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and therefore, (Art. 147), multiplying by 18, 2y---4x+1=18-24-6y + 3x – 3y ;

.' by transposition, 7=7x-lly. But from the second equation, 7x=12y.

Substituting therefore this value in the preceding equation, it becomes 12y-117=7, or y=7.

12y

84 and .. x=

=12, 7 7

154+

47
Ex. 3. Given -
3y--2+2

3
=1+
11

33
and 347

11x +152 3y +1
3x + 2y y-5
6
4
12

2
to find the values of u and y.
Multiplying the first equation by 33,

4y 33x-9y+-6--3x=33+15x+

;

3 multiplying again by 3, and transposing, we shall have 45x-319=31. Multiplying the second equation by 12, 6x+4y-3y +15=114+152-18y-6;

.. by transposition, 19y-5x=131. Multiplying this by 9, 1714--45x=1179;

but 45x - 3y= 81;

.. by addition, 140y=1260;

and by division, y=9. Now, 5x=197 - 131=171—131=40;

.. by division, x=8.

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=18

43

80+-3x 40+3y-8
Ex, 4. Given
15

7
6x-35
and 10y + =55-+-10x,

5 to find the values of x and

y. Multiplying the first equation by 105, the least common multiple of 3, 7, and 15,

560+212=1925-60x—45y + 120;
.. by transposition, 812+55y=1485;

and dividing by 9, 9x+ 5y=165. From the second equation,

50y +6x-35=275+50x ;
.. by transposition, 50y-44x=310;

and dividing by 2, 25y-22x=155 ; but multiplying the equation found above, by 5,

25y+45x=825;

• by subtraction, 67x=670,

and by division, x=10. Now 5y=165-9x=165--90=75, ..y=15.

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4x, 5y_9 Ex. 5. Given --

+

1 ya y

to find the values 5 4 7 3 of x and and at +

y. y

2 Reducing the first equation to lower terms,

4.5 9

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1;

X

у у
.. by transposition,

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morf eht 2nd equation, by trans",

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+

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a

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2 1 .. by addition,

2 and, consequently, x=4. Now 4+1=2; 4

.. 2y=4, and y=2. y

6
Ex. 6. Given -+-=m,

y

to find the values of d tes

-=R,

y Multiplying the first equation by c, and the second by a, we shall have

ac

bc
+==mc,

y

ad and +

=na, y

x and y..

с

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ac

[ocr errors]

1 ... by subtraction, (bc-ad).-=mc--na;

y

bc-ad Y

тс па a

b mbc-nab And

- m
y

bc-ad mbc-mad-mbct nab_nab-mad bc-ad

bc-ad
1 nb-md

be ad
and =
bc-ad

nb-md
2x
7+
y

5x +9 Ex. 17. Given 3

-5 5

Зу

107

2xy 4+15y.

8

6x-2 2c +5 to find the values of c and y.

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and y

OC

-2wy

8

Multiplying the first equation by 15y,

.. 457--217-6x=754-25x--45;

and by transposition, 517–19x=45. Multiplying the second equation by 2x+5, 8x +20+30xy+75y

107 2xy +5y

; 6.-2

8

107_8x +20+30xy+75y. ::. (Art. 186) 5y+

6x-2 and multiplying by 6x-2, we shall have

321x107 30xy-10y+

=8x+20+30xy +-75y 4

3213-107 c. (Art. 186),

=8x +85y + 20,

4 and 3213-107=32x+340y +30; .. by transposition, 340y-289x=-187. The coefficients of y in this case, having aliquot parts ; multiplying the first by 20, and the last by 3,

1020y--380x= 900, and 1020y-867x=-561;

=

... by subtraction, 487x=1461,

and x=3; consequently, 5ly=45+19x=45+57=102;

..y=2

16+60x _ 16xy-107 Ex. 8. Given Ex

3y-1

5+2y

27x2-12y +38 and 2+by+9x=

3x-2y+1 to find the values of u and y.

Multiplying the first equation by 5+2y, 10x +-16xy 80+300x+32y+120xy

=16xy-107;

3y-1

80+300x +32y+120xy ... trans" 40x+107 =

3y-1 and multiplying by 3y-1, we shall have

120xy~40x +3217-107=80+300x * 32y + 120xy ;

.. by transposition, 289y— 340x=187. And from the second equation, 27x% -12y +15x+2y+2=27x02 - 12y2 +38;

... by transposition, 15x+2y=36; whence, the coefficients of x having aliquot parts, multiplying the first equation by 3, and the second by 68,

867y-1020x=561, and 136y+1020x=2448 ;

.. by addition, 1003y=3009,

and y=3; consequently, 15x=36 - 2y=36-6=30;

and .. by division, x= 2.

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2y—

59- 2x Ex. 9. Given x

-20 23-3

2

73— 3y 8-18

3 to find the values of x and y.

Ans. x=21, and y=20*

and yt

Y-3

= 30

33 -1
Ex. 10. Given +3y-4=15,

5
and
3y-5
+2x~-8=73,

=

6 to find the values of x and y.

Ans. x=7, and y=5.

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