Solve the following systems of equations:

The given equations are:

$\frac{22}{x+y}+\frac{15}{x-y}=5$

$\frac{55}{x+y}+\frac{45}{x-y}=14$

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

$22 u+15 v=5 \ldots(i)$

$55 u+45 v=14 \ldots(i i)$

Multiply equation $(i)$ by 3 and subtracting (ii) from (i), we get

$\Rightarrow u=\frac{1}{11}$

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

$22 \times-\frac{1}{11}+15 v=5$

$\Rightarrow 15 v=3$

$\Rightarrow v=\frac{1}{5}$

Then 

$\frac{1}{x+y}=\frac{1}{11}$

$\Rightarrow x+y=11$

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

$\Rightarrow x-y=5$

Add both equations, we get

$x+y=11$

Put the value of  in second equation, we get

$8-y=5$

$\Rightarrow-y=-3$

$\Rightarrow y=3$

Hence the value of $x=8$ and $y=3$.