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

48—34,

2 42—24 _48—34, 3

2
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

42-24 18

6. 3

3

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

47--9y From the first equation, 2=

4

1 and from the second, x=

8 47—9y_1+134

;

8 And multiplying both sides of this equation, by 8,

94–18y=1+134 ;

4

.. by transposition, -18y-13y=-94+1; Changing the signs, or what amounts to the same thing, multiplying both sides by -1, and we shall have 18y +- 13y=94-1, or 3ly=93;

93

3; 1+134 1+39 40 whence a

5. 8

8

... y=31

8

Ex. 4. Given a + y=a, ? to find the values

bx+cy=de, ) of æ and y. From the Girst equation, š=Q-Y;

de and from the second, i=

b

-cy

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de-cy

Oly

i

b and multiplying by b; we shall have

ab-by=de-cy;

by transposition, cy-by-de--ab; by collecting the coefficients, (cb) y=de-ab;

de-ab ,, by division, y=

-b

de- ab whence w=Q-Ya

6 -ab- de tab -de that is,

c-b

ca

ca

250. If in the above equations, there existed, between the coefficients, these relations, cab, and ca> or <de; then,

de- ab =, and y=

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

ca-de

3C

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y= =

and x=

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In order to give a numerical example, let c=h= 4, a=3, and de=10; then, by substituting these values, we shall have 10-12 2

12-10 2 0

0 09 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, æty =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.

Again, if c=b=4; a=3, and de=12; then <= 0

0 0

and y=;; therefore, under these relations, the 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. 3. Given 30+7y=79, and 2y =9, to find the values of x and y.

Ans. x=10, and y=7.

c Ex. 6. Given

and 3

7 find the valaes of x and y.

Ans. <=11, and y=45

to

9

y=3.

2x-3

67 Ex. 7. Given ty=7, and 5x-13y= 2

2 find the values of x and y.

1 Ans. x=8, and y

2 3r7y_2x+y +1, and 3Ex. 8. Given

+ X-Y 3 5

5 =6, to find the values of u and y.

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. x=7, and Ex. 10. Given 3.0-5y=13, and 2x + 7y=81, to find the values of x and y.

Ans. x=16, and y=7.

+2 Ex. 11. Given

= and +10x=

3
192, to find the values of x and

y.
Ans. =19,

and

y= Ex. 12. Given

=, 2

3 to find the values of x and y.

Ans. x=5, and y=3. Ex. 13. Given

to find the va. 6

3'

lues of x and 7y - 3.1 and

y 2

Ans. x=6, and y=8.

y+5
4

2x-Y+14=18, and

2y+* = 3;

2x+3y=8

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=11+y,

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

8-Y=24

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2+3

3x-2y Ex. 1. Given 2y-

=7+ 4

5

to find

2x+1
and 43
3

2
ibe values of x and y.
Multiplying the first equation by 20,

40y-5x-15=140+12x ---8y;

.. by transposition, 487–17x=155. Multiplying the second equation by 6,

24x --16+2y=147-6.0-3;

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

48y +720x=3840; but 487 17= 155;

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.. by subtraction, 737x=3685,

and by division, x=5. From equation (A), 2y=160-30x ; :. by substitution, 2y=160—150,

10 by division, y =

jy=5.

2 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.

4+

2y 8.0 -2 Ex. 2. Given

1+y,x-o :1

X-ry?

+
18 36

3 6
and : 3y :: 4:7,
to find the values of x and
Redacing the first equation to lower terms,

4+y*-Y

1

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y_4x — 1

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