ax+by=c a^2x+b^2y=c^2 Solve simultaneous linear equations in two variables by the method of elimination

ax+by=c a^2x+b^2y=c^2 Solve simultaneous linear equations in two variables by the method of elimination

ax+by=c —-(i)

a²x+b²y=c² —-(ii)

Equation (i) is multiplies by a and equation (ii) by 1 we get,

a²x+aby= ac —-(iii)

Now equation (iii) is subtracted from (ii) we get,

(a²x+b²y)-(a²x+aby)=c²-ac

⟹ a²x +b²y -a²x -aby = c² -ac

⟹ b²y -aby = c(c-a)

⟹ by(b-a) = c(c-a)

⟹ y=$\frac{c(c – a)}{b(b – a)}$

⟹ y =$\frac{c(a – c)}{b(a – b)}$

Substituting the value of y in equation (i) we get,

$ax + b\left[\frac{c(a – c)}{b(a – b)}\right]$ = c

$\Rightarrow ax + \frac{c(a – c)}{(a – b)}$ = c

$\Rightarrow ax = c – \frac{c(a – c)}{(a – b)}$

$\Rightarrow ax$ = $\frac{c(a – b) – c(a – c)}{(a – b)}$

$\Rightarrow ax$ = $\frac{ca – bc – ca + c^2}{(a – b)}$

$\Rightarrow ax$ = $\frac{c^2 – bc}{(a – b)}$

$\Rightarrow x$ = $\frac{c(c – b)}{a(a – b)}$

∴ Required Solutions are x = $\frac{c(c – b)}{a(a – b)}$ and y =$\frac{c(a – c)}{b(a – b)}$