Solve the following systems of equations:

The given equations are:

$\frac{5}{x+y}-\frac{2}{x-y}=-1$

$\frac{15}{x+y}+\frac{7}{x-y}=10$

Let $\frac{1}{x+y}=u$ and $\frac{1}{x-y}=v$ then equations are

$5 u-2 v=-1 \ldots(i)$

$15 u+7 v=10 \ldots(i i)$

Multiply equation $(i)$ by 7 and equation $(i i)$ by 2 and add both equations, we get

Put the value of $u$ in equation $(i)$, we get

$5 \times \frac{1}{5}-2 v=-1$

$\Rightarrow-2 v=-2$

$\Rightarrow v=1$

Then

$\frac{1}{x+y}=\frac{1}{5}$

$\Rightarrow x+y=5$

$\frac{1}{x-y}=1$

$\Rightarrow x-y=1$

Add both equations, we get

Put the value of $x$ in first equation, we get

$3+y=5$

$\Rightarrow y=2$

$\Rightarrow y=2$

Hence the value of $x=3$ and $y=2$.