Solve the following systems of equations:

The given equations are:

$x+y=2 x y \ldots(i)$

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

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

Add both equations we get

$x+y=2 x y$

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