Solve for x and y:

$a^{2}x+b^{2}y=c^2,b^{2}x+a^{2}y=d^2.$

Let $a^{2}x+b^{2}y=c^2$---(1)

$b^{2}x+a^{2}y=d^2$---(2)

Multiply (1) with $b^2$ and (2) with $a^2$

$b^2{a}^{2}x+b^{4}y=b^2{c}^2$---(3)

$a^2{b}^{2}x+a^{4}y=a^2{d}^2$---(4)

Subtract (4) from (3), we get

$(b^4-a^4)y=b^2{c}^2-a^2{d}^{2}y=\frac{b^2{c}^2-a^2{d}^2}{b^4-a^4}$

Multiply (1) with $a^2$ and (2) with $b^2$

$a^{4}x+a^2{b}^{2}y=a^2{c}^2$---(5)

$b^{4}x+b^2{a}^{2}y=b^2{d}^2$---(6)

Subtract (6) from (5), we get

$(a^4-b^4)x=a^2{c}^2-b^2{d}^{2}x=\frac{a^2{c}^2-b^2{d}^2}{a^4-b^4}$