If $ax+by=a^{2}-b^{2}$ and $bx+ay=0$, then find the value of $( x+y)$.

Given: $ax+by=a^{2}-b^{2}$ and $bx+ay=0$.

To do: To find the value of $( x+y)$.

Solution: 

Given equations are $ax+by=a^{2}-b^{2}$ and $bx+ay=0$

On adding both equations,

$ax+by+bx+ay=a^{2}-b^{2}+0$

$\Rightarrow ax+ay+bx+by=a^{2}-b^{2}$

$\Rightarrow a( x+y)+b( x+y)=( a+b)( a-b)$

$\Rightarrow ( a+b)( x+y)=( a+b)( a-b)$

$\Rightarrow ( x+y)=( a-b)$

Thus, $x+y=a-b$.