Question

Solve for $x$ and $y$:
$\frac{x}{a}+\frac{y}{b}=a+b$
$\frac{x}{a^2}+\frac{y}{b^2}=2$

Answer 1

$\frac{x}{a}+\frac{y}{b}=a+b$

$\Rightarrow bx+ay=a^{2}b+b^{2}a.....\left(1\right)$

$\frac{x}{a^2}+\frac{y}{b^2}=2.....\left(2\right)$

$\Rightarrow b^{2}x+a^{2}y=2a^2{b}^2.....\left(3\right)$

Multiplying equation $\left(1\right)$ by $a$, we get

$abx+a^{2}y=a^{3}b+a^2{b}^2.....\left(4\right)$

Subtracting equation $\left(4\right)$ from $\left(3\right)$, we have

$(b^2-ab)x=a^2{b}^2-a^{3}b$

$\Rightarrow x=\frac{a^2(b^2-ab)}{(b^2-ab)}=a^2$

Putting $x=a^2$ in equation $\left(2\right)$, we have

$\frac{a^2}{a^2}+\frac{y}{b^2}=2$

$\Rightarrow y=b^2$

Hence $x=a^2,y=b^2$