Solve the following simultaneous equations:

x + y = 2xy, x − y = 6xy

Given,

Equations :

⇒ x + y = 2xy     ….(1)

⇒ x - y = 6xy     ….(2)

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

⇒ x + y + (x - y) = 2xy + 6xy

⇒ x + y + x - y = 8xy

⇒ 2x = 8xy

⇒ 2 = 8y

⇒ y = $\dfrac{2}{8} = \dfrac{1}{4}$.

Substituting $y = \dfrac{1}{4}$ in equation (1), we get :

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

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