Let x+3e-15 To find x and y. x—y-7 By subtmetion, 2 y-8, and By- addition, 2 x-22, and x11.
If the co-efficients of the unknown quantity to be exterminated be different, multiply the terms of the first equa tion by the co-efficient of the unknown quantity in the second, and the terms of the second equation by the co-efficient of the same unknown quantity in the first; then add, or subtract, the resulting equa tions, as in the former mise.
S 3x-5y=13 1 To find x Ex. 1. Let '? 2x+7,y-=81 5 and y.
Multiply the terms of the first equation by 2, and the terms of the other by 3, then 6 x--10y=26 .
6 x+21 y=243 .
By subtraction, — 31 y = — 217 217 and y=--3-i- =7 ; also, 3 x— 5 y = 13, or 3 x —35 = 13, therefore 3 x = 13 +35 = 48 and x = 48 =16 3 S a x+b y=c ?. To find x Ex. 2. Let im x—n y=d 5 and y.
From the first, In a x + 171 b y --- in c from the other, ma x—n ay = a d . by subtraction, mb y ± 71 a y=mc--a d, In d 'therefore, y =--.
ntb+n a Again, n a x+n b y=nc 711 b x--n b y=b d by addition, n a b .x = 71 C ± b d, ne±bd therefore x = — .
na+mb •If there be three independent simple equations, and three unknown quantities, reduce two of the equations to one, con taining only:two of the unknown quanti ties, by the preceding rules; then reduce the third equation and either of the form er to one, containing the same two un known quantities; and from the two equations thus obtained, the unknown quantities which they involve may be found. The third quantity may be found
by substituting their values in any of the proposed equations.
2 x+3 y+4z----16 To find x, Ex. Let. 3 x+2 y-5 z --8 y, and 5 x--6 y+3 z -- 6 z. From the 21st equa. 6 x+n y+ 12 z=48 6x+4 y-10z=16 by subtr. 5 y— 22 z=32 from the 1, tanc13,d 10 x+ 15 y +20 z-80 10x-12y+ 6 z=12 by subtr.27 y + 14 .z--= 68 and 5 y + 22 z=32 hence 135 y + 70 z=340 and 135 y +594 z=864 by subtr. 524 z =524 z -- 1 5y+22 z= 32 that is, 5 y + 22 =32 5 y = 32 —22 --,---- 10 10 y =-3—=2 2 x+3 y+4 z 16 thatis,2x+ 6+4 16 2 x = 16 — 6 — 4 -= 6 x 3.
The same method may be applied to ally number of simple equations.
That the unknown quantities may have definite values, there must be as many independent equations as unknown quan titles.
Thus, if x+y=a,x.,---a—y; and assuming y at pleasure, we obtain a value of s, such that x+ y = a.
These equations must also be inde pendent, that is, not deducible one from another.
Letx+y=a,and 2x+2 y--- 2 a; this latter equation being deducible from the former, it involves no different sup position, nor requires any thing more for its truth, than that x+ y = a should be a just equation.