Solve :

x + y = 2xy

x - y = 6xy

Given, equations : x + y = 2xy and x - y = 6xy

Dividing both the sides of first equation by xy, we get :

$\Rightarrow \dfrac{x + y}{xy} = \dfrac{2xy}{xy} \\[1em] \Rightarrow \dfrac{x}{xy} + \dfrac{y}{xy} = 2 \\[1em] \Rightarrow \dfrac{1}{y} + \dfrac{1}{x} = 2 \text{ …….(1)}$

Dividing both the sides of second equation by xy, we get :

$\Rightarrow \dfrac{x - y}{xy} = \dfrac{6xy}{xy} \\[1em] \Rightarrow \dfrac{x}{xy} - \dfrac{y}{xy} = 6 \\[1em] \Rightarrow \dfrac{1}{y} - \dfrac{1}{x} = 6 \text{ …….(2)}$

Adding equations (1) and (2), we get :

$\Rightarrow \Big(\dfrac{1}{y} + \dfrac{1}{x}\Big) + \Big(\dfrac{1}{y} - \dfrac{1}{x}\Big) = 2 + 6 \\[1em] \Rightarrow \dfrac{2}{y} = 8 \\[1em] \Rightarrow y = \dfrac{2}{8} = \dfrac{1}{4}.$

Substituting value of y in equation (1), we get :

$\Rightarrow \dfrac{1}{\dfrac{1}{4}} + \dfrac{1}{x} = 2 \\[1em] \Rightarrow 4 + \dfrac{1}{x} = 2 \\[1em] \Rightarrow \dfrac{1}{x} = 2 - 4 \\[1em] \Rightarrow \dfrac{1}{x} = -2 \\[1em] \Rightarrow x = -\dfrac{1}{2}.$

Hence, $x = -\dfrac{1}{2} \text{ and } y = \dfrac{1}{4}$.