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Sni^m'n — 1) (3;«'« + 1) (2;a/';/ — 1)

{2rn'n — 1) {2m' fi - 1) '3n

Canceling common factors, we have 2m*n -\- 1. Ans.

(.) 9 + ^ . (3 + ^) simplified = ?^ -

3x-\-2y

X — y

9;tr* — 4:1'' X y

Inverting the divisor, we have — 5 7- X

jr>_y dx-\-2y
Canceling common factors, the result equals — .
Ans.

(187) According to Art. 456, the trinomials 1 — 'Hx*
-|- X* and ^r" -f- 4.r + 1 are perfect squares. (See Art.
458.) The remaining trinomials are not perfect squares,
since they do not comply with the foregoing principles.

24 24

(188) (a) By Art. 481, the reciprocal of ^ = 1 -4-

49 49

, 49 49 .
= ^><24=2-4- ^^^-

(d) Since, by Art. 481, a number may be found from
its reciprocal by dividing 1 by the reciprocal, the number

= 1 -^ 700 = .0014^. Ans.

102 ALGEBRA.

(189) Applying the method of Art. 474,

x-y
3xy

2, ^(^+j), J(^-J), 1

Whence, L. C. M. = {x -\- y) {x —y) S xyx2x2x
r{x + y)xy{x-y)= 12x>' {x+yy {x -y)\

(190) {a) 2+4^-5tf'-6tf'

7a'

nxy{x-y), %x\x+y).

Sy{x-y)\ 6x

Uxy 2x''{x-^y),

3y(^-^), 6x

4, 2x{x+y),

y{x-yl 2

14<z' + 2Sa* - 35tf' - 42^* Ans. (Art. 423.)
(*) 4^-4y + 6-s»

12vr*j' — 12jry + ISx'ys' Ans.

(<:) 3^ + 5r - 2^/
6a

(191) («) See Arts. 359 and 361.
(d) See Arts. 419 and 440.
(c) See Art. 416.

(192) {a) On removing the vinculum, we have

2a - [Sd -{- {4c - 4a - {2a -^ 26) } -\- {Sa - d - c} ].

(Art. 405.)
Removing the parenthesis,

2a- [3d -\- {Ac - Aa-2a-2d\ + {da - d-c]].
Removing the braces,

2a — [S6 + 4c~4a — 2a — 2d-\-3a — d — c].

(Art. 406.)
Removing the brackets,

2a — Zd-4c+4a-\-2a+2d — Sa + 6+c.
Combining like terms, the result is 5a — 3c. Ans.

ALGEBRA. 103

(d) Removing the parenthesis, we have

7a- [da—{2a — 5a-\-^a\].
Removing the brace,

7a — [3a — 2a -{- 5a — 4«].
Removing the brackets,

7a — da -\- 2a — 5a -{- Aa.
Combining terms, the result is 5a. Ans.
(c) Removing the parentheses, we have

a—\2b-\- {3c — 3a — a —b\-{-\2a ~ b — cW
Removing the braces,

a—\2b-\-3c — 3a — a — b-\-2a — b— c\.
Removing the brackets,

a — 2b —3c -\- 3a -\- a -\- b — 2a -\- b -]- c.
Combining like terms, the result is 3a — 2c. Ans.

(1 93) {a) {x' + 8) = (,r + 2) {x' - 2x -\- 4). Ans.

{b) X* - 27 y* = {^- 37) {x^ + 3xy +9/). Ans.
(c) xm— nm -j- xy — ny = in (,f —ri) -\- y{x — «),

or (^ — n) {in + j). Ans.

(Arts. 466 and 468.)

(194) Arrange the terms according to the decreasing
powers of x. (Art. 523.)

4x* -f- 8a.r» + 4a»-r » + \U^x'^ + Xioab^x -+- \W (2^« + 'i.ax + 4<J». Ans.
{2.r»)« = 4-r«.

4.r« 4- 2a.jr

8a^3 ^ 4^j^»
8a;r» + \d*x*

4x* + Aax+U^

/-•r^^N c{a + b)-\-cd ac-{-bc-\-cd ^ ,. , . ,

(195) / , , J = — r-T — . Cancehng c, which

^ ' (« -|- <7)r ac -\- be '

IS common to each term, we have — . ' — = 1 + ■ — r— .. Ans.

a-Y b a+b

104 ALGEBRA.

(196) {a) ^+_;/ + ^-(^ - j)-(j+^) _(-j)be.
comes x-{-f-\-s — x-\-y—y — 2-{-jyon the removal of the
parentheses. (Art. 405.) Combining like terms,

X — X -\-f-\-f — f -\-f-{-2— 2 = '2y. Ans.

(d) (24r - J + 4^) + ( - ;r - / - 4x:) - {3x - 2y - z) be-
comes "Hx — jf -\- 4:S — X — ^ — 4^ — dx -\- 2}^ -^ s, on the
removal of the parentheses. (Arts. 405 and 406.) Com-
bining like terms,

2x — X — 3x— y— y-^Zj'-^Az— 4:2-\-z=::s— 2x. Ans.
(c) a— [2a-{- {Sa — 4:a)] — 5a— {6a— [(7^+8^) -9^]}.

In this expression we find aggregation marks of different
shapes, thus, [, (,and {. In such cases look for the corre-
sponding part (whatever may intervene), and all that is in-
cluded between the two parts of each aggregation mark
must be treated as directed by the siga before.it (Arts. 405
and 406), no attention being given to any of the other aggre-
gation marks. It is always best to begin with the inner-
most pair ^ and remove each pair of aggregation marks in
order. First removing the parentheses, we have

a—\2a-\-Za — 4^] — 5« — 1 6^ — [7« -f 8^ — 9/?] }.

Removing the brackets, we have

^ — 2« — 3« -f 4^ — 5« — {6« — 7^ - 8«+ 9tf|.

Removing the brace, we have

a — 'i.a — 3a -\- 4a — ha — %a -\- '^a ■\- ^a — ^a.

Combining like terms, the result is— 5rt. Ans.

(197) {a) A square x square, plus 2^ cube b fifth,
minus the quantity a plus b.

{hi) The cube root of jr, plus j into the quantity a minus

2
n square to the - power.
o

(f) The quantity ;;/ plus n, into the quantity in minus
n squared into the quantity vt minus the quotient of n
divided by 2.

ALGEBRA. 105

(198)

a» - a» - 2a - 1 ) 2a« - 4a» - 5a* + 3fl» + 10a* + 7a 4- 3 ( 2a» - 2a» - 3a - 2
2a« — 2a* — 4a* — 2a» Ans.

-2a*- a* + 5a3 + 10a«
— 2a* + 2a* + 4a» + 2a»

-3a»+ a« + 8a» + 7a
— 3a* 4- 3a3 + 6a« + 3a

— 2a» 4- 2a« + 4a + 2

— 2a3 4- 2a« + 4a + 2

(199) (a) Factoring according to Art. 452, we have
^y {x' — 64). Factoring {x" — 64), according to Art. 463,
we have

{x* + 8) {x' - 8).

Art. 466, rule. ;tr' + 8 = (^ + 2) {x"" - %x + 4).

Art. 466, rule, x* -^={x - 2) {x^ + 2.r + 4).

Therefore, xY — 64^y = xy {x + 2) (-r'-2^+4) (^'—2)
{x' + 2.r + 4), or xy {x + 2) {x - 2) {x' +2^+4)
(V -2x+4). Ans.

(/^) a' — d" — c' -\-l — 2^ + 2<5f. Arrange as follows
(Art. 408):

(a* - 2a + 1) - {b' - %bc + c") = {a - 1)' - {b - cy.
(Art. 455.)

By Art. 463, we have

(a-\Jrb-c){a-\-\b-c^\
or {a — \ -\- b — c) {a — \ — b -\- c). Ans.

{c) 1 — 16^' + 8^^ — ^'. Placing the last three terms in
parentheses (Art. 408), 1 — (16^' — %ac + c"").

\W - Sac +c' = (4« - r)^ (Art. 455.)

1 - {Ua' - Sac -\-c') = l- {Aa - c)'.

l_(4rt: - cy=[l + (4rt - c)] [l-(4rt -r)]. (Art. 463.)

Removing parentheses, and writing parentheses in place
of the brackets,

l-{Aa-cy={i-\-Aa-c) {l-4a + c). Ans.

(200) See Art. 482.

106 ALGEBRA-

(201 ) The subtraction of one expression from another,
if none of the terms are similar, may be represented only by
connecting the subtrahend with the minuend by means of
the sign — . Thus, it is required to subtract 5a*d — 7<?'^' +
5ad* from a* — d*, the result will be represented by a* — d*—
{5a*d — 7a*d* -\- bab*), which, on removing the parentheses
(Art. 405), hecom^sa^—b' — 6a^b~\-ld'b' — 5ab\ From
this result, subtract Za* — ia'b + Ga'b' + 5ab* — db\

a*— b* — 5a^b + 7a'b' — 5ab' minuend.

— 3/z* + W 4- \a'b — ^a^b* — oab^ subtrahend, with signs

— "ia* + 2b' — a^b + d'b'' — lOab^ changed. (Art. 401 .)
Or, — 2rt* — a*b + a*b* — lOab* + 2^*. Answer arranged

according to the decreasing powers of a.

(202) (tf) 3«-23 + 3^ ^-2b + ^

'ia — %b— c becomes — 2^z + 8^ + ^

a + 6^ + 4r
when the signs of the subtrahend are changed. Now, add-
ing each term (with its sign changed) in the subtrahend to
its corresponding term in the minuend, we have (— 2/z) +
(3^z) = «; ( + 83) + (-2^)=+6/^; (+ ^ + (3^) = +4^-
Hence, a-\-^b-\- \c equals the difference. Ans.

{J}) 2vr»-3<)'+2j-y

.r* + J'* — ^y becomes

' 24^- 3-r> + 2-ry'

X* — Sjr'jf + 2-ry — j' + x^
when the signs of the subtrahend are changed. Adding
each term in the subtrahend (with its sign changed) to its
corresponding term in the minuend, we have x^ — Sx'y +
2-rK* — >^ + xy^ which, arranged according to the decreasing
powers of jr, equals x* — 3-r*^ + ^J* + ^-ry* —y, Ans.
{e) 14a + Ab-6e-3d

Ua — 2^ + 4^ — 4^

On changing the sign of each term in the subtrahend, the
problem becomes

ALGEBRA. 107

l^ + U- 6c- 3d
— lla + 2d-^-\-4:d

3a + 6d— 10c + d
Adding each term of the subtrahend (with the sign
changed) to its corresponding term in the minuend, the dif-
ference, or result, is 3a-{- Qd — lOc -\- d. Ans.

(203) The numerical values of the following, when a
= 16, ^ = 10, and ;r = 5, are:

{a) {ab'x + %abx) 4^ = (16 X 10' X 5 + 2 X 16 X 10 X 5)
X 4 X 16. It must be remembered that when no sign is
expressed between symbols or quantities, the sign of multi-
plication is understood.

(16 X 100 X 5 -H 2 X 16 X 10 X 5) X 64 = (8,000 + 1,600)
X 64 = 9,600 X 64 = 614,400. Ans.

{b) 2^4^ -li^ + ^lHf:.. 2^/64-

a- b ' X " 16 - 10 '

10-5 ,^ 100 , , 96-100 + 6 2 1 .
-^- = 16-— +1 = _^_ = _^_ Ans.

{c) {b - \^) {x' - b') {a' - b") = (10 - VT6) (5' - 10')
(16' - 10') = (10 - 4) (125 - 100) (256 - 100) = 6 X 25 X
156 = 23,400. Ans.

(204) (a) Dividing both numerator and denominator

by 15 m xy', ■— — '-fj- = . Ans.

•' "^ ' 75 ;« x'y 5 xjy

.,. x'-l (x-\-l){x-l) , ^, ^ .

lo) - — 7 — -— - = -^^ — H — / , ^. when the numerator is
^' 4;ir(;r+l) 4^'(;ir+l)

factored.

Canceling {x -]- 1) from both the numerator and denom-

X — 1
inator (Art. 484), the result is — j — . Ans.

(c) ; ' ,.: ; ' , ,,! when factored becomes

^ ' {a' — b^) {a' — ab-\- b)

{a + b){a^-ab-^b^){a^ + ab + b*)

{^a-b){d' + ab-\- b') {a^-ab + bj ^ ^^^''

108 ALGEBRA.

Canceling the factors common to both the numerator and
denominator, we have

(a -\- I?) {a" — a b -\- d^) {a" -\- ab ^ b") _ a -^ b
\a - b) {a" -\- a b -\- b") {a^ - a b + b^)~ a - b'
a-\.b)a'-^b' {a^-ab-^b* a-b)a'-b' {a^-{.ab+b'
a* + a^b a* — d'b

- a'b + b" a'b - b'

-a^b-a b^ a'b - ab*

ab' + b* ab-b'

ab* + b' ab'-b'

(205) (a)

1 + x

-1 + x

1

-x'

l + ,r

+ 1-X

1- X 1 + X 1- x'' 2x

_, _ , _ x^

l-.r+l + ^ l-x"

YZT^ = Y^ ^ ~T^ "= ""• ^''^- (^^^ ^'■^- ^^^-^

b''^ a ab' ab"

a a-b ifb - b\a — b)~ cfb - ab + b

b^ ab ab ab

a^^b" , a^b - ab'' J^b _ a'-^b ab _ a + b

ab • ab ~ ab ^ b{a' - ab + b') ~ b '

Ans.
(0 1 - 1 i_ Ans

"^ 1 , -^ + 1 ^ * (Art. 509.)

'^ 3- X

/o^^A 3 + 2,1- 2-3.r, iar-jr» ,, ,, ,

(206) ^ r— 5 r - If the denommator

^ ^ 2 — X 2 + -r X — 4:

of the third fraction were written 4 — a', instead of x' — 4,

the common denominator would then be 4 — a".

o V ^oo IG'^'-'^'u Ux-x* IGx - x'
By Art. 482, — .^^j- becomes — _ ^., ^ = ^_^ •

3 + 2.r 2—3^ IGx — x' , -, j *

Hence, — -^ r— ; 3-, when reduced to a

' 2 — X 2 + 4r 4-jr

common denominator, becomes

ALGEBRA. 109

(3 4- 2x){2 4- ;r) - (2 - dx){2 - x) - {\(Sx - x^) _

^ {Q + tx + 2;r*) - (4 - 8^ + 3.r') - {Ux - x*)
4 — ;tr"

Removing the parentheses (Art. 406), we have
6 +lx + 2.y' - 4 + 8.y - 3;tr* - HiX -f- x^

4:-X'

Combining like terms in the numerator, we have

2-x
4:-x'''

Factoring the denominator by Art. 463, we have

2-x

(2 + ^)(2-,r)-

Canceling the common factor (2 — jr), the result equals

, or — ^. Ans. (Art. 373.)

2 + x' x-\-2'
(207) {a)

, , „ 4:X-4: 5X + lOx' - 4:X + 4: lOX^ + X + 4: .

l + 2.r = = ■ = ■ — = ■ — . Ans.

5x 5x 5x

(Art. "604.)

3^ + 2-r+1 ^3^_.^Q _41 ^^g (Art. 505.)

X -\- 4: X -\- 4: '

;r + 4 ) 3^-' + 2;i: + 1 ( 3;ir - 10 + ^^

4r4-4
3ar' + 12;t' ^

- lOx + 1

- \^x - 40

41
(r) Reducing, the problem becomes

^' + 4^ — 5 .tr - 7

;i^ ^ ;f " - 8;r + 7'

Factoring, we have

(^ + 5)(;r-l) .y-7

;^» ^(^-l)(^-7)'

Canceling common factors, the result equals ^ . Ans.

110 ALGEBRA.

(208) (a) Writing the work as follows, and canceling
common factors in both numerator and denominator (Arts,
496 and 497), we have

9 X 5 X 24 X vi^ifp'qx'y^ _ Smnxy .

8 X 2 X 90 X wz X « X /' X ^' X ;i^ X J 4/^'
(d) Factoring the numerators and denominators of the
fraction (Art. 498), and writing the factors of the numer-
ators over the factors of the denominators, we have

{a — x) (a' -f ax -{- x^) {a + •^) (^ + -^) _
{a + x) {a' — ax -\- x') [a — x) {a — x)

{a + x) {a'' -\-ax^ x') ^^^

{a — x) {a^ — ax -\- x^)'
(c) This problem may be written as follows, according to

Art. 480 :

Sax + 4 a^

a{'dax -j- 4) {Sax -f 4)"
Canceling a and (3«;r-|-4), we have- — — . Ans.

(209) (a) - Imy ) Zhm^y + 28;;?y - 14w/
— hm^ — 4;«y -f ^j'

Ans. (Art. 442.)
(d) a* ) W - ^a^b - a*b ^

4 — 'dab — d'b'' Ans.
{c) 4;r' ) 4,r' - S.r' + 12^' - 16;r'

X - 2x' + Sx' - Ax' Ans.-
(21 0) {a) IGa'b' ; a* + 4.ab ; 4^' - IGa'b + 5^* + 7 ax.

(b) Since the terms are not alike, we can only indicate
the sum, connecting the terms by their proper signs.
(Art. 389.)

(c) Multiplication: 4«rV means 4 X ^ X ^' X ^. (Art.
358.)

^' 4- c' + ac a^-^c' — b' - 2ac

^-^1 * ) a'^b'-c'- 2ab ^ a'c - ac*

Arranging the terms, we have

ALGEBRA. Ill

a^j^ac-\-^ a" - "la c ■{- c" - U"

a" - lab + ^' - c' a'c - ac*

which, being placed in parentheses, become

a" J^ac-\-c^ {a' - 2ac + c') - ^

{a' - 2ad -\- d')- c' a*c - ac*

By Art. 456, we know that «" — 2ad + ^', also a" — 2ac
-\- c^, are perfect squares, and may be written {a — by and
{a-c)\

Factoring a*c — ac* by Case I, Art. 452, we have
a^ ^acJ^ c" {a - c)' - b"" _
(rt _ by -c'^ ac {a' -c') ~
a' -{• ac -\- c' {a — c — b) {a — c -\- b)

{a — b— c) {a -^b -\- c) ac {a — c) (a" -\- ac -\- c')'

(Arts. 463 and 466.)
Canceling common factors and multiplying, we have

a — c -\- b a-\- b — c .

or — ; ;!— — ^-, r. Ans.

{a — b -{- c) ac {a — cy ac {a — b -\- c) {a — cy

(212) The square root of the fraction a plus b plus c
divided by n, plus the square root of a, plus the fraction b
plus c divided by 7i, plus the square root of a plus b, plus
the fraction c divided by «, plus the quantity a plus b, into
c, plus a plus be.

(213) {a)-^ 5:r- + i2P-

We will first reduce the fractions to a common denomina-
tor. The L. C. M. of the denominator is 60-r', since this is
the smallest quantity that each denominator will divide
without a remainder. Dividing 00,1'" by 3, the first denom-
inator, the quotient is 20;r' ; dividing GO;r' by 5x, the second
denominator, the quotient is 12x; dividing GO;ir' by 12x', the
third denominator, the quotient is 5. Multiplying the cor-
responding numerators by these respective quotients, we
obtain 20^'(4;r + 5) for the first new numerator ; V2x{3x — 7)
for the second new numerator, and 5 X 9 = 45 for the third
new numerator. Placing these new numerators over the
common denominator and expanding the terms, we have

112 ALGEBRA.

20j^(4^+5)-12.r(3:r-7)+45 _ 80^-'-|-100-t-*-36x*+84;r+45

Collecting like terms, the result is

SOx' + 64^' + 84.r + 45 .

TT^ » • AJIS.

(d) In ■—-, — ; — r + -r—, r, the L. C. M. of the denomi-

^ ' '■Za{a-\-x) 2a{a — x)

nators is 2a{a'' — x'), since this is the smallest quantity that
each denominator will divide without a remainder. Dividing
2a{a* — x^) by 2a{a -f- x), the first denominator, we will
have, a — x; dividing 2a{a'' — x') by 2a{a — x), the second
denominator, we have a -^ x. Multiplying the correspond-
ing numerators by these respective .quotients, we have
(a — x) for the first new numerator, and {a + x) for the
second new numerator. Arranging the work as follows :

IX (a — x) = a — X = 1st numerator.

1 X {a -}- x) = a -\- X = 2d numerator.

or 2a = the sum of the numerators.

Placing the 2a over the common denominator 2a{a^ — x*),
we find the value of the fraction to be
2a 1

2a{a''—x') a' - x''

Ans.

y x-\-y x* -\- xy y x-\-y x-\-y
The common denominator =■ y {x-{-y). Reducing the
fractions to a common denominator, we have

y{^+y) A^-^y) y

(214) {a) Apply the method of Art. 474 :

6ax

ISax', 72ay\

12xy

2y

dx, ny.

2y

3

Sx, ey,

1

X, 2y, 1
Whence, 6ax X2yx3xxx2y = 72ax'y*. Ans.

ALGEBRA.

113

(^) 2(1 + .t-)
2(1-^-)

4(l + x), 4(1-A-), 2(l-.0

2,

2(1-^), 1-x

1, 1, 1

Hence, L. C. M. = 2(1 + x) x 2(1 - x) = 4(1 - x'). Ans.

(r) a-d
b-c

(^a-d) {b-c), {b-c) {c-a), {c-a) {a-b)

{1,-c), {b-c) {c-a), {c-a)

1,

c—a, c—a

1, 1, 1

Hence, L. C. M. = {a — b) {b — c) {c — a). Ans.

(21 6) 3,r» - 3 +rt -«;!'• = (3 - a)x* - 3 + ^ =
(3 — a) {x" — 1). Regarding jr' — 1 as {x^Y — 1, we have, by
Art. 462, x'-\ = {x^ _ i = (^' _ i) {x' + i). ^» - i z=
(4.- - 1) (jr' + ;r + 1) ; x^-\-\ = {x-\- 1) (;.;'- ^r + 1). Art.
466.

Hence, the factors are
(jr» + ;r + 1) (^'" - ;r + 1) {x + 1) {x _ 1) (3 - a). Ans.

(216) Arranging the terms according to the decreasing
powers of x, and extracting the square root, we have

x' + x^y + 4i^'y + ^xy^ + ^y' {x' + ^;r/ + 2y\ Ans.

2;^* + ^xy

X y -\- 4.^x y

y
»,.,»

2^' + ^J + 27'

4^y + 2-rj/' + 4j'
44r>» + 2ay/' + Ay'

(217) The arithmetic ratio of ^* — 1 to ;r + 1 is .r* — 1 —

{x-\-l) = x' - x-'l. Art. 381.

x' — \
The geometric ratio of -r* — 1 to ;r + 1 is — ^tt- = x* -~ x*

-^x-l = {x* + l) {x - 1). Ans.

x^l

ALGEBRA.

(QUESTIONS 218-257.)

(218) {a) According to Art. 628, .arl expressed radi-
cally is V^;

Zx^y'^ expressed radically is 3 \/xy-* ;
3^^ Jo-l _ 3^;fjj/-V, since s' = -cl Ans.

*' . 1 .1

3^4

a c

a c' {a -{- d) in — n b* '
{c) ^'=:x\ Ans. ^' = x'\ Ans.
{{/^^y = i^^^'^y = b'^xK Ans.

Ans.

(219) 3i/^=i/l89. Ans. (Art. 542.)
d'b'/Ifc = \/ a*b'c . Ans. %x\G = \/^a'\ Ans.

(220) Let ;r = the length of the post.

X

Then, — = the amount in the earth.

^x

-jr- = the amount in the water.

^i.±- + 13 = x.

Itx + 15.r + 455 = 35^:.

- 13.t' = - 455.

;f = 35 feet. Ans.

In order to transform this formula so that /, may stand
alone in the first member, we must first clear of fractions.
Clearing of fractions, we have

116 ALGEBRA.

Transposing, we have

Factoring (Arts. 452 and 408), we have

whence, /, = '-^—^ ^^^ ^-^-i. Ans.

yy.s.

» «

(222) Let ,r = number of miles he traveled per hour.

48
Then, — = time it took him.

X

48

= time it would take him if he traveled 4

X-{-4:

miles more per hour.
In the latter case the time would have been 6 hours less ;
whence, the equation

48 48
x-\- 4 X

Clearing of fractions,

48^ = 48.*- + 192 - ^x^ - 24;r.
Combining like terms and transposing,

6;r' + 24^ =192.
Dividing by 6, x' -{- ^x = 32.

Completing the square, x' -{- 4:X -\- 4: = 36.
Extracting square root, x -{-2= ± 6 ;

whence, ;r=— 2 + 6 = 4, or the

number of miles he traveled per hour. Ans.

(223) (a) S=V . ly,. =V "- /^ .

Cubing both members to remove the radical,

CPD*

S* =

2/+^^-

CPiyd*
Simphfymg the result, 5' = ^^^ ^^ .

ALGEBRA. 117

Clearing of fractions,

2 S'/d' + S'/D' = CPITd}.
Transposing, CPITd^ = 2 S'/d' + SyB' ;

whence, P= cj)^^^' =- c D^d' ' ^""'

{d) Substituting the values of the letters in the given
formula, we have

(2 X 18' 4- 30') X 864 X 6' _ (648+900) X 864 X 21G __
~ 10 X 30' X 18' . 9,000 X 324 "

288,893,952 ^^ , , .

^,916,000 -^^-l^ nearly. Ans.

(224) (a) 3;r + 6 — 2^ = Hx. Transposing 6 to the
second member, and 7-t- to the first member (Art. 561),

3^ — 2^ — 7^ = — 6.
Combining like terms, — 6^ = — 6 ;

whence, x = 1. Ans.
(d) bx - (3;tr - 7) = 4^ - {<dx - 35).

Removing the parentheses (Art. 405),

5^ - 3.r + 7 = 4jr - 6^ + 35.
Transposing 7 to the second member, and ^x and — Qx to
the first member, 5x — dx — 4^ -^ 6x = 35 — 7. (Art. 561 .)
Combining like terms, 4^: = 28 ;

whence, ;tr = 28 -h 4 = 7. Ans.
(c) {x + 5)' - (4 - xy = 21x.

Performing the operations indicated, the equation becomes

x' + 10^ + 25 - 16 + 8;r - ;r' = 2Lr.
Transposing, x^ — x* -{- 10.;r -\-8x — 21x =16 — 25.
Combining like terms, — 3;r = — 9.

Dividing by — 3, x = 3. Ans.

(225) (a) Simplifying by Art. 538,

/%r= /t" X 4/3 = 34/3.

24/T8" = 2VT6 X'/3= 8-/3.

3/108 = 34/36 X 4/3 = I84/3!

Sum = 294/3. Ans. (Art. 544.)

118 ALGEBRA.

Sum = 13v^. Ans. (Art. 544.)
<')^ = ^^i^=^^I^- (Art. 540.)

'37 ' 27 3 '81 9'
Sum = (i + 1 + ^)4/0 = 124/6. Ans.

(226) Let X = the capacity.

Then, x — 42 = amount held at first ;
7{x-42) = x;
7x-294. = x;
6^=294;
;r = 49 gallons. Ans.

(227) (a) 2 V'dx + 4 - ;r = 4.

Transposing, Art. 579, so that the radical stands alone
in the first member, 2 ^dx -{- 4 = x -{- 4:.

Squaring both members, since the index of the radical
is understood to be 2, 4(3-r -|- 4) = (.r + 4)',

or 12-r + 16 = ;r' + 8;r + 16.
Transposing and uniting terms,

-X* -8x + 12;tr = IG - 16.
— x'-\-4x = 0.
Dividing by — ;r, x— 4 = 0;

whence, x = 4. Ans.

{6) i^dx - 2 = 2(jr - 4).

Squaring, 3.r — 2 = 4{x — 4)\

or 3^ — 2 = 4-ir' — 32jr + 64

ALGEBRA. 119

Transposing, — A:X* -\- S2x + 3^ = 64 + 2.
Combining terms, — ix^ + d5x = 66.

Dividmg by — 4, x — — = — j-.

Completing the square,

, S5x , /35\' 66 , 1,225

^ - 4-+UJ = -T + -6r-

^ 4"'"\8/ 64 "^64" 64"

Extracting the square root,

35 , 13

n. ■ 35 , 13 ' „3 .

Transposmg, ^ = — - ± — = 6, or 2-. Ans.

(c) i^x-\- 16 = 2 + i/I becomes jr + 16 = 4 + 4V^ + x,
when squared. Canceling ^ (Art. 662), and transposing,

— 4^/^ = 4- 16.

- 4i/]r = - 12.

|/:?=3;
whence, ;ir = 3' = 9. Ans.

(328) {a) /3J^=lSl£.

Clearing of fractions,

x^^dx — 5 = 4/74:' + 36"J.

;»r»(3.r - 5) = 7.r' + 36;tr.

3x* - 5x' = 7x* + 36;r.
Dividing by x, 3.r' — bx — 1x-{- 36.
Transposing and uniting,

3x' - nx = 36.
Dividing by 3, x^ — A:X = 12.

130 ALGEBRA.

Completing the square,

x* — 4tx-\-A= 16.
Extracting the square root,

x-2= ±4;
whence, ^ = 6, or —.2. Ans.

(^) x' — (d — a)c = ax — bx -\- ex.

Transposing, x* — ax -\- bx — ex = {b — a)e.
Factoring, x''— {a — b + e)x = be — ae.

Regarding (a — b -\- e) as the coefficient of x, and com-
pleting the square,

^ -(.-*+.).+ (i:z|±£y =,,_.,+ (i^^y.
^_(,_^+,),+ (fL=i±£y=

a* — 2ab + b^ — 'Zae-^-^be + e'.
4
a — b -{- e a — b — c

2 "^ 2

la — 2^ 2r

X = a — b, or <:. Ans.
{e) (;r - 2) (;r - 4) - 2 (^ - 1) (^ - 3) = 0, becomes

^' — 6jr + 8 — 2^' + 8^ — 6 = 0, when expanded.
Transposing and uniting terms,

- ;r' + 2x = - 2.
Changing signs, x'' — 2.r = 2.

Completing the square, x* — 2x -}- 1 = 3.
Extracting the square root, x — 1 = ± ^/d;

whence, x= 1 ± i/sT Ans.

(229) (a) v^Tr4^=ti±i)l^ZL^.

^x
Expanding and clearing of fractions,

i/x" - iabx = a'- b\
Squaring both members,

x' - ^abx = a'- 2a'b' + b\

ALGEBRA. 131

Completing the square,

x" - ^abx + Aa^b'' = a* -\- 2a* b^ + b\
x-2ab= ± {a' -\- b').

x={a'-^2ab-\-b'),

or- {a^.-2ab-\-b*).
x={a + by, or -{a — by. Ans.

Ill

(b) y ' ■ 4- , = , , becomes

■'' V^+l l/JUT ^x'-l

— Vx — 1 + V^-\- 1 = 1 when cleared of fractions.
Squaring,

X-1 — 2i/-r' - 1 + ;ir + 1 = 1.

— 24/^' - 1 = 1 - 2;ir.

Squaring again, 4jr' — 4 = 1 — 4^: -j- 4ar".

Canceling 4;r' and transposing,

4;jr= 5.

;r = - = 1- Ans.
4 4

(230) 5;ir - 2j = 51. (1)

19a' - 3j = 180. (2)
We will first find the value of ;ir by transposing — 2j to the
second member of equation (1), whence 5x = 51 + 2y, and

- = 5i±^. (8)

This gives the value of x in terms of f. Substituting the
value of X for the x in (2), (Art. 609.)
19(51 + 2f)

5

3/ = 180.

T^ J- 969 + 38 y
Expandmg, ^ ^ 3 j = 180.

Clearing of fractions, 9G9 + 38j — 15^ = 900.
Transposing and uniting, 23 j = — 69.

J = — 3. Ans.
Substituting this value in equation (3), we have

X = — z — = 9. Ans.

122

ALGEBRA.

(231) (a)

2^:'- 27^ = 14.

^-^ = 7.

x"-

T+rTT-+(T7=^

27 . 29

;r = - = 14,

2 1

or ^= - = - .

Hence,

:r = 14, or — -• Ads

A)

(^)

" 3 +12 = 0-

Transposing,

, 2;r 1
■^ 3 ~ 12'

Completing the square,

3 ^\3/ 12^9 36

Extracting the square root,

^-|=±i

Transposing,

1 . 1
^=3 + 6

1

-2'

1 1
^^ -^=3-6

1

~6

Therefore,

1 1

A]

Transposing and factoring,

X* -\- {a — d) X = a b.

^:ab-\-d' — 'lab-^-b' _ a' -\-'lab-\-b^
4 ~ 4 '

ALGEBRA. 123

Extracting

square
X -

root.

-b _

a + b
"^ 2 •

X —

a-b
2 '

a + b

2

K

or

X =

a-b
2

a-{-b

2

Therefore,

X =

b, or — a.

Ans.

(232)

Lei

: X =

rate of current.

y = rate of rowing.
Down stream, the rowers are aided by the current, so

Since it takes them twice as long to row a given distance

up stream as it does down stream, they will go only - as far

1 '^

in 1 hour, or — of 12 = 6 miles per hour up stream.

^+j=12. (1)
-x-\-y= 6. (2)

Subtracting, 2x = 6, and x = 3 miles per hour.

Ans.

(233) (,)iO£±3_6£_^^10(.-l).

Reducing the last member to a simpler form, the equation
becomes

o Z

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