« ΠροηγούμενηΣυνέχεια »
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=
2 42—24 _48—34, 3
• 84-4y=144 -9y;
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
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=
1 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;
3; 1+134 1+39 40 whence a
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 and multiplying by b; we shall have
by transposition, cy-by-de--ab; by collecting the coefficients, (c—b) y=de-ab;
de-ab ,, by division, y=
de- ab whence w=Q-Ya
6 -ab- de tab -de that is,
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.
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
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
7 find the valaes of x and y.
Ans. <=11, and y=45
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 3r—7y_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=
y= Ex. 12. Given
3 to find the values of x and y.
Ans. x=5, and y=3. Ex. 13. Given
to find the va. 6
lues of x and 7y - 3.1 and
Ans. x=6, and y=8.
2y+* = 3;
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-
.. by transposition, 487–17x=155. Multiplying the second equation by 6,
by transposition, 2y+30x=160 . , (A). Multiplying this by 24, we have
48y +720x=3840; but 487 – 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 =
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.
2y 8.0 -2 Ex. 2. Given
y_4x — 1