Hunting The Variables!

We can rewrite the expression as:

$\begin{aligned} \dfrac xy + x &= \dfrac yx + y\\ \dfrac xy - \dfrac yx &=(y - x)\\ \end{aligned}$

Now we can solve this expression to get our desired result:

$\begin{aligned} \dfrac{x^2 - y^2}{xy} &= (y - x)\\ \dfrac{(x+ y)(x-y)}{xy} &= (y - x)\\ \dfrac{(x+ y)}{xy} &= -1\\ \dfrac 1x + \dfrac 1y &= -1\\ \end{aligned}$