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

Multiplying both equations by 6, and we shall bave 3x +2y=42, and 2x + 3y=47,

42-2y From the first of these equations, x=

3

48 - 3y and from the second, w=

2 42—2ý _48—34,

-3y 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 haye

42-24 18

6. 3 3

.

Ex. 3. Given 8x+18y=94, and 82—13y=1, to find tbe values of x and y.

47-9y From the first equation, u=

4

1+13y and from the second, x=

8 47-9y_1+13y

; 4

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

94-18y=1+13y;

=1+134

... 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

· ye

31 1 +13y 1+39 40 8

8 8

whence a

5.

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

bx+cy=de, ) of æ and y. From the first equation, x=a--y; and from the second, x=

b

de-cy

de-cy

...a-Y=

;

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-y=a

-b ca--ab-detab -de that is, a

cob

C

C

ca

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C

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

=b, and ca> or <de; then,
-de

de- ab
co, and

y= 0

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

ca

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a

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
y?

and 26
0
0

0 0 Where the values of x and y are both infinites and therefore, under these relations, there can be no finite values of u 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.

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

0 and g=%; 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.

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

Ans. x=10, and y=7.
-Y

,
3
find the valacs of x and y.

Ans. =11, and y=45

C

+y

Ex. 6. Given **4+1=6, and 7+3=4, to

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to

2x3

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

2' find the values of u and y.

1 Ans. x=8, and y=

2 3x74_2x+y+1, and 8

- y Ex. 8. Given

3
5

5 =6, to find the values of x 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 y=3. Ex. 10. Given 3x~-5y=13, and 2x +-7y=81, find the values of x and y.

Ans. x=16, and y=7. =

+10x= 3

4 192, to find the values of x and y.

y

Ans. x=19, and y= Ex. 12. Given 24,-4+14=18

, and 2y+ x

=3. 2

3 to find the values of « and y.

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

8

to find the va 6

3

lues of x and 7y-3x and

=11+y, yo 2

Ans. x=6, and y=8.

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

y+5

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2x + 3y =

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=

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

8-Y

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2+3

3x-2y Ex. 1. Given 2y- =7+

4

5

to find

2x+1 and 40

= 241 3

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

40-5x-15=140+120---By ;

... by transposition, 487–170=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 48 - 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=0; 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.

2

4十1 十

2y 8x-2 Ex. 2. Given

1.

X --y 2 18 36

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

3-
Bl
9 18

8 6

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4+y +

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