Solve: x/a+y/b=a+b , x/a^2 +y/b^2=2 ; (a,b ≠ 0) [Linear equations in two variables]
$\frac{x}{a} + \frac{y}{b}$ = a + b —–(i)
$\frac{x}{a^2} + \frac{y}{b^2}$ = 2 —–(ii)
Multiplyig equation (i) with $\frac{1}{a}$ we get,
$\Rightarrow \frac{x}{a^2} + \frac{y}{ab}$ = $\frac{a + b}{a}$
$\Rightarrow \frac{x}{a^2} + \frac{y}{ab}$ =$ \frac{a}{a} + \frac{b}{a}$
$\Rightarrow \frac{x}{a^2} + \frac{y}{ab}$ = $1 + \frac{b}{a}$ —-(iii)
Subtracting equation (ii) from equation (iii) we get,
($\frac{x}{a^2} + \frac{y}{ab}$)-($\frac{x}{a^2} + \frac{y}{b^2}$)=$1 + \frac{b}{a}$ – 2
$\Rightarrow\frac{y}{ab} – \frac{y}{b^2} = \frac{b}{a} – 1$
$\Rightarrow \frac{by – ay}{ab^2}$ = $\frac{b – a}{a}$
$\Rightarrow \frac{y(b – a)}{b^2}$ = b – a
$\Rightarrow y$ = $b^2$
Substituting the value of y in equation (i) we get ,
$\frac{x}{a} + \frac{\left(b^2\right)}{b}$ = a + b
$\Rightarrow \frac{x}{a}$ + b = a + b
$\Rightarrow \frac{x}{a}$ = a
$\Rightarrow x$ = $a^2$
∴ Required Solutions are x = a2 and y = b2