Question

Add:$\frac{5x}{xy-y^2}+\frac{2x}{x^2-y^2}$

Short answer

The addition of the rational expression is$\frac{x\left(5x+7y\right)}{\left(x+y\right)\left(x-y\right)y}$

Step-by-step solution

Given information 

The given rational expression $\frac{5x}{xy-y^2}+\frac{2x}{x^2-y^2}$.

Simplify the rational expression 

The LCD of the expression is$\left(x+y\right)\left(x-y\right)\left(y\right)$$$\begin{gathered} =\frac{5x}{y\left(x-y\right)}+\frac{2x}{\left(x+y\right)\left(x-y\right)} \\ =\frac{5x\left(x+y\right)}{y\left(x-y\right)\left(x+y\right)}+\frac{2xy}{\left(x+y\right)\left(x-y\right)y} \\ \mathrm{Add}\mathrm{the}\mathrm{rational}\mathrm{expression} \\ \frac{=5x^2+5xy+2xy}{\left(x+y\right)\left(x-y\right)y} \\ =\frac{5x^2+7xy}{\left(x+y\right)\left(x-y\right)y} \\ \mathrm{Simplify} \\ =\frac{x\left(5x+7y\right)}{\left(x+y\right)\left(x-y\right)y} \end{gathered}$$