Solve the following systems of equations:

The given equations are:

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

$\frac{6}{x-1}-\frac{3}{y-2}=1$

Let $\frac{1}{x-1}=u$ and $\frac{1}{y-2}=v$ then equations are

$5 u+v=2 \ldots(i)$

$6 u-3 v=1 \ldots(i i)$

Multiply equation $(i)$ by 3 and add both equations, we get

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

$5 \times \frac{1}{3}+v=2$

$\Rightarrow v=\frac{1}{3}$

Then 

$\frac{1}{x-1}=\frac{1}{3}$

$\Rightarrow x-1=3$

$\Rightarrow x=4$

$\frac{1}{y-2}=\frac{1}{3}$

$\Rightarrow y-2=3$

$\Rightarrow y=5$

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