Solve the following systems of equations:

The given equations are:

$\frac{x+y}{x y}=2$

$x+y=2 x y$ ..( $(i)$

$\frac{x-y}{x y}=6$

$x-y=6 x y$$\ldots($ ii $)$

Adding both equations, we get

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

$x+\frac{1}{4}=2 x \times \frac{1}{4}$

$\Rightarrow \frac{-x}{2}=\frac{1}{4}$

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

Hence the value of $x=-\frac{1}{2}$ and $y=\frac{1}{4}$