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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=
48 - 3y and from the second, w=
2 42—2ý _48—34,
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
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=
1+13y and from the second, x=
8 And multiplying both sides of this equation, by 8,
... 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;
31 1 +13y 1+39 40 8
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=
' by collecting the coefficients, (c—b) y=de-ab;
de- ab :.. by division, y=
de-ab whence w= =Q-y=a
-b ca--ab-detab -de that is, a
250. If in the above equations, there existed, between the coefficients, these relations,
=b, and ca> or <de; then,
0 And therefore, (Art. 233), the two proposed equations would be contradictory.
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
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.
Ex. 5. Given 3x+7y=79, and 2y— =9, to find the values of x and y.
Ans. x=10, and y=7.
Ans. =11, and y=45
Ex. 6. Given **4+1=6, and 7+3=4, to
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 3x—74_2x+y+1, and 8
- y Ex. 8. Given
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. =
4 192, to find the values of x and y.
Ans. x=19, and y= Ex. 12. Given 24,-4+14=18
, and 2y+ x
3 to find the values of « and y.
Ans. x=5, and y=3. Ex. 13. Given
to find the va 6
lues of x and 7y-3x and
=11+y, yo 2
Ans. x=6, and y=8.
Ex. 11. Given **2 +8y=31, and
2x + 3y =
251. Examples in which the preceding Rules are applied, in the Solution of Simple Equations, Involv. ing two unlonown Quantities.
3x-2y Ex. 1. Given 2y- =7+
2x+1 and 40
= 241 3
2 the values of x and y. Multiplying the first equation by 20,
... 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;
.. 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.
2y 8x-2 Ex. 2. Given
X --y 2 18 36