Question

Express the following as rational expression.
$\frac{x}{x-y}+\frac{y}{x+y}+\frac{2xy}{y^2-x^2}$

Answer 1

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

The expression can be re-written as,

$\frac{x}{x-y}+\frac{y}{x+y}+\frac{2xy}{\left(y-x\right)\left(y+x\right)}$

$\Rightarrow \frac{x}{x-y}+\frac{y}{x+y}-\frac{2xy}{\left(x-y\right)\left(x+y\right)}$ $(\therefore \left(y-x\right)=-\left(x-y\right))$

$\Rightarrow$ Now, the LCM of $\left(x-y\right),\left(x+y\right)\&\left(x-y\right)\left(x+y\right)$ is $\left(x-y\right)\left(x+y\right)$

Hence, the expression is $\left(\frac{x\left(x+y\right)+y\left(x-y\right)+\left(-2xy\right)}{\left(x+y\right)\left(x-y\right)}\right)$

$=\frac{x^2+xy+xy-y^2-2xy}{\left(x+y\right)\left(x-y\right)}$

$=\frac{x^2-y^2}{\left(x+y\right)\left(x-y\right)}$

$=\frac{x^2-y^2}{x^2-y^2}$

$=1$

Hence, the expression reduces to $\frac{1}{1}=1.$

Hence, the answer is $1.$