Question

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

Solve for a

$\left\{\begin{matrix}a=2x-b\text{; }a=0\text{, }&x\neq b\text{ and }x\neq 0\\a\neq x\text{, }&b=0\text{ and }x\neq 0\end{matrix}\right.$

Solve for b

$\left\{\begin{matrix}b=-\left(a-2x\right)\text{; }b=0\text{, }&x\neq 0\text{ and }a\neq x\\b\neq x\text{, }&a=0\text{ and }x\neq 0\end{matrix}\right.$