To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $x-y$ and $x+y$ is $\left(x+y\right)\left(x-y\right)$. Multiply $\frac{1}{x-y}$ times $\frac{x+y}{x+y}$. Multiply $\frac{1}{x+y}$ times $\frac{x-y}{x-y}$.
Since $\frac{x+y}{\left(x+y\right)\left(x-y\right)}$ and $\frac{x-y}{\left(x+y\right)\left(x-y\right)}$ have the same denominator, subtract them by subtracting their numerators.
Since $\frac{2y}{\left(x+y\right)\left(x-y\right)}$ and $\frac{2x}{\left(x+y\right)\left(x-y\right)}$ have the same denominator, add them by adding their numerators.
