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Ex. 2. Given 4x+y=7, and x+y=8, to find the values of x and y.

Multiplying both equations by 6, and we shall

have

3x+2y=42, and 2x+3y=47,

42-2y

From the first of these equations, x=- 3

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Multiplying each member by 6, we shall have 84-4y=144-9y;

by transposition, 9y-4y-144-84,

or 5y=60;.. y=12. And, by substituting this value of y, in one of the values of x, the first, for instance, we shall

have

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Ex. 3. Given 8x+18y=94, and 8x-13y=1, to find the values of x and y.*

From the first equation, x=;

47-9y

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4

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4

8

And multiplying both sides of this equation, by 8, 94-18y=1+13y;

by transposition,

18y-13y=—94+1; Changing the signs, or what amounts to the same thing, multiplying both sides by -1, and we shall 18y+13y=94-1, or 31y=93;

have

1+13y 1+39 40

93

y=

3

31

whence x

5.

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Ex. 4. Given x+y=a,

to find the values

bx+cy=de, of x and y.

From the first equation, x=a—y ;

de-cy

and from the second, x= b

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ab-by-de-cy;

by transposition, cy-by-de-ab ;

by collecting the coefficients, (c—b) y=de-ab;

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de-ab

.. by division, y=

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whence x=

ca-ab-de-ab ca -de

that is, x

C

c-b

250. If in the above equations, there existed, between the coefficients, these relations,

cb, and ca> or <de; then,

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And therefore, (Art. 233), the two proposed equations would be contradictory.

In order to give a numerical example, let c=b 4, a=3, and de-10; then, by substituting these values, we shall have

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Where the values of x and y are both infinite, and therefore, under these relations, there can be no finite values of x and y, which would fulfil both equations at once; this is what will still appear more evident, if we substitute these values in the proposed equations; for then, we shall have, x+y =3, and 4x+4y-10; which are evidently contradictory; since, if we multiply the first by 4, and subtract the second from the result, we should have 0=2.

0

0'

Again, if cb=4, a 3, and de=12; then a

0 and y; therefore, under these relations, the

0

two proposed equations would be indeterminate ; and, in fact, this appears evident by inspection only; for the second furnishes no condition, but what is contained in the first, since the two proposed equations, in this case, would become

x+y=3, and 4x+4y-12.

Ex. 5. Given 3x+7y=79, and 2y-1-9, to find the values of x and y.

Ans. 10, and y=7.

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Ex. 6. Given ++1=6, and "="+3=4, to

3

7

find the values of x and y.

Ans. 11, and y=4

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Ex. 8. Given

Ans. x= = 8, and y

X

3x—7y_2x+y+1, and 8—*—y

3

5

=6, to find the values of x and y.

5

Ans. x 13, and y=3,

Ex. 9. Given x+y=10, and 2x-3y=5, to find

the values of x and y.

Ans. x7, and y=3.

Ans. x=16, and y=7.

Ex. 10. Given 3x-5y=13, and 2x+7y=81, to find the values of x and y.

Ex. 11. Given *+2+8y=31, and +5

3

192, to find the values of x and y.

4

+10x=

Ans. x=19, and y=

Ex. 12. Given 2x+14=18, and 2y+x

2

to find the values of x and y.

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and 7y=32=11+y, √ y.

Ans. x 6, and y=8.

251. EXAMPLES in which the preceding Rules are applied, in the Solution of Simple Equations, Involv. ing two unknown Quantities.

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the values of x and y.

Multiplying the first equation by 20,

40y-5x-15=140+12x-3y;

.. by transposition, 48y-17x=155.

Multiplying the second equation by 6,
24x-16+2y-147-6x-3;

... by transposition, 2y+30x=160. . . (A). Multiplying this by 24, we have

48y+720x=3840;

but 48y- 17x= 155;

.. by subtraction, 737x=3685, and by division, x=5.

From equation (A), 2y=160-30x ; ... by substitution, 2y-160-150,

10
2

by division, y==;•'•y=5.

The values of x and y might be found by any of the methods given in the preceding part of this Section; but in solving this example, it appears, that Rule I, is the most expeditious method which <we could apply. ~

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