— Y)^ (7)
In this equation the index of the higher power is exactly twice
that of the lower. Completing the square we have (2) ; extract-
ing the square root of each member of (2), we have (3) ; trans-
posing, we have (4), which divided by 6 gives (5); extracting the
cube root of (5), we have (6), which involved to the fourth power
gives (7).
3. Reduce ^ + ^ = A. Ans. x = ^J, or — ^J.
it 4 o 2i
4. Reduce -^a;^ + f/^x=:l. Ans. a; = j, or — 8.
EQUATIONS OF THE SECOND DEGREE. 189
5. Reduce a: — § >/ x = 44^. Ans. x = 49, or 40|.
6. Reduce x* — x^ = 0. Ans. a; = 1, or 0.
7. Reduce 3ar* — 2ar»+3 = 228.
Ans. X =z ± S, or ± 5 \/ — ^.
8. Reduce 3 x^- — 2 x" = 8.
Ans. x = aC^2, or <^ — J.
9. Reduce ^A^-±J = t±ll . Ans. x = 64, or 4.
10. Reduce x* — a x^ = i.
Ans.x = ±y/(|±^^ + J).
180. A polynomial may take the place of the unknown
quantity in an affected quadratic equation. In this case
the equation can be reduced by considering the polyno-
mial as a single quantity.
1. Reduce (x — 4)^ — 2 (x — 4) = 8.
OPERATION.
(x — 4)2 — 2 (x — 4) = 8 (1)
(x-4)2-() + l = l-f 8 = 9 (2)
X — 4--l=±3 (3)
X = 6 ± 3 = 8, or 2 (4)
Considering x — 4 as a single term, and completing the square,
we have (2) ; extracting the square root, transposing, &c., we have
(4), or X =« 8, or 2.
Note. — We might put (x — 4)=y; then (x — 4)* = 3^, and
the equation becomes y* — 2 y = 8. After finding the value of y
in this equation, x — 4 must be substituted for y.
2. Reduce V5 + x + /^5 + x = 6.
Ans. x= 11, or 7ft,
3. Reduce 4 + 2x — x3 + j^>v/44- 2x — x» = i.
Ans. X = 1 ± ^\/T9, or 3, or — 1.
190 ELEMENTARY ALGEBRA.
4. Reduce x + 7 — T s/ x -f- Y = 8 — hs/x + t.
Ans. a; = 9, or — 3.
6. Reduce ix — 6)^ — Z s/ x — 5 = .
Ans. itr = 9, or 5 + ^25.
6. Reduce 7?-\-Zx-\- s/ x^ _[_ 3 a: -f- 6 = 14.
Note. — Add 6 to both members.
Ans. a; = 2, or — 5, or — f ±1 V^.
1, Reduce 4 + aj2 — 2a: — 2 V6 — 2a: + x^=l.
Ans. a: = 3, or — 1, or 1 ± 2 \/ — 1.
60
8. Reduce Vx^ + a: + 6 = , — 4.
Ans. a: = 5, or — 6, or — \±.\ \/377.
181. Of the methods given for completing the square,
the first is the best when the coefiScient of the less power
of the unknown quantity is even, and the coefficient of the
higher power is unity, or when these become so by reduc-
tion ; the second method is better than the third when-
ever the coefficient of the less power of the unknown quan-
tity is even. When the equation cannot be reduced by
the first method without introducing fractions, if the co-
efficient of the higher power of the unknown quantity
is a perfect square, and the coefficient of the less power
is divisible without remainder by twice the square root
of the coefficient of the higher power, the method given
in Note 2, Art. 177, is the best. Let each of the follow-
ing equations be reduced by the method best adapted
to it.
1. Reduce 4x2— 14 _ 3^:2 __ i2a; — 1.
Ans. a: = 1, or — 13.
2. Reduce 36ar^ + 24 a; =1020.
Ans. a: = 6, or — 5|.
EQUATIONS OF THE SECOND DEGREE. 191
3. Reduce x — i + ? = 21. Ans. x = 21, or f .
„ , 9, — x x — 2 2a:— 11
4. Reduce —2 6~ = x-S *
5. Reduce g "~ 5 "" To ~ ^^*
6. Reduce — ^ — = ~^ — ^*
Ans. a; = 3(c — d), or 3rf.
7. Reduce |±^ = ^ + 2§.
8. Reduce — = 9 —
X — 4 ^
9. Reduce - — y = ^
10. Reduce ^ + ^ = ?• Ans. x == 1 ± Vl— «"•
11. Reduce 3x + 3 = 13 + -•
3a: — 3 « , 3x — 6
12. Reduce 5 X — — ^==2xH ^
13. Reduce ^-1^ + 1 = ^-
U. Reduce 2 >v/^ — ^x — *l = 5.
Ans. X = 16, or 7^.
15. Reduce 2/s/«^^+3V2^= -^j^=-
Ans. X = 9 a, or — a.
15
16. Reduce 4>\/^ — \/2x+ 1 = ,^^ ■
Ans. X == 4, or — 2f .
17. Reduce 5 V25 — x = 6 \/26 — x -f x — 13.
Ans. X = 16, or 9.
192 ELEMENTARY ALGEBRA.
18. Reduce ^ii = V 4~+ V 2^H^^'.
Ans. X = 12, or 4.
19. Reduce 6 + 4x-i — 12 a:-^ = 100 ar-^.
20. Reduce i ^~^' + 6 .^x — y = y •
21. Reduce 3 a;* — 24 a;^ — 80 = 304.
22. Reduce ^ -J- 10 = 1 + 4:r».
Ans. a: = ± 4, or ± 2 V— 2.
Ans. a: =: \/"9, or .^.
23. Reduce 5 x* — 3 a:^ + ?1 = 27,
' 4
Ans. a;=^ ± iv 6, or ± 3/v/— ^V-
24. Reduce 2 x^ — 5 a:^ + 4 = 2.
25. Reduce 6 a;^ + 1184 = 5a;i
26. Reduce \/^ + 3 — ^x -f 3 = 2.
Ans. X = 13, or — 2.
27. Reduce x^ — x/ar^ -j- a; — 5 = 25 — a:.
Note. — By transposing — x and subtracting 5 from each mem.
ber, make the expression without the radical in the first member like
that under the radical; then complete the square, &c.
28. Reduce a:^ — 2ar + 3V2a:2 — 6ar~ll=x + 33.
Ans. a; = 6, or — 3, or f ± ^ \/^T3.
29. Reduce 21 a:* + 11 a:^ _ 69 =:= 321 — (11 ar* + 5 a;^).
Ans. x= ± ^ VT3, or ± ^ \/ — lb.
30. Reduce 7— J—— = ^ 4-
(2 a: — 4/ 8 ' (2a: — 4)*
31. Reduce (a:2— 4a:)2z=12x — 3a:2.
32. Reduce a; + (a;2 — ar)2 = a:2 -I- 6112.
EQUATIONS OF THE SECOND DEGREE. 193
PROBLEMS
PRODUCING AFFECTED QUADRATIC EQUATIONS WITH
BUT ONE UNKNOWN QUANTITY.
182. Though the numerical negative values obtained in
solving the following Problems satisfy the equations formed
in accordance with the given conditions, they are prac-
tically inadmissible, and are therefore not given in the
answers.
1. Divide 40 into two parts such that the sum of their
squares shall be 1042.
SOLUTION.
Let X = one part ;
then 40 — x = other part
Then, x« -}- (40 — x)' = 1042
Expanding, a^ _|- 1600 — 80 a: -j- x» = 1042
Transposing and uniting, a:" — 40a: = — 279
Whence, x = 20 ± 1 1 = 81, or 9
And, 40 — X = 9, or 31
2. Divide 20 into two parts such that their product
will be 99J. Ans. ^ and 10^.
3. The ages of two brothers are such that the age of
the elder plus the square root of the age of the younger
is 22 years, and the sum of their ages is 34 years. What
is the age of each? • Ans. Elder, 18; younger, 16.
Note. — The other answers found by reducing the equation, viz.
25 and 9, satisfy the conditions of the equation only upon consid-
ering y' 9 = — 3. To make the problem correspond to these an-
swers, the word "plug** must be changed to "minus."
4. A merchant had two pieces of cloth measuring to-
gether 96 yards. The square of the number of yards in the
194 ELEMENTARY ALGEBRA.
longer is equal to one hundred times the number of yards
in the shorter. How many yards are there in each piece ?
Ans. 60 and 36.
5. Find two numbers whose difference is 3, and the
sum of whose squares is 117. Ans. 9 and 6.
6. A merchant having sold a piece of cloth that cost
him S42, found that if the price for which he sold it were
multiplied by his loss, the product would be equal to the
cube of the loss. What was his loss ?
Note. — If the word " loss " were changed to gain; the other an-
swer, — 7, or as it would then become, -}- 7> would be correct.
Ans. $6.
T. Find two numbers whose difference is 5, and prod-
uct 176. Ans. 11 and 16.
8. There is a square piece of land whose perimeter in
rods is 96 less than the number of square rods in the
field. What is the length of one side ? Ans. 12 rods.
9. Find two numbers whose sum is 8, and the sum of
whose cubes is 152.
10. A man bought a number of sheep for $240, and
sold them again for $6.75 apiece, gaining by the bargain
as much as 5 sheep cost him. How many sheep did he
buy ? Ans. 40.
11. Find two numbers whose difference is 4, and the
sum of whose fourth powers is 1312.
Note. — Let x — 2 and x -|- 2 be the numbers.
Ans. 2 and 6.
12. A man sold a horse for $312.50, and gained one
tenth as much per cent as the horse cost him. How
much did the horse cost him? Ans. $250.
13. The difference of two numbers is 5, and the less
minus the square root of the greater is 7. What are the
numbers? Ans. 11 and 16.
EQUATIONS OF THE SECOND DEGREE. 195
14. A and B started together for a place 300 miles dis-
tant. A arrived at the place 7 hours and 30 minutes be-
fore B, who travelled 2 miles less per hour than A. How
many miles did each travel per hour?
Ans. A, 10 ; B, 8 miles.
15. A gentleman distributed among some boys $15;
if he had commenced by giving each 10 cents more, 5
of the boys would have received nothing. How many
boys were there? Ans. 30.
16. Find two numbers whose sum is a, and product h.
Ans. "f-^ and ^-^^
17. A merchant bought a piece of cloth for $45, and
sold it for 15 cents more per yard than he paid. Though
he gave away 5 yards, he gained $4.50 on the piece.
How many yards did he buy, and at what price per yard ?
Ans. 60 yards, at 75 cents per j'^ard.
18. A certain number consists of two figures whose
sum is 12; and the product of the two figures plus 16 is
equal to the number expressed by the figures in inverse
order. What is the number? Ans. 84.
19. From a cask containing 60 gallons of pure wine a
man drew enough to fill a small keg, and then put into
the cask the same quantity of water. Afterward he drew
from the cask enough to fill the same keg, and then there
were 41 j gallons of pure wine in the cask. How much
did the keg hold ? Ans. 10 gallons.
20. There is a rectangular piece of land 75 rods long
and 65 rods wide, and just within the boundaries there is
a ditch of uniform breadth running entirely round the
land. The land within the ditch contains 29 acres and
96 square rods. What is the width of the ditch ?
Ans. .5 of a rod.
196 ELEMENTARY ALGEBRA.
SECTION XX.
QUADRATIC EQUATIONS CONTAINING TWO
UNKNOWN QUANTITIES.
183. The Degree of any equation is shown by the sum
of the indices of the unknown quantities in that term in
which this sum is the greatest. Thus,
^xy — 2a: = 7 is an equation of the second degree,
hx'y'-^- xif^a'c " '' " fourth
5^4 _ 14^ _ ^2^.3 .. Ci i^ fifth
Note. — Before deciding what degree an equation is, it must be
cleared of fractions, if the unknown quantities appear both in the
denominators and in the numerators or integral terms; and also
from negative and fractional exponents.
184. A Homogeneous Equation is one in which the sum
of the exponents of the unknown quantities in each term
containing unknown quantities is the same. Thus,
^x^ — ^xy -\- y"^ z=:i\^
or x^-\- 3:r^2 + 3ar2y + / = 2T
or x^ — ^.x^yAr Gx^/ — 4a;/ + ^^ = 256
is a homogeneous equation.
185. Two quantities enter Symmetrically into an equa-
tion when, whatever their values, they can exchange places
without destroying the equation. Thus,
x^ — 2xy + / — 25
or a:»+3ar2y + 3:r/4- /= 8
or x' + 2xy-\-y''^ 2x + 2y = 24
QUADRATIC EQUATIONS. 197
186. Quadratic equations containing two unknown quan-
tities can generally be solved by the rules already given,
if they come under one of the three following cases: —
I. When one of the equations is simple and the other
quadratic.
II. When the unknown quantities enter symmetrically
into each equation.
111. When each equation is quadratic and homogeneous.
CASE I.
187. When one of the equations is simple and the other
quadratic.
1. Given 1^^+^^,"= ^- I, to find X and y.
OPERATION.
2x-f2y = 22 (1) 3x'-{-if=in (2)
3/=ll— x(3) 3z»-f-121 — 22a:-|-3:»=lll (4)
42^— 22x = — 10 (5)
4«a:«—()-fll«= 121— 40 = 81 (6)
4x= 11 ±9 = 20, or 2 (7)
y=6,orl0^(9) a:=5, or^ (8)
From (1) we obtain (3), or y = 11 — x. Substituting this value
of y in (2), we obtain (4), an affected quadratic equation, which
reduced gives (8) ; and substituting these values of x in (3), we
obtain (9).
In this Case the values of the unknown -quantities can
generally be found by substituting in the quadratio eqvaHon
the value of one unknown quantity found by reducing the
siviple equation.
198 ELEMENTARY ALGEBRA.
2. Given | ^^ " ^^ | , to find a: and y.
xy = 2^ (1)
OPERATION,
X — y = 3
(2)
x' — 2xy + 7f= 9
(3)
4:xy ==112
(^)
x^ + 2xy + y'=l2l
(S)
x + y =±n
(6)
2x = 14, or —
8
a)
2y = 8, or —
14
(8)
X = 1, or —
4
(9)
y = 4, or —
7
(10)
Subtracting four times (1) from the square of (2), we obtain (5) ;
extracting the square root of each member of (5), we obtain (6);
adding (2) to (6), we obtain (7) ; subtracting (2) from (6), we ob-
tain (8) ; and reducing (7) and (8), we obtain (9) and (10).
Note. — Though Example 2 can be solved by the same method
as Example 1, the method given is preferable.
By this method find the values of x and y in the fol-
lowing equations : —
3. Given |^ "^ = H. Ans. j ^ = ^•
4. Given |^+^=13|. Ads. j^ = ^^or6.
6. Given < ^ ^ >■ •
\5x + y = 29>
6. Given j ^y-24>
iSx — 2yz=lQi
Zxy
= 45
^y
= 15
7?-
-2xy+y«
= 4
^ — y
= ±2
QUADRATIC EQUATIONS. 199
CASE II.
188. When tlie unknown quantities enter symmetrically
into each equation.
1. Given | ^ + ^ = ^ J | , to find x and y.
(x'-j-y*= 152)
OPEIIATION.
x+y = % (I) 2:»+y=152 (2)
a:^ + 2x^ + 3/^ = 64 (3)
ar«- xy + y^=19 (4)
(5)
(6)
a)
(8)
2x = 10, or 6 (9)
2y = 6, or 10 (10)
X = 5, or 3 (11)
y = 3, or 5 (12)
Squaring (1), we obtain (3); dividing (2) by (1), we obtain (4);
subtracting (4) from (3), we obtain (5), from which we obtain (6) ;
subtracting (6) from (4), we obtain (7) ; extracting the square root
of each member of (7), we obtain (8) ; adding (8) to (1), wc ob-
tain (9); subtracting (8) from (1), we obtain (10); and reducing
(9) and (10), we obtain (11) and (12).
Note 1. — It must not be inferred that x and y are equal to
each other in these equations ; for when x = 5, y = 3 ; and when
X = 3, y = 5. In all the equations under this Case the values of
the two unknown quantities are interchangeable.
Note 2. — Although 3* -^ r^ = \b2 is not a quadratic equation,
yet as we can combine the two given equations in such a manner
as to produce at once a quadratic equation, we introduce it here.
200 ELEMENTARY ALGEBRA.
2. Given i„.„ ^^ r»to find x and y.
U2-fy2_2a; — 2y = 3> ^
OPERATION.
a:y=6 (1) a:2_|_^_2a;_2y= 3 (2)
2a:y =12 (3)
(^ + 2//-2(a: + 2/) = 15 (4)
(3; + yy- () + 1 = 16 (5)
x + 3/=l±4 = 5,or — 3 (6)
a: = 3,or2,or ""^^/~^\ 7)
y=2,or3,or^:iiHyj:iL^(8)
Adding twice (1) to (2), we obtain (4) ; completing tlie square
in (4), we obtain (5) ; extracting the square root of each member
of (5), and transposing, we obtain (6) ; and combining (1) and (6)
as the sum and product are combined in the preceding example,
we obtain (7) and (8).
In Case II. the process varies as the given equations
vary. In general the equations are reduced by a proper
combination of the sum of the squares, or the square of the
sum or of the difference, with multiples of the product of
the two unknoiun quantities; and finally, of the sum, with
the difference of the two unknown quantities.
Note 3. — When the unknown quantities enter into each equation
symmetrically in all respects except their signs, the equations can be
reduced by this same method ; e. g. ar — ?/ = 7, and x^ — t/' = 511.
In such equations the values of the unknown quantities are not inter-
changeable.
Note 4. — The signs ± ^ standing before any quantity taken in-
dependently are equivalent to each other ; but when one of two quan-
tities is equal to ± a while the other is equal to ^ &, the meaning is
that the first is equal to -f- a, when the second is equal to — b; and
the first to — a, when the second is equal to -|- ^«
QUADRATIC EQUATIONS. 201
By this method find the values of x and y in the follow-
ing equations : —
3. Given i2-+2y= IH . Ans. j- = 4'Or3.
l3x»+3/ = 273) (y = 3,or4.
(^^y— 8) .^„ (x = 9,or— 1.
4. Given ^ . •^, w«« f • -^^s* "^ , o
(ar» — / = 728) Cy= 1, or — 9.
6. Given K + ^^t ^'^ «tl '
Note. — Divide the second equation by the first.
6. Given |^-^^ + y = U.
CASE III.
189. When each equation is quadratic and homogeneous.
1. Given pa:y+ y^ = 5) to find a: and y.
KZt? — ary = 10 )
OPERATION.
2xy+y» = 5 (1) Sa:'* — xy=10 (2)
Let x=zvy
2vy^J^f = b (3) 3t;2y2 — vy2=io (4)
5 10
a)
2i;-^-l ~ Zv'—v
. 15t;» — 5v=20v+ 10
(8)
3r^ — 6t; = 2
(9)
r = 2, or — ^
(10)
y=±l,or±Vl5 (12)
x = ry= ±2, or qp i\/T5 (13)
202 ELEMENTARY ALGEBRA.
Substituting vy for x in (1) and (2), we obtain (3) and (4) ; from
(3) and (4) we obtain (5) and (6) ; putting these two values of y^
equal to each other, we obtain (7), which reduced gives (10) ; sub-
stituting this value of v in (5), we obtain (11), which reduced gives
(12) ; and substituting hx x = vy the values of v and y from (10) and
(12), we obtain (13).
Examples under Case III. can generally be reduced best
by substituting for one of the unknown quantities ih^ product
of the other by some unknown quantity, and then finding the
value of this third unknown quantity. When the value of
this third quantity becomes known, the values of the given
unknown quantities can be readily found by substitution.
Note. — Whenever, as in the example above, the square root is
taken twice, each unknown quantity has four values ; but these values
must be taken in the same order, i. e. in the example above, when
y = -j-l, x = -f~2; when y = — l,a: = — 2; when y = -[- y^Ts,
X •«= — \ y/Ts ; and when y = — y/ 15, a; = -]- ^ V^ 15.
By this method find the values of x and y in the follow-
ing equations : —
2. Given j ^^ - ^^ = 1^
a:= ± T, or ± 4\/— -J.
y= ±5, or dbllV— ~i.
Ans. <
3. Given \-'' + ^-y = ^n.
'•1
Ans. ^-=±3,or±9^/-i.
= ± 2,.or TS-i/— i-
ixy — ^y = 3 — x^ )
Ans j^=±2,or±24V-V^.
\y=± l,or qillV— T^T-
5. Given j'' ": '^^ = '/ ^^^l •
( x2 — 3=y2 + 2 >
QUADRATIC EQUATIONS. 203
190« Find the values of x and y in the following
Examples.
Note. — Some of the examples given below belong at the same
time to two Cases. Thus in Example 1 both the equations are
symmetrical, and both are (juadratic and homogeneous, and there-
fore it belongs both to Case II. and Case III. Example 3 belongs
both to Case I. and Case II.
1. Given I , -y = 20) ^^^ Cx=±6,or±4.
a; = 7, or 3.
y = 3, or 7.
ar» + ^-^ = 41 ) (y = ± 4, or ± 5.
2. Given I ^y= 6 )
U^+7x?/ = 55— y2)
3. Given i^+y= "^ I.
Ans. j^ = 4^^^8-
(y = 3, or 4.
4. Given | ^^=12 >
(a:» + x = 32— y— y^>
5. Given \ ^T^'^^^l- Ans. |
(sxy — 7 = 56)
6. Given \ ^-y= H.
(a^y''-\-2xyz= 1295)
Note. — Considering xy a single quantity, find its value in the
second equation.
7. Given i^y--f=m.
Note. — Subtract from the second equation three times the first,
and extract the cube root of each member of the resulting equation.
8. Given j2x=y + 2x/= 168)
-Ans. 1^ = 1'°' !•
- (« =3, or4.
204
9. Given
10. Given
11. Given
12. Given
13. Given
ELEMENTARY ALGEBRA.
\ a:y=10 j
na;2 — 2a:y=:88)
Ans. |^ =
= ±5.
±2.
Ans.
{
a:=±4, or± 66 a^
y= ±3, or q: Its V^.
FJrg--
{
2a: + 2y = 30~y2.
4xy = 60 >
(6a:^-— 23/^= 10 )
Ans.
fa-rr: 1000, or 8.
14. Given | 3a:^:^18)
(0:^ — 2/^ = 65)
Ans. ■!
625, or 1
a:= ± 3, or ± 2/^— 1,
3^=: ± 2, or ± 3^"^=^.
16. Given J^ (o:-^/) == 3 (V^^ + Vy) )
{
a;y = 36
Ans. -
±\/-
-23-
-11
Tv/-
2
-23-
-11
, or 9, or 4.
, or 4, or 9.
16. Given
IT. Given
I x + y = Al )
Jo:*— /= n
(x — 2^ =19)
QUADRATIC EQUATIONS. 205
18. Given -{^ ^ ^ :f l .
i X + y = 65 >
19. Given i^-'-^^^i^^
20. Given i-* + 2x^y + 2xy^ + / = 95|
(x«— x'f/— xy^ + y»= 5>
21. Given ■[ ^y= H.
U* + y* = 272)
22. Given j^ +.y = H-
U* + y* = 626)
PROBLEMS
PRODUCING QUADRATIC EQUATIONS CONTAINING TWO
UNKNOWN QUANTITIES.
191. Though the numerical negative values obtained
in solving the following Problems satisfy the equations
formed in accordance with the given conditions, they are
practically inadmissible, and, except in Example 4, are
not given in the answers.
1. The sum of the squares of two numbers plus the
sum of the two numbers is 98 ; and the product of the
two numbers is 42. What are the numbers ?
Ans, 7 and 6.
2. If a certain number is divided by the product of its
figures the quotient will be 3 ; and if 18 is added to the
number, the order of the figures will be inverted. What
is the number "^ Ans. 24.
3. A certain number consists of two figures whose
X roduct is 21 ; and if 22 is subtracted from the number.
206 ELEMENTARY ALGEBRA.
and the sum of the squares of its figures added to the
remainder, the order of the figures will be inverted. What
is the number? Ans. 37.
4. Find two numbers such that their sum, their prod-
uct, and the difierence of their squares shall be equal to
one another. Ans. f ± ^ V5 and ^ ± ^ V5.
6. There are two pieces of cloth of difterent lengths;
and the sum of the squares of the number of yards in
each is 145 ; and one half the product of their lengths
plus the square of the length of the shorter is 100. What
is the length of each ?
Ans. Shorter, 8 ; longer, 9 yards.
6. Find two numbers such that the greater shall be to
the less as the less is to 2f, and the difference of their
squares shall be 33.
7. The area of a rectangular field is 1575 square rods ;
and if the length and breadth were each lessened 5 rods,
its area would be 1200 square rods. What are the length
and breadth ?
8. Find two numbers such that their sum shall be to
6 as 9 is to the greater, and the sum of their squares
shall be 45. Ans. 9 Vl and 3 VT, or 6 and 3.
9. The fore wheels of a carriage make 2 revolutions
more than the hind wheels in going 90 yards ; but if the
circumference of each wheel is increased 3 feet, the car-
riage must pass over 132 yards in order that the fore
wheels may make 2 revolutions more than the hind wheels.
What is the circumference of each wheel ?
Ans. Fore wheels, 13^ feet ; hind wheels, 15 feet.
10. Find two numbers such that five times the square
of the greater plus three times their product shall be 104,
and three times the square of the less minus their prod-
uct shall be 4.
KATio AXD r::o?onTio-M. 207
SECTION XXI.
RATIO AND PROPORTION.
102. Ratio is the relation of one quantity to another of
the same kind ; or, it is the quotient which arises from
dividing one quantity by another of the same kind.
Ratio is indicated by writing the two quantities after
one another with two dots between, or by expressing the
division in the form of a fraction. Thus, the ratio of a to
b is written, a : h, or ^ ; read, a is to b, or a divided by b,
193. The Terms of a ratio are the quantities compared,
whether simple or compound.
The first term of a ratio is called the antecedent, and the
other the consequent; and the two terms together are called
a covplet.
194. An Inverse, or Reciprocal Ratio, of any two quan-
tities is the ratio of their reciprocals. Thus, the direct ratio
of a to ft is a : b, i. e. r; and the inverse ratio of a to ft is
1 1 . 1 1 ft ,
- : Tf 1. e. - -T- 7 = -» or ft : a.
a b aba
195. Proportion is an equality of ratios. Four quan-
tities are proportional when the ratio of the first to the
second is equal to the ratio of the third to the fourth.
The equality of two ratios is indicated by the sign
of equality (==) or by four dots (: :).
Thus, a : b = c : d, or a : b: : c : d, or -r=z-,; read, a to ft
h a
equals c to d, or a is to ft as c is to d, or a divided by ft
equals c divided by d.
208 ELEMENTARY ALGEBRA.
196i In a proportion the antecedents and consequents
of the two ratios are respectively the antecedents and con-
sequents of the proportion. The first and fourth terms are
called the extremes, and the second and third the means.
197. When three quantities are in proportion, e. g.
a : b = b : c, the second is called a mean proportional be-
tween the other two ; and the third, a third proportional
to the first and second.
198. A proportion is transformed by Alternation when
antecedent is compared with antecedent, and consequent
with consequent.
199. A proportion is transformed by Inversion when
the antecedents are made consequents, and the conse-
quents antecedents.
200. A proportion is transformed by Composition when
in each couplet the sum of the antecedent and consequent
is compared with the antecedent or with the consequent.
201. A proportion is transformed by Division when in
each couplet the diff'erence of the antecedent and conse-
quent is compared with the antecedent or with the con-
sequent.
THEOREM I.
202. In a proportion the product of the extremes is equal
to the product of the means.
Let a : b = c : d
i. e.
Clearing of fractions,
a c
h~d
RATIO AND PROPORTION. 209
THEOREM II.
203t If the product of Ivjo quantities is equal to the prod-
uct of two others, the factors of either product may he made
the extremes^ and the factors of the other the means of a
proportion.
Let ad=:bc
Dividing by hd, h^^d
i. e. a : b = c : d
THEOREM III.
204. If four quantities are in proportion, they will he in
proportion by alternation.
Let a : b = c : d
By Theorem I. ad=bc
By Theorem II. a -. c=^b : d
THEOREM IV.
205. If four quantities are in proportion, they mil he in
proportion by inversion.
Let a : b = c : d
By Theorem I. ad= be
By Theorem 11. b : a = d : c
THEOREM V.
206. If three quantities are in proportion, the product of
tJie extremes is equal to the square of tJie mean.
Let a : b = b : c
By Theorem I. ac = If^
THEOREM VI.
207. If four quantities are in proportion, they will he in
proportion hy composition.
a
b —
c
'd
t + ^ =
a '
b ~
c + rf
d
a + b: b =
c + d:
d
210 ELEMENTARY ALGEBRA.
Let a \h:=.c \ d
\. e.
Adding 1 to each member;
or
i. e.
THEOREM VII.
208. If four quantities are in proportion, they will he in
proportion by division.
Let a : b =z c : d
i. e.
Subtracting I from each member
or
i. e. a
THEOREM VIII.
209. Two ratios respectively equal to a third are equal
to each other.
Let a : b=:m '. n and c : d=m : n
. ^ a m . c m
1. e. - = - and ,== -
on d n
Hence (Art. 13, Ax. 8), |= f
i. e. a '. b = c : d
THEOREM IX.
210. If four quantities are in proportion, the sum and
difference of the terms of each couplet will he in proportion.
a c
b~~d
l-^-l-^
a — b c — d
b ~ d
-b: b = c — d:
d
RATIO AND PROPORTION. 211
Let a : h:=c : d
By Theorem VI. a -\- b : b = c -{- d : d {I)
and by Theorem VII. a^—b . b = c — d : d (2)
From (1 ), by Theorem III. a-{'b:c-\-d=b:d
From (2), by Theorem III. a — b:c — dz=b:d
By Theorem VIII. a + b:c + d=a — b : c — d
Hence, by Theorem III. a -\- b : a — b = c -\- d : c — d
TUEOREM X.
211* Equimultiples of two quantities have tJw same ratio
as tfie quantities themselves.
For by Art. 83, ? = ^
•^ mo
i. e. a : b = ma : mb
Cor. It follows that either couplet of a proportion may
be multiplied or divided by any quantity, aiid the result-
ing quantities will be in proportion. And since by Theo-
rem III. if a : b = ma : mb, a : ma == b : mb, or ma : a
= mb : b, it follows that both consequents, or both ante-
cedents, may be multiplied or divided by any quantity,
and the resulting quantities will be in proportion.
THEOREM XI
212, If four quantities are in proportion, like powers or
like roots of these quantities will he in proportion.
Let a : b = c : d
a c
b=d
Hence, ^=^
i. e. t^ : l^ = c" : d^
Since n may be either integral or fractional, the theorem
is proved.
212 ELEMENTARY ALGEBRA.
THEOREM XII.
213. If any number of quantities are proportional, any
antecedent is to its consequent as the sum of all the antece-
dents is to the sum of all the consequents.
Let a : b = c : d=ze :f
Now ab=zab (1)
and by Theorem I. ad=zbc (2)
and also af^=:be (3)
Adding(l), (2), (3), a(H-^ + /) = b {a -\- c -^ e)
Hence, by Theorem II. a : b =z a-\- c -\-e •.b-\- d -\-f
THEOREM XIII.
214. If there are two sets of quantities in proportion, their
products, or quotients, term by term, wilt be in proportion.
Let
a : b = c : d
and
e:f=g :k
By Theorem I.
ad = bc
(1)
and
eh=fy
(2)
Multiplying (1) by (2),
adeh = b cfg ;
(3)
Dividing (1) by (2),
ad be
(4)
From (3), by Theorem IL
ae : bf=zcg : dh
and from (4),
a b c d
PROBLEMS IN PROPORTION.
215. By means of the principles just demonstrated, a
proportion may often be very much simplified before
making the product of the means equal to the product
of the extremes ; and a proportion which could not oth-
erwise be reduced by the ordinary rules of Algebra may
often be so simplified as to produce a simple equation.
RATIO AND PROPORTION. 213
1. The cube of the smaller of two numbers multiplied
by four times the greater is 96 ; and the sum of their
cubes is to the difference of their cubes as 210 : 114.
What are the numbers ?
SOLUTION.
Let X = the greater and y = the less.
Then 4x/ = 96 (1) x'+y :ar» — / = 210 : 114 (2)
From (2), by Theo. X^ Cor. ar' + y' : a:* — y* = 35 : 19
By Theorem IX. 2a:» : 2y» = 64 : 16
By Theorem X., Cor. a:« : / = 27 : 8
By Theorem XI. a: : y = 3 : 2
By Theorem I. 2x = 3y (3)
From (1) and (3) we find x = 3 and y = 2.
2. The product of two numbers is 78 ; and the differ-
ence of their cubes is to the cube of their difference as
283 : 49. What are the numbers ?
SOLUTIOX
I^t X = the greater and y = the less.
Thenxy=78 (1) I* — 3/»: a:» — 3x«y + 3a:y» — y» = 283 : 49 (2)
From (2), by division, Sx^y — 3 x^ : (x — y)' = 234 : 49
Dividing 1st couplet by a: — y, 3xy :(x — y)' = 234 : 49
Dividing antecedents by 3, xy : (x — y)' = 78 : 49
Substituting the value of zy, 78 : (x — y)* = 78 : 49
Dividing antecedents by 78, 1 : (x — y)' = 1 : 49
Extracting the square root, 1 : x — y = 1 : 7
Whence, x — y = 7 (3)
From (1) and (3) we find x = 13 and y = 6.
3. The sum of the cubes of two numbers is to the cube
of their sum as 13 : 25 ; and 4 is a mean proportional be-
tween them. What are the numbers?
214 ELEiMENTARY ALGEBRA.
4. The difference of two numbers is 10 ; and their prod-
uct is to the sum of their squares as 6 : 3*7. AVhat are
the numbers ?
SOLUTION.
Let X = the greater and y = the less.
Then a;— 2/= 10 (1) xy\ t? -\- f = ^ -. ^1 {^^
From (2), by Theorem X., Cor. 2xy\ o? -{- f = \1 -.^1
By Theorem IX. o? -\- Ixy -{- y^ \ x^ — 'Ixy -\- f = 4.^ -. lb
By Theorem XL x ■\- y \ x — y = 7:5
By Theorem IX. ' 2a::2^/=12:2
By Theorem X., Cor. x\y =6:1
By Theorem I. a; = 6 y (3)
From (1) and (3) we find a; = 12 and y = 2.
5. The product of two numbers is 136 ; and the dif-
ference of their squares is to the square of their differ-
ence as 25 : 9. What are the numbers ? Ans. 8 and IT.
6. As two boys were talking of their ages, they dis-
covered that the product of the numbers representing
their ages in years was 320, and the sum of the cubes
of these same numbers was to the cube of their sum as
T : 27. What was the age of each?
Ans. Younger, 16; elder, 20 years.
Y. As two companies of soldiers were returning from
the war, it was found that the number in the first multi-
plied by that in the second was 486, and the sum of the
squares of their numbers was to the square of the sum as
13 : 25. How many soldiers were there in each company?
Ans. In 1st. 27; in 2d, 18.
8. The difference of two numbers is to the less as 100
is to the greater ; and the same difference is to the greater
as 4 is to the less. What are the numbers ?
J^OTE. — Multiply the two proportions together. (Theorem XIII.)
PROGRESSION. 215
SECTION XXII.
PROGRESSION.
216. A Progression is a series in which the terms in-
crease or decrease according to some fixed law.
217. The Terms of a series are the several quantities,
whether simple or compound, that form the series. Tlie
first and last terms are called the extremes, and the others
the means.
ARITHMETICAL PROGRESSION.
218. An Arithmetical Progression is a series in which
each term, except the first, is derived from the preced-
ing by the addition of a constant quantity called the com'
mon difference.
219. When the common difference is positive, the series
is called an ascending series, or an ascending progression ;
when the common difference is negative, a descending se-
ries. Thus,
a, a -\- d, a -\- 2d, a -\- Zd, &c.
is an ascending arithmetical series in which the common
difference is d] and
a, a — d, a — 2d, a — Zd, &c.
is a descending arithmetical series in which the common
difference is — d.
220. In Arithmetical Progression there are five elements,
any three of which being given, the other two can be
found : —
1. The first term.
2. The last terra.
216 ELEMENTARY ALGEBRA.
3. The commou difference.
4. The number of terms.
5. The sum of all the terms.
221. Twenty cases may arise in Arithmetical Progres-
sion. In discussing this subject we shall let
a = the first terra,
I = the last term,
d = the common difference,-
n = the number of terms,
S=. the sum of all the terms.
CASE I.
222. The first term, common difference, and number of
terms given, to find the last term.
In this Case a, d, and n are given, and / is required. The suc-
cessive terms of the series are
a, a -\- d, a -\- 2d, a -{- Sd, o -j- 4 rf, &c. ;
that is, the coefficient of d in each term is one less than the number
of that term, counting from the left ; therefore the last or nth term in
the series is
a-\- (n—1) d
•or Z = a -J- (n — 1) d
in which the series is ascending or descending according as d is posi-
tive or negative. Hence,
RULE.
To the first term add tJie product formed by multiplying the
common difference by the number of terms less one.
1. Given « = 4, d =z 2, and w = 9, to find /.
l=:a-\-(n — I) d = 4: + (9 — 1) 2 = 20, Ans.
2. Given a = t, d=3, and n = 19, to find /.
Ans. /=61.
PROGRESSION. 217
3. Given a = 29, rf = — 2, and n = 14, to find I
Ans. / = 3.
4. Given a = 40, d=z 10, and n = 100, to find /.
5. Given a = 1, d=z ^, and n =. 17, to find /.
6. Given a = |^, rf = — ^^, and n = 13, to find /.
7. Given a = .01, d = ^ .001, and n = 10, to find /.
CASE II.
223. The extremes and the number of terms given, to
find the sum of the series.
In this Case a, /, and n are given, and S is required.
Now 5 = o + (o + d) + (a + 2d) + (a + 3d) + + Z
or, inrerting the seriee, 5= i + ( i — d) + {l — 2d) -\- (l — Sd) + -fa
Adding these together, 2 5= (a + + (a +0 + (o + + (a + + + (a +
And since (a -j- /) is to be taken as many times as there are terms,
hence 2 5 = n (a -f-
or S = - (a -\- I). Hence,
RULE.
Find one half (he product of the sum of the extremes and
the number of term^.
Note. — If in place of the last term the common difference is
given, the last term must first be found by the Rule in Case I.
1. Given a = 3, /= 141, and n = 26, to find S.
^= ^ (a + = IT (3 + 141) = 1872, Ans.
2. Given a = |, / = 25, and n = 63, to find S.
Ans. ^=793f.
3. Given a = 4, <f = 2, and n = 24, to find S.
Ans. aS'=648.
4. Given a = — 3, c? = 2, and n = 4, to find S.
Ans. 5=0.
10
218 ELEMENTARY ALGEBEA.
5. Given a = ^, c? = — ^, and n-=z 3, to find S.
6. Given a =z .07, I = .17, and n = 11, to find aS'.
7. Given «= — 4J-, c?= |, and w = 25, to find aS.
CASE III.
224. The extremes and number of terms given, to find
the common difference.
In this case a, I, and n are given, and d is required.
From Case I. we have Z = a -f- (« — 1) <i
Transposing and reducing, d = . Hence,
RULE.
Divide the last term minus the first term by the number
of terms less one, and the quotient will be the common dif-
ference.
1. Given a = 5, I=z4z1, and n == 7, to find d.
I —a 47 — 5 ^ ,
2. Given a = 27, 1= 148, and n = 12, to find d.
Ans. d=z 11.
3. Given a = 41, ^=3, and n = 20, to find d.
Ans. d = — 2,
4. Given a = -^l, 1 = ^^, and » = 6, to find d.
Ans. d = — 2%.
5. Given a = .09, /= .9, and w = 10, to find d.
Note. — This rule enables us to insert any number of arithmet-
ical means between two given quantities ; for the number of terms
is two greater than the number of means. Hence, if m = the num-
ber of means, m4-2 = n, or m 4- 1 == n — 1, and d = — p-r*
m-j- 1
Having found the common difference, the means are found by add-
ing the common difference once, twice, &c., to the first term.
PROGRESSION. 219
6. Find 6 arithmetical means between 3 and 38.
Ans. 8, 13, 18, 23, 28, 33.
T. Find 3 arithmetical means between 3 and 27.
8. Find 5 arithmetical means between 1 and 3Y.
9. Find 7 arithmetical means between 2 and 26.
Note. — When m= 1, the formula becomes
-'-^
Adding a to each member,
But a -\- d is the second term of a series whose first term is a and
common difference </, or the arithmetical mean of the series a, a -|- rf,
a -\- 2d. Hence, the arithmetical mean beticeen two quantities is one
half of their sum.
10. Find the arithmetical mean between 7 and 17.
Ans. 12.
11. Find the arithmetical mean between ^ and J.
12. Find the arithmetical mean between 4 and — 4.
225. From the formulas established in Arts. 222 and
223, viz.
lz=a + {n — l)d (1)
S=l(a + l) (2)
can be derived formulas for all the Cases in Arithmetical
Progression.
From (1) we can obtain the value of any one of the four quanti-
ties, /, a, n, or d, when the other three are given ; and from (2)
the value of any one of the four quantities, «S, n, a, or /, when the
other three are given. Formulas for the remaining twelve Cases
which may arise are derived by combining the two formulas (1)
and (2), so as to eliminate that one of the two imknown quantities
whoM value is not sought
220 ELEMENTARY ALGEBRA.
1. Find the formula for the value of n, when a, rf, and
S are given.
OPERATION.
Z =« + (w — l)rf (1) 8z=^
n .
+
I) (2)
ln = an + dn^ — dn (3) 2S—anz=
-.In
(4)
an-{-dn'^ — dn = 2S^q,n
(5)
^2 (d-2a\ 2S
(6)
"" \ d )''— d
(d-2ay_(d-2ay 2S
a)
?2
2d
(8)
To obtain the formula required in this example, I must be elim-
inated from (1) and (2). From (1) and (2) we obtain (3) and (4).
Placing these two values of Zn equal to each other, we form (5),
which reduced gives (8), or the value of n in known quantities.
2. Find the formula for the value of n, when d, I, and
S are given. An« « — 21+ d ±vj2T-^ dy -8dS
Ans. n — 2^
3. Find the formula for the value of *S', when a, d, and
n are given. Ans. S=in [2 a -)- (w — I) d].
4. Find the formula for the values of S, when a, d, and
/ are given. ^^^ ^ __ (^ + «) (l — a-^d)
z d
5. Find the formula for the value of S, when d, n, and
I are given. Ans. S=^n[2l — (n — 1) d].
6. Find the formula for the value of I, when a, d, and
S are given. * z ^ i fl ^Y
Ans. «=-i±y/(«-|) +2<^'S-
T. Find the formula for the value of I, when d, n, and
S are given. * ^_ 2 S -^n (n — i) d ^
2n
PROGRESSION. 221
8. Find the formula for the value of d, when a, n, and
S are given. . , 2 6' — 2 an
~ n (n — 1) *
9. Find the formula for the value of d, when. a, I, and
S are given. ^ ^ ^ ibpll^l. '
2 5 — (/ -f- a)
10. Find the formula for the value of d, when n, I, and
5 are given. Ans. rf = ?^?^^.
n (n — 1)
11. Find the formula for the value of a, when rf, n, and
5 are given. . 2 5 — n (n — 1) rf
2n
12. Find the formula for the value of a, when d, I, and
S are given. Ans. a = f ± J g + /)' _ 2 rf 5.
226. To find any one of the five elements when three
others are given.
RULE.
Substitute the given values in that formula whose first mem-
ber is the required term, and whose second contains the three
given terms.
1. Given d = 2, 1=21, and *S'= 120, to find a.
OPERATION.
^=l±\j{l+^^y-^'^'i^o (2)
a = 3, or — 1 (3)
In Example 12, Art 225, we find (I), the required formula; substi-
tuting the given values of d, /, and 5, we obtain (2), which reduced
gives (3), or a = 3, or — 1.
Note. — If a = 3, n= 10; butifa = — 1, n = 12.
222 ELEMENTARY ALGEBRA.
2. Given d=^, 1=21, and *S'=:392, to find n.
Ans. n = 147, or 16.
Note. — When n = 147, a = — 21| ; but when n = 16, a = 22.
3. Given d=:1, n = Q, and S=: 135, to find I
Ans. 1:= 40.
4. Given c? = — 2, ti = 6, and a = 5, to find aS'.
Ans. ,^=0.
5. Given « = ^, / = 15, and S =z 2881, to find d.
Ans. c? = f .
6. Find the 100th term of the series 3, 10, 17, &c.
Ans. 696.
T. Find the sum of 100 terms of the series 3, 10, 17, &c.
Ans. 34950.
8. Find the common difference and sum of the series
whose first term is 25, last term 95, and number of
terms 15. Ans. c?=5; 1^=900.
9. Find the sum of the natural series of numbers from
1 to 100, inclusive.
10. Find the sum of 10 of the odd numbers 1, 3, 5, &c.
11. Find the sum of 10 of the even numbers 2, 4, 6, &c.
12. How many strokes does a clock strike in 12 hours?
13. If 100 trees stand in a straight line 10 feet from
one another, how far must a person, starting from the
first tree and returning to it each time, travel to go to
every tree ? Ans. 18J miles.
14. If a person should save a cent the first day, two
cents the second, three the third, and so on, how much
would he save in 365 days? Ans. $667.95.
15. If a person should save $25 a year and put this
sum at simple interest at 5 per cent at the end of each
year, to how much would it amount at the end of 25
years ?
PROGRESSION. 223
PROBLEMS
TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY.
227. Sometiraes in examples in progression the terms
are not directly given, but are implied in the conditions of
the problem. In this case the formulas cannot be directly
used, but the terms can be represented by unknown quan-
tities, and equations formed according to the given con-
ditions.
228. If X = first term and y = the common difference ;
then
X, X + y, X -f 2 y, x -f 3 y, &c.
will represent the series.
It will often be found more convenient when the num-
ber of terms is odd to represent the middle term by x
and the common difference by y ; then the series for three
terms will be
X — y, X, X + y ;
and for five terms,
X — 2 y, X — y, x, x + y, x -f- 2 y ;
and when .the number of terms is even, to represent the
two middle terms by x — y and x -[- y» and the common
difference by 2y; then the series for four terms is
X — 3y, X — y, x + y, x + 3y.
The advantage of this latter method is, that the sum of
the series, or the sum or difference of an}' two terms
equally distant from the mean, or means, will contain but
one unknown quantity.
1. The sum of three numbers in arithmetical progres-
sion is 15, and the sum of their squares is 83. What
are the numbers ?
224 ELEMENTARY ALGEBRA.
Let X =: the mean term and y = the common diiference ;
then the series will be x — y, x, and x -\- y.
By the conditions, 3 a; =15 (1)
and Bx'' + 2f = SS (2)
Ans. 3, 5, 1.
2. The sum of four numbers in arithmetical progres-
sion is 44, and the sum of the cubes of the two means
is 2926. Ans. 5, 9, 13, 17.
3. Find seven numbers in arithmetical progression such
that the sum of the first and fifth shall be 10, and the
difference of the squares of the second and fourth 40.
4. There are four numbers in arithmetical progression ;
the product pf the first and third is 20 ; and the product
of the second and fourth 84. What are the numbers ?
Ans. 2, 6, 10, 14.
6. The sum of four numbers in arithmetical progression
is 32 ; and their product 3465. What are the numbers ?
Ans. 5, Y, 9, 11.
6. The sum of the squares of the extremes of four num-
bers in arithmetical progression is 461 ; and the sum of
the squares of the means 425. What are the numbers ?
Ans. 10, 13, 16, 19.
Y. A certain number consists of three figures which
are in arithmetical progression ; if the number is divided
by the sum of its figures, the quotient will be 15 ; and
if 396 is added to the number, the order of the figures
will be inverted. What is the number i* Ans. 135.
8. Find four numbers in arithmetical progression such
that the sum of the squares of the first and. third shall be
104, and of the second and fourth 232.
9. Find four numbers in arithmetical progression such
that the sum of the squares of the first and second shall
be 29, and of the third and fourth 185.
PROGRESSION. 225
SECTION XXIII.
GEOMETRICAL PROGRESSION.
229. A Geometrical Progression is a series in which
each term, except the first, is derived by multiplying the
preceding term by a constant quantity called the ratio.
230* If the first term is positive, when the ratio is a
positive integral quantity, the series is called an ascending
series, and when the ratio is a positive proper fraction^ a
descending series ; but if the first term is negative, the sc-
ries is ascending when the ratio is a positive proper frac-
tion, and descending when the ratio is a positive integral
quantity. Thus,
2, 6, 18, 54, &c. ) J.
' ' ' ' > are ascendine: series :
-^64,-18,— 6,— 2, &c.) ^
64, 32, 16, 8, &c.> , ,.
' ' ' ' V are descending series.
— 8, — 16, — 32, — 64, &c. > ^
If the ratio is negative, the terms of the progression are
alternately positive and negative. Thus, if the ratio is
— 2 and the first term 3, the series will be
3, — 6, + 12, — 24, + 48, &c. ;
but if the first term is — 3,
— 3, +6, —12, +24, —48', &c.
The positive terms of these two series constitute an as-
cending progression whose ratio is the square of the given
ratio ; and the negative terms a descending progression
having the same ratio.
231* In Geometrical Progression there are five elements,
any three of which being given, the other two can be found.
These elements are the same as in Arithmetical Progres-
sion, except that in place of the common difference we have
the ratio.
10* o
226 ELEMENTARY ALGEBRA.
232t Twenty cases may arise in Geometrical Progres-
sion. In discussing these cases we shall preserve the
same notation as in Arithmetical Progression, except
that instead of d = the common difference we shall use
r = the ratio.
CASE I.
233. The first term, ratio, and number of terms given,
to find the last term.
In this Case a, r, and n are given, and I required.
The successive terms of the series are
a, ar, ar^^ ar^, a?-*, &c.
That is, each term is the product of the first term and that power
of the ratio which is one less than the number of that term count-
ing from the left; therefore the last or nth term in the series is
or l=sar^-^. Hence,
RULE.
Multiply the first term by that power of the ratio whose
index is one less than the number of terms.
1. Given a = Y, r = 3, and w = 5, to find /.
/=ar"-^ = 1 X 3* =567, Ans.
2. Given a = 3, r = 2, and w = 9, to find I.
Ans. /=768.
3. Given a = 64, r == J, and n = 10, to find /.
Ans. I = ^.
4. Given a = — T, r = — 4, and w = 3, to find /.
Ans. Z = — 112.
5. Given a = — ^, r = ^, and w = 5, to find /.
Ans. 1==. — y¥s-
6. Given a = 5, r = — \, and n = 10, to find I.
7. Given a == — ^, r=^, and w = 8, to find /.
8. Given a = — 10, r = — 2, and n = 6, to find /.
PROGRESSION. 227
CASE II.
234. The extremes, and the ratio given, to find the sum
of the series.
In this Case ct, /, and r are given, and S is required.
Now S == a -{- ar -{- ar^ -\- at^ -{- -f- / (1)
Multiplying (1) by r, rS=ar-\- aH -f ar* -f -|-^+^'* (2)
Subtracting (1) from (2), rS — S= Ir — a
Whence, <?»= . Hence,
RULE.
Multiply the last term hy the ratio, from tJie product sub-
tract the first term, and divide iJie remainder by the ratio
less one.
1. Given a = 2, /== 20000, and r = 10, to find S.
S='^ = 'O^lXJl - ' = 22222, Ans.
2. Given a = 7, / = 45927, and r = 3, to find S.
Ans. 5=68887.
3. Given a = — 5, l = — 405, and r r= 3, to find S.
4. Given a = — ttVjj ^ = i* ^^^ ^ = — *?» to ^^^ *S'.
Ans. S=^%o^.
CASE III.
235. The first term, ratio, and number of terms given,
to find the sum of the series.
In this Case a, r, and n are given, and S required.
The last term can be found by Case I., and then the sum of the
series by Case II. Or better, since
lr = ar*
Substituting this value of Z r in the formula in Case II. we have
r" 1
S a« — — — X a- Hence,
228 ELEMENTARY ALGEBRA.
RULE.
From the ratio raised to a power whose index is equal to
the number of terms subtract one, divide the remainder by
the ratio less one, and multiply the quotient by the first term.
1. Given a =: 4, r =z^, and ?i == 5, to find aS'.
^= ^^-=4 X « = ^^ X 4 = 11204, Ans.
r — 1 7 — 1
2. Given a=.^, r = 6, and n = 6, to find S.
V Ans. aS'= 558.
3. Given a =: ^, r = i, and n = Y, to find *S'.
Ans. S=iU.
4. Given a=. — 5, r = — 4, and n =: 4, to find S.
Ans. ^5^=255.
6. Given a = — |, r = Q, and w = 5, to find *S'.
6. Given a = §, r = — 3, and w = 6, to find S.
Y. Given a = — }, r = 2, and n = 8, to find aS'.
». In a geometrical series whose ratio is a proper frac-
tion the greater the number of terms, the less, numeri-
cally, the last term. If the number of terms is infinite, the
last terra must be infinitesimal ; and in finding the sum
of such a series the last term may be considered as noth-
ing. Therefore, when the number of terms is infinite, the
formula
S =z becomes
Sz=
r — 1 1 — r
Hence, tg find the sum of a geometrical series whose ra-
tio is a proper fraction and number of terms infinite,
RULE.
Divide the first term by one minus the ratio.
PROGRESSION. 229
1. Find the sum of the series 1, J, i, &c. to infinity.
•5=f^ = r^ = 2, Ans.
2. Find the sum of the series f, |, y^* ^c. to infinity.
Ans. ^jj.
3. Find the sum of the series -, ^, -j, &c. to infinity.
Ans. r-
c — 1
4. Find the sum of the series 6, 4, 2 J,' &c. to infinity.
Ans. 18.
6. Find the value of the decimal .4444, &c. to infinity.
Note. — This decimal can be written ^ -J- ^^ -f~ njW» ^*
Ans. |.
6. Find the value of .324324, &c. to infinity.
7. Find the value of .32143214, &c. to infinity.
CASE IV.
237. The extremes- and number of terms given, to find
the ratio.
In this Case a, /, and n are ^ven, and r is required.
From Case I. l=:at*-^
Whence, r= 4/-. Hence,
RULE.
Divide the last term hy the firstj and extract that root of
the quotient whose index is one less than the number of
terms.
1. Given a = 7, I = 567, and n = 5, to find r.
,-1/7
■^i = ^iiI==3.A„s.
2. Given a = 6f , / = i, and n = 6, to find r.
Ans. r =z }.
230 ELEMENTARY ALGEBRA.
3. Given a = — I, 1= 31^, and n = 4:, to find r,
Ans. r =. — 5.
Note. — This rule enables us to insert any number of geometri-
cal means between two numbers ; for the number of terms is two
greater than the number of means. Hence, if in == the number of
means, m-(-2 = n, orm-}-l == n — 1; and r = t /-• Having
found the ratio, the means are found by multiplying the first term
by the ratio, by its square, its cube, &c.
4. Find three geometrical means between 2 and 512.
Ans. 8, 32, 128.
5. Find four geometrical means between 3 and 3072.
Ans. 12, 48, 192, 768.
6. Find three geometrical means between 1 and y'g^.
Ans. ^, i, i.
Note. — When m = 1 , the formula becomes
= \/^
Multiplying by a, ar = a^/-=i/aL
sjl-^'-
But ar is the second term of a series whose first term is a and
ratio r ; or the geometrical mean of the series a, ar, a 1^. ' Hence,
the geometrical mean between two quantities is the square root of their
product.
7. Find the geometrical mean between 8 and 18.
Ans. 12.
8. Find the geometrical mean between ^ and 343.
Ans. 7.
9. Find the geometrical mean between \ and i-^2b'
10. Find the geometrical mean between — J and — s-tVt-
PROGRESSION. 231
238. From the formulas established in Arts. 233 and 234,
l=ar^-^ (1)
5='rl" • (2)
can be derived formulas for all the Cases in Geometrical
Progression.
From (1) we can obtain the value of any one of the four terms, /,
a, n, or r, when the other three are given; from (2), the value of 5,
/, r, or a, when the other three are given. Formu\^s for the remaining
twelve Cases which may arise are derived by combining the formulas
(1) and (2) so as to eliminate that one of the two unknown terms
whose value is not sought.
1. Find the formula for the value of *S', when /, n, and
r are given.
From (1), ,3^11 = «
Substituting this value of a in (2), Sz
or • ' S =
r— 1
/(r--l)
(r_i)r«-i
Note. — The four formulas for the value of n cannot be derived or
used without a knowledge of logarithms ; and four others, when n ex-
ceeds 2, cannot be reduced without a knowledge of equations that can-
not be reduced by any rules given in this book.
239. To find any one of the five elements when three
others are given.
RULE.
Subsliiufe in that one of the formulas (1) or (2) that con-
tains the four elements, viz. the three given and the one re-
quired, the given values, and reduce the resulting equation.
If neither formula contains the four elements, derive a for-
mula that will contain them, then substitute and reduce the
resulting equation ; or substitute the given values before deriv-
ing the formula, then eliminate tJie superfluous element and
reduce the resulting equation.
232 ELEMENTARY ALGEBRA.
1. Given r = 3, w =
5,
and
[ S=126, to find /.
/=a/^-i (1)
^ Ir — a
(2)
l=Sla (3)
V26 = ^^-«
<4)
8l = « (')
a = 3/— U52
(6)
3 Z— 1452
I
81
(7)
242?
1452 X 81 (8)
? = 486 (9)
Substituting the given values of r, n, and 5 in (1) and (2), we
obtain (3) and (4) ; finding the value of a, the superfluous element,
from (3) and (4), and putting these values equal to each other, we
form (7), an equation containing but one unknown quantity. Re-
ducing (7) we obtain (9), or I = 486.
2. Given a == 4, r = b, and ASf= 15624, to find I.
Ans. 1= 12500.
3. Given a = 2, n = 5, and /= 512, to find S.
Ans. 5=682.
4. Find the formula for the value of a, when r, n, and
S are given. . (r — l) 'S'
Ans. a — ^ _— •
5. A gentleman purchased a house, agreeing to pay one
dollar if there was but one window, two dollars if there
were two windows, four if there were three, and so on,
doubling the price for every window. There were 14
windows. How much must he pay? Ans. $8192.
6. A man found that a grain of wheat that he had sown
had produced 10 grains. Now if he sows the 10 grains
the next year, and continues each year to sow all that is
produced, and it increases each year in tenfold ratio, how
many grains will there be in the seventh harvest, and how
many in all ? j^ ( In Yth harvest, 10000000 grains.
(In all, lUlllU grains.
PROGRESSION. 233
PROBLEMS
TO WHICH THE FORMULAS DO NOT DIRECTLY APPLY.
240. In solving Problems in Geometrical Progression,
if we let X ■=. the first term and y ^ the ratio, the series
will be
a:,-ary, xf, xf, &c.
It will often be found more convenient to represent the
series in one of the following methods : —
1st When the number of terms is odd,
a^» ^^!h y
for three terms;
3» 7/*
-, I*, xy, y*, ~ for five terms.
2d. When the number of terms is even,
a" «»
-, X, y, — for four terms;
y a;
X* X" y" t^ r. '
-0 1 -» a:, y, -, -^ for six terms.
f y -^ x' x*
Which method is most convenient in any case will de-
pend upon the conditions that are given in the problem.
1. There are three numbers in geometrical progression,
the greatest of which exceeds the least by 32 ; and the
difierence of the squares of the greatest and least is to
the sum of the squares of the three as 80 : 91. What are
the numbers ?
SOLUTION.
Let X, xy, and xy* represent the scries. Then
xy«~x = 32(l) x«y*— x»:x» + x«y«-f-x»y« = 80:91 (2)
y«— l:l-|-y»4-y*==80:91 (3)
9ly* — 91 = 80-f-80y»-|-80y* (4)
lly« — 80y»=171 (5)
x=4 (7) y=3 (6)
234 ELEMENTARY ALGEBRA.
Dividing the first couplet of (2) by x^ we obtain (3) ; from (3)
vre form (4), which reduced gives (G), or y = 3. Substituting the
value of 1/ in (i), we obtain (7)^ or a; = 4.
Ans. 4, 12, 36.
2. The sum of three numbers in geometrical progres-
sion is 39, and the sum of their squares 819. What are
the numbers ?
SOLUTION.
Let X, s/ xy, and y represent the series. Then
a: + /v/^y + yr=39 (1) a;^ + xy + ^^ ^ 819 (2)
ar~>v/^7 + y=r21 (3)
2x^2y — m (4)
x- \- y = 3 (5)
2\/^=18 (6)
xy = ^\ (1)
Dividing (2) by (1), we obtain (3); adding (3) to (1), we ob-
tain (4), which reduced gives (5) ; subtracting (3) from (1), we
obtain (6), which reduced gives (7). Combining (5) and (7) as
the sum and product are combined in Example 1, Art. 188, we
obtain x = 27 and y = 3.
Ans. 3, 9, 21.
3. Of four numbers in geometrical progression the dif-
ference between the fourth and second is 60 ; and the sum
of the extremes is to the sum of the means as 13 : 4.
What are the numbers ?
SOLUTION.
Let X, xy, xy^, and xy^ represent the series. Then
xf
64a:
xy=:60 (1)
x/ + x:xy' + xy=l3:i
(2)
if-!/ + l :.y=:13:4
(3)
4:f — iy + 4,=z 13y
(i)
4a: = 60 (7)
if—11y^ — i
(5)
x=l (8)
y = 4
(6)
PROGRESSION. 235
Dividing the first couplet of (2) by xy -\- x,yre obtain (3) ; from
(3) we form (4), which reduced gives (G), or y = 4. Substituting
this value of y in (1) and reducing, we obtain (8), or ar== 1.
Ans. 1, 4, 16, 64.
4. Of four numbers in geometrical progression the sum
of the first two is 10 and of the last two 160. What are
the numbers ? Ans. 2, 8, 32, 128.
5. A man paid a debt of S310 at three payments. The
several amounts paid formed a geometrical series, and the
last payment exceeded the first by $240. Wliat were the
several payments? Ans. $10, $50, $250.
6. In the series x, \/xi/, and y what is the ratio?
A„s./f.
7. In the series -, x, y, and - what is the ratio?
y' ' ^' X
8. There are four numbers in geometrical progression
whose continued product is 64 ; and the sum of the series
is to the sum of the means as 5 : 2. What are the num-
bers ? Ans. 1, 2, 4, 8.
9. There are five numbers in geometrical progression ;
the sum of the first four is 156, and the sum of the last
four 780. What are the numbers ?
10. There are three numbers in geometrical progression
whose sum is 126; and the sum of the extremes is to the
mean as 17 : 4. What are the numbers?
11. The sum of the squares of three numbers in geo-
metrical progression is 2275 ; and the sum of the ex-
tremes is 35 more than the mean. What are the numbers ?
12. Of four numbers in geometrical progression the sum
of the first and third is 52 ; and the difference of the means
is to the difference of the extremes as 5 : 31. What are the
numbers ?
236 ELEMENTARY ALGEBRA.
SECTION XXIV.
MISCELLANEOUS EXAMPLES.
1. From 6ac— 5ab-{-c^ take Sac— [Sab — (c — c^)
+ 1c}. Ans. Sac — 2ab + 2c^-^6c.
2. Reduce x^ y^ — ( — xi/'^-{-x^ ) xi/ — x"^ f — {3/^
— y {^y — ^^) } ) t^ ^^s simplest form.
Ans. 2x^y^ -\- x^.
3. Reduce {a — h -\- cf — fa (c — a — h) -^ [b {a + h
-f- c) — c (a — b — ^) I ) to its simplest form.
Ans. 2(a'-\-b'-{-c^).
4. Reduce (x -\- a) a -{- 1/ — { & + ^) {'^ + ^) — y
(x -\- a — 1) — {^ -\- y) {b — ct)} to its simplest form.
Ans. a^ — b"^.
5. Reduce (a^ — b^)c — {a — b) [a {b^ c) — b {a — c)\
to its simplest form. Ans. 0.
6. Reduce {a -\- b) x — {b — c) c — ^{b — x)b—{b — c)
{b -\- c)\ — ax to -its simplest form. Ans. 2b x — be.
1. Multiply a^ + 2aH — SaP by — {—SaH-{- a^ b^).
8. Multiply a"^ -|- 6 a^ _f- 9 by a^ — 6 a^ + 9.
9. Multiply a -\- b — c by n — b -\- c.
10. Divide 2% a- — 6 a^ — Ga^ — 4a^ — 96 a + 264 by
3a2 — 4a + 11.
11. Divide 1 — 18x2+810:^ by 1 -\-Qx + ^x\
12 Divide ^ a^ -\- I —Aa^ — Q a hy 1 -|- 2 «2 — 3 a.
13. Divide 9 ^^ — 7 x^/ -\-2y^ by 3 x^ + 2 ^^^ — /.
MISCELLANEOUS EXAMPLES. 287
14. Divide 23 a — 30 — T a» + 6 a< by 3 a — 2 a^ — 5.
15. Find the prime factors of a* — b*.
16. Find the prime factors of 4m'n'^ — 49 m^n^".
17. Find the prime factors of x^ — 2xi/'\-y^,
18. Find the prime factors of x^ — y".
19. Find the greatest common divisor of 6 a:* — lOx^y
+ ISy* and 4x« + 8 x'^y + 8 xy^ _|_ 4y8. Ans. x + y.
20. Find the greatest common divisor of 8 a 6* + 24 a 6*
+ 16 a 6 and 7 i« + 7 i« + 7 6* — 7 6^ Ans. ^ + b.
21. Find the greatest common divisor of 6 a:^ -|- ^ ^y —
Bt/^ and 12x2 _^ 22 ary + 6y^
22. Find the greatest common divisor of 4x -|- 4ar'' — 40
and 3a:r*y — 48 y. Ans. x — 2.
23. Reduce 7 — i,x / « . » ^ . mv to its lowest terms.
(a — 6) (a* 4- 2 a 6 -f- ^)
24. Reduce , ,. to its lowest terms.
a' — IT
25. Reduce ., . .^ ^ ~ \, ; — ir to its lowest terms.
26. Find the least common denominator and reduce
z r— i ,~i « — -^ 15 to a single fraction.
1 — a 1 + a 1-f-o 1 — <^ n
Ans. —
1 +a«
27. Find the least common denominator and reduce
■r-^ — 4 — r— i — i to a single fraction.
28. Find the least common denominator and reduce
4a»+3a6 _ 48a'6 ^ • 1 r ^•
4a«-3a6-^- 16a^-9a«y *^ ^ «^°^^^ ^'^^*^^'^-
29. Reduce to one fraction with the least possible de-
. . a y — a*-|-a6 3ft — a , c
nominator j- j—f U — .
.bed cd ' bd
238 ELEMENTARY ALGEBRA.
30. Reduce to one fraction with the least possible de-
a + & h 4- c , a -\- c
nominator yr ^ r — 7 X7 h\ 1
(b — c){c-'a) {a — c){a — h) ' (6 — a)(c — 6)
Ans. j-j r-7 r-7 77 = 0.
(h — c) (c — a) (a — h)
4- 2 a: /2 — Zx (\Q — x)x\^ . , „ ^.
31. Find the least common denominator and reduce
3j-2a:
Y . .
Ans. z~.
2 -\- X
32. Reduce to one fraction with the least possible de-
. ^ l+rc Ax \—x . 2x-\-Q7?
nommator ^j^, - ^^^^ - j^^j^,- Ans. -^^--^, •
33. Reduce a — c ~ ^^ ^ to its simplest form.
34. Reduce ^~^ to its simplest form.
35. Reduce — —- \- x and — —, 2y each to a
X — 2?/ ' X -\-2y ^
single fraction and find their product. Ans. , „ »
36. Subtract ~^— from
2^ — a; a;-h?/
37. Subtract 3a:-[-^ from a; —
&«
88. Multiply (-i^,y by ^-3!
39. Divide — — by f , , and multiply the result by a'.
40. Divide ^--^^--^, by -^— ^.
41. Divide T by (— ri; + z r)'
m -f- n '' \a-\- a — 0/
iUSCELLANEOUS EXAMPLES. 239
Je^ rk- -J 4(a» — aft) , Sab
42. Divide -^, — f— ,;,- by -5 rj-
b{a -\- by "^ a} — b*
43. Divide --^ i — by — ■ j — , and give
a — X ' a 4- X '' a — x a -+- x °
0*4- z"
the answer in its lowest terms. Aus. —^ — •
2ax
44. Reduce r-r = 1 • What is the value of
a a -{- b a — o
X, if a = — 2 and 6 = 3?
3x — 1
45. Reduce x
1 +
46. 'Reduce(a-{-x){b — x) — a{b — c)-~^-^ = 0.
What is the value of a:, if a = 2, b = — 3, and c = — 1 .
47. Reduce -^^^ = — ~ What is the value of
b a
ar, if a = 2, 6 = — 1, and c = 3 ?
48. Reduce a — J-±^ = 0.
1 — x
ab
49. Find the value of x in the equation x = ^^r| ^^rj
in its simplest form. a — b a-t-b
60. A man spends $2. He then borrows as much money
as he has left, and again spends S2. Then borrowing
again as much money as he has left, he again spends $2,
and then has nothing left. How much money did he have
at first?
61. If 5 is subtracted from a certain number, two thirds
of the remainder will be 40. What is the number ?
62. Having a certain sum of money in my pocket, I
lost c dollars, and then spent one ath part of what re-
mained and had loft one />th part of what I had at first.
What was the original sum ? What does the answer be-
come if a = 3, 6 = 9, and c = 5 ?
240 ELEMENTARY ALGEBRA.
63. If I buy a certain number of pounds of beef at $0.25
a pound, I shall have $0.25 left; but if I buy the same
number of pounds of lard at $0.15 a pound, I shall have
$1.25 left. How much money have I?
54. Divide 84 into three parts so that one third of the
first, one fourth of the second, and one fifth of the third
shall be equal.
65. In a certain orchard 25 more than one fourth of
the trees are apple trees, 2 less than one fifth are pear
trees, and the rest, one sixth of the whole, are peach
trees. How many trees are there in the orchard ?
56. A merchant spent each year for three years one
third of the stock which he had at the beginning of the
year ; during the first year he gained $ 600, the second
$500, and the third $400. At the end of the three
years he had but two thirds of his original stock. What
was his original stock ?
57. From a cask of wine out of which a third part had
leaked, 84 liters were drawn, and then the cask was half
full. What is the capacity of the cask ?
58. A gentleman has two horses and a chaise. The
chaise is worth a dollars more than the first horse and h
dollars more than the second. Three fifths of the value of
the first horse subtracted from the value of the chaise is
the same as seven thirds of the value of the second horse
subtracted from twice the value of the chaise. What is
the value of the chaise and of each horse ? What are the
answers if a = — 50 and J = 50 ?
59. A had twice as much money as B. A gained $30
and B lost $40. Then A gave B three tenths as much as
B had left, and had left himself 20 per cent more than he
had at first. How much did each have at first?
MISCELLANEOUS EXAMPLES. 241
60. A number of men had done one third of a piece of
work in 9 days, when 18 men were added and the work
completed in 12 days. What was the original number of
men?
61. A boatman can row down the middle of a river 14
miles in 2 hours and 20 minutes ; but though he keeps
near the shore where the current is one half as swift
as in the middle, it takes him 4 hours and 40 minutes to
row back. What is the velocity of the water in the
middle of the river ? Ans. 2 miles an hour.
62. A had three fifths as much money as B. A paid
away $80 more than one third of his, and B $50 less
than four ninths of his, when A had left one third as much
as B. What sura had each at first?
63. A farmer hired a man and his son for 20 days,
agreeing to pay the man $3.50 a day and the son $1.25
for every day the son worked ; but if the son was idle,
the farmer was to receive $0.50 a day for the son's board.
For the 20 days' labor the man received $67. How
many days did the son work ?
64. I purchased a square piece of land and a lot of three-
inch pickets to fence it. I found that if I placed the pick-
ets 3 inches apart, I should have 50 pickets left ; but if I
placed the pickets 2^ inches apart, I must purchase 60
more. How much land and how many pickets did I pur-
chase ? Ans. 18906J square feet and 1150 pickets.
65. A criminal having escaped from prison travelled 10
hours before his escape was discovered. He was then
pursued and gained upon 3 miles an hour. When his
pursuers had been on the way 8 hours, they met an ex-
pressman going at the same rate as themselves, who had
met the criminal 2 hours and 24 minutes before. In what
time from the commencement of the pursuit will the crimi-
nal be overtaken ? Ans. 20 hours.
11 F
242 ELEMENTARY ALGEBRA.
66. In February, 1868, a man being asked the time,
answered that the number of hours before the close of the
month was exactly one sixth of 10 less than the number
that had passed in the month. What was the exact
time ? Ans. February 25th, 10 o'clock, p. m.
6*?. A and B owned adjoining lots of land whose areas
were as 3 : 4. A sold to B 100 hectares of his, and after-
ward purchased of B two fifths of B's entire lot ; and then
the original ratio of their quantities of land had been re-
versed. How much land did each own at first ?
Ans. A, 300 ; B, 400 hectares.
68. A laborer Was hired for TO days ; for each day he
wrought he was to receive $2.25, and for each day he was
idle he was to forfeit $0.75. At the end of the time he
received $118.50. How many days did he work?
69. A sum of money was divided equally among a num-
ber of persons by giving to the first $100 and one sixth of
the remainder, then to the second $200 and one sixth of
the remainder, then to the third $300 and one sixth of the
remainder ; and so on. What was the sum divided and
wjiat the number of persons ?
70. A besieged garrison had a quantity of bread which
would last 9 days if each man received two hectograms a
day. At the end of the first day 800 men were lost in a
sally, and it was found that each man could receive 2f
hectograms a day for the remainder of the time. What
was the original number of men ?
71. Find a fraction such that if 1 is added to the de-
nominator its value will be ^ ; but if the denominator is
divided by 3 and the numerator diminished by 3, its value
will be f .
72. If 7 years are added to A's age, he will be twice as
old as B ; but if 9 years are subtracted from B's age, he
will be one third as old as A. What is the age of each ?
MISCELLANEOUS EXAMPLES. 213
Y3. A, B, and C compare their fortunes. A says to
B, " Give me S700 of your money, and I shall have twice
as much as you retain." B says to C, "Give me $1400,
and I shall have three times as much as you retain."
C says to A, "Give me S420, and I shall have five times
as much as you retain." How much has each?
74. An artillery regiment had 39 soldiers to every 5
guns, and 4 over, and the whole number of soldiers and
oflScers was six times the number of guns and officers.
But after a battle in which the disabled were one half of
those left fit for duty, there lacked 4 of being 22 men to
every 4 guns. How many guns, how many officers, and
how many soldiers were there ?
Ans. 120 guns, 44 officers, 940 soldiers.
75. A car containing 5 more cows than oxen was started
from Springfield to Boston. The freight for 4 oxen was
$2 more than the freight for 5 cows, and the freight for
the whole would have amounted to $30 ; but at the end
of half the journey 2 more oxen and 3 more cows were
taken into the car, in consequence of which the freight of
the whole was increased in the proportion of 6 to 6. What
was the original number of cows and oxen, and what was
the freight for each ?
. j 9 cows and 4 oxen.
I Freight for a cow, $2 ; for an ox, $3.
76. Two sums of money amounting together to $1600
were put at interest, the less sum at 2 per cent more than
the other. If the interest of the greater sum had been in-
creased 1 per cent, and the less diminished 1 per cent, the
interest of the whole would have been increased one fif-
teenth ; but if the interest of the greater had been increased
1 per cent while the interest of the other remained the
same, the interest of the whole would have been increased
one tenth. What were the sums, and the rates of interest?
Ans. $1200 at 7 per cent ; $400 at 9 per cent.
244 ELEMENTARY ALGEBRA.
TT. A and B can perform a piece of work together in
lYf days. They work together 10 days, and. then B fin-
ishes the work alone in 16| days. How long would it
take each to do the work ?
T8. The Emancipation Proclamation of President Lincoln
was promulgated on the 1st day of January in a year rep-
resented by a number that has the following properties:
the second (hundred's) figure is equal to the sum of the
third and fourth minus the first ; or to twice the sum of
the jfirst and fourth ; the third is a third part of tlie sum
of the four; and if 1818 is added to the number, the order
of the figures will be inverted. What was the year ?
79. A and B can do a piece of work in a days ; A and
C in b days ; B and C in c days. In how many days can
each do it ?
80. A can do a piece of work in a days, B in ^ days,
and C in c days. In how many days can A and B together
do it? B and C together? A and C together? All three
together ?
81. A market-man bought some eggs for $0.28 a dozen,
and sold some of them at 3 for 8 cents and some at 5 for 12
cents, receiving for the whole $6.24, and clearing $0,64.
How many did he sell at each rate ?
82. One cask contains 56 liters of wine and 40 of water,
and another 96 of wine and 16 of water. How many liters
taken from each cask will make a mixture containing 52
liters of wine and 24 of watei- ?
83. A and B are travelling on roads which cross each
other. When B is at the point of crossing, A has 120 me-
ter's to go before he arrives at this point, and in 4 minutes
they are equally distant from this point ; and in 32 minutes
more they are again equally distant from it. What is the
rate of each? Ans. A's, 100; B's, 80 meters a minute.
MISCELLANEOUS EXAJIPLES. 245
84. Multiply tT by a:".
85. Multiply x' by x"'.
86. Divide y^* by y-^
87. IMvide a""* by 0*+*.
88. Transfer the denominator of — ^5 to the numerator.
89. Free -zr-f-^ ^^^^^ negative exponents.
90. Expand (— 2a«)*.
91. Expand (a^i)"*.
92. Expand (— 3a:-V)*-
93. Expand (x'" X a:")'.
94. Expand {x — >v/y)'.
96. Expand (a» — 2 6)».
96. Expand {1x—ff.
97. Expand (2x — 3)*.
98. Expand (3 a — 2 6)".
99. Find five terms of (x — y)*^.
100. Expand (2 — x — y)«.
101. Expand (3 ■— a — 6 + cf.
102. Find ^C^"^.
103. Find y/^;;r^-
104. Find V— 16a:«.
105. Find the square root of | — (2a + 2x — 4j i/^
+ a«__4a + 2ax + 4 — 4a: + a:«.
106. Reduce </ 256 a* 6» — 768 a* 6' c^ to its simplest form
246 ELEMENTARY ALGEBRA.
101. Reduce ^ — ^nd tV | to equivalent radicals hav-
ing a common index.
108. Add VA. V^> and Vif.
109. From a^IM take a^JO.
110. Multiply i^f by i>?^J.
111. Divide —Sa/TO by V'5.
112. Divide a^'6 by /</'6.
113. Find the cube of S a/2x.
114. Find the square root of 6 -^"3^
115. Multiply 6 + \/3 by 3 — a^~3.
116. Expand (x^ — 2Ai/lcy.
111. Expand (^^-s/-^y.
118. Expand (^ I -ly.
119. The area of a rectangular field is 4 acres and 35
square rods ; and the sum of its length and breadth is
equal to twice their difference. What are the length and
breadth ?
120. Two travellers, A and B, set out to meet each
other. They started at the same time and travelled on the
direct road toward each other. On meeting it appeared
that A had travelled 18 miles more than B, and that A
could have travelled B's distance in 9 days, while it would
have taken B 16 days to travel A's distance. How far
did each travel ? Ans. A, 12 miles ; B, 54 miles.
121. Find three quantities such that the product of the
first and second is a ; of the second and third, b ; and of
the first and third, c
MISCELLANEOUS EXAMPLES. 247
122. A and B invest in stocks. At the end of the
year A sells his stocks for S108, gaining as much per
cent as B invested; B sold his for $49 more than he
paid, gaining one fourth as much per cent as A. What
sum did each invest? Ans. A, $45; B, $140.
123.
Reduce ISar* — 33ar — 40 = 0.
124.
r. J X x-fl 1
Reduce ^_^ T ^e*
125.
Reduce (^-y) (- !/)=,,-
Ans. y = — 2.1 ±3 V 1.49, or 6, or -
-f
126.
Reduce (x* — x" + 4:y + x" = 5T04 + x\
Ans. x=±3, or ±2^—2, or ii/^^^v^"
91
^^
l^T
RpfliiPA A./2 r -J- 1 _L 2 ^/a: —
128.
Reduce y* — 2 A/f — 3y + 5 = 3y — 2.
129.
Reduce > ■ » = ^^ —
130.
131.
132.
Given 1 . l'^*, ^ !^ 1 . to find z and y.
133. Given \'\'^^~\l},to&ndx and y.
134. Given } IT^y "^^^V, to find x and y.
C 5xy=T5)
248 ELEMENTARY ALGEBRA.
135. Given i ^ "^^ 7 ^! "^ !!H > to S^d x and y.
l2x^i/ — 2xf= 140) ^
136. Given j^x^y - 2:.^^ = 875 ) ^^ ^^^ ^ ^^^
13Y. Given j^''^' "" ^^^, == ^H , to find x and y.
( a;2 4- y2 __ 25 i ^
138. Given j ^ ^' 7 f ^^ = ^^^ I , to find a: and y.
(3/_j-5a;2/z= 132) ^
139. Given 1^' + 2^' + ^ + ^ == ^^l , to find a; andy.
C X -\- 1/ — xy= 0)
Ans. 1^=2, orH-3±V2T).
(y = 2, or i(— 3TV21).
140. Given 1^^ + ^ = "^U , to find x and y.
(a:* + 3^* = 1921) ^
141. A drover sold a number of sheep that cost him
$297 for $1 each, gaining $3 more than 36 sheep cost
him. How many sheep did he sell ?
142. A merchant sold a piece of cloth for $75, gaining
as much per cent as the piece cost him. What did it
cost him?
143. A drover bought 12 oxen and 20 cows for $920,
buying one ox more for $160 than cows for $6Q. What
did he pay a head for each?
144. A started from C towards D and travelled 4 miles
an hour. After A had been on the road 6^ hours, B
started from D towards C, and travelled every hour one
fourteenth of the whole distance, and after he had been
on the road as many hours as he travelled miles an hour,
he met A. What was the distance from C to D ?
MISCELLANEOUS EXAMPLES. 249
145. A person bought a number of horses for S1404.
If there had been 3 less, each would have cost him $39
more. What was the number of horses and the cost of
each ?
146. Find a number of four figures which increase
from left to right by a common difference 2, while the
product of these figures is 384. Ans. 2468.
147. A rectangular garden 24 rods in length and 16 in
breadth is surrounded by a walk of uniform breadth which
contains 3996 square feet. What is the breadth of the
walk? Ans. 3 feet.
148. A square field containing 144 ares has just within
its borders a ditch of uniform breadth running entirely
round the field and covering 381.44 centares of the area.
What is the breadth of the ditch ? Ans. 0.8 meter.
149. A and B hired a pasture into which A put 5 horses,
and B as many as cost him $5.50 a week. If B had put
in 4 more horses, he ought to have paid $6 a week. What
was the price of the pasture a week? Ans. $8.
150. A father dying left $3294 to be divided equally
among his children. Had there been 3 children less, each
would have received $ 183 more. How many children
were there ?
151. A merchant bought a quantity of tea for $66. If
he had invested the same sum in coffee at a price $0.77
less a pound, he would have received 140 pounds more.
How many pounds of tea did he buy ?
152. Find two quantities such that their sum, product,
and the sum of thoir squares shall be equal to one an-
other. Ans. i (3 ± \/'^^^) and i {S T \/^^^).
163. Find two numbers such that their product shall
be 6, and the sum of their squares 13.
U*
250 ELEMENTARY ALGEBRA.
154. A and B talking of their ages find that the square
of A's age plus twice the product of the ages of both is
3864 ; and four times this product, minus the square of
B's age, is 3575. What is the age of each?
Ans. A's, 42 ; B's, 25.
155. Find two numbers such that five times the square
of the less minus the square of the greater shall be 20 ;
and five times their product minus twice the square of the
greater shall be 25. .
156. A and B purchased a wood-lot containing 600
acres, each agreeing to pay $11500. Before paying for
the lot, A offered to pay $20 an acre more than B, if B
would consent to a division and give A his choice of situ-
ation. How many acres should each receive, and at
what price an acre ?
Ans. A, 250 acres at $ TO an acre ; B, 350 at $50.
157. A merchant bought two pieces of cloth for $175.
For the first piece he paid as many dollars a yard as there
were yards in both pieces ; for the second, as many dol-
lars a yard as there were yards in the first more than in
the second ; and the first piece cost six times as much as
the second. What was the number of yards in each
piece? ^ Ans. In 1st, 10 yards ; in 2d, 5.
158. Two sums of money amounting to $14300 were
lent at such a rate of interest that the income from each
was the same. But if the first part had been at the same
rate as the second, the income from it would have been
$532.90 ; and if the second part had been at the same
rate as the first, the income from it would have been
$490. What was the rate of interest of each?
Ans. First, 7 per cent ; second, 7fV P^r cent.
159. Divide 29 into two such parts that their product
will be to the sum of their squares as 198 : 445.
MISCELLAKEOUS EXAMPLES. 251
160. What is the length and breadth of a rectangular
field whose perimeter is 10 rods greater than a square
field whose side is 50 rods, while its area is 250 square
rods less than the area of the square field ?
Ans. Length, To rods ; breadth, 30.
161. A rectangular piece of land was sold for $5 for
every rod in its perimeter. If the same area had been
in the form of a square, and sold 'in the same way, it
would have brought S90 less; and a square field of the
same perimeter would have contained 272^ square rods
more. What were the length and breadth of the field ?
Ans. Length, 49 ; breadth, 16 rods.
162. A starts from Springfield to Boston at the same
time that B starts from Boston to Springfield. When
they met, A had travelled 30 miles more than B, having
gone as far in IJ days as B had during the whole time;
and at the same rate as before B would reach Springfield
in 6f days. How far fi-om Boston did they meet?
Ans. 42 miles.
163. The product of two numbers is 90 ; and the dif-
ference of their cubes is to the cube of their difference as
13 : 3. What are the numbers?
164. A and B start together from the same place and
travel in the same direction. A travels the first day 25
kilometers, the second 22, and so on, travelling each day
3 kilometers less than on the preceding day, while B
travels 14^ kilometers each day. In what time will the
two be together again ? Ans. 8 days.
165. A starts from a certain point and travels 5 miles
the first day, 7 the second, and so on, travelling each day
2 miles more than on the preceding day. B starts from
the same point 3 days later and follows A at the rate of
20 miles a day. If they keep on in the same line, when
will they be together ? Ans. 8 or t days after B Btarts.
252 ELEMENTARY ALGEBRA.
166. A gentleman offered his daughter on the day of
her marriage $1000; or $1 on that day, $2 on the next,
$3 on the next, and so on, for 60 days. The lady chose
the first offer. How much did she gain, or lose, by her
choice ?
167. The arithmetical mean of two numbers exceeds
the geometrical mean by 2 ; and their product divided by
their sum is 3^. What are the numbers ?
168. A father divided $130 among his four children in
arithmetical progression. If he had given the eldest $25
more and the youngest but one $5 less, their shares would
have been in geometrical progression. What was the share
of each ?
169. The sum of the squares plus the product of two
numbers is 133 ; and twice the arithmetical mean plus
the geometrical mean is 19. What are the numbers ?
110. The sum of three numbers in geometrical progres-
sion is in ; and the difference of the second and third
minus the difference of the first and second is 36. What
are the numbers ?
171. There are four numbers in geometrical progression,
and the sum of the second and fourth is 60 ; and the sura
of the extremes is to the sum of the means as 7 : 3. What
are the numbers?
Cambridge : Electrotyped and Printed by Welch, Bigelow, & Ca
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