Solve the following systems of equations:

The given equations are:

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

$\frac{10}{x+1}+\frac{2}{y-1}=\frac{5}{2}$

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

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

$10 u+2 v=\frac{5}{2} \ldots(i i)$

Add both equations, we get

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

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

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

$\Rightarrow-2 v=-\frac{1}{2}$

$\Rightarrow v=\frac{1}{4}$

Then 

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

$\Rightarrow x+1=5$

$\Rightarrow x=4$

$\frac{1}{y-1}=\frac{1}{4}$

$\Rightarrow y-1=4$

$\Rightarrow y=5$

Hence the value of $x=4$ and $y=5$.