Equation of Variation y with x

Question

$y-x= 1/x - 1/y$ where x and y are not equal to 0 then y varies ? (Complete)
I am supposed to know if it is direct or inverse variation but I did not manage to get it

Answer 1

If $y - x = 1/x - 1/y$, then we can rearrange as follows: \begin{align*} xy(y - x) &= xy(1 / x - 1 / y) \\ \implies xy(y - x) &= y - x \\ \implies (xy - 1)(y - x) &= 0 \\ \end{align*} So either $xy = 1$ (ie $x = 1/y$, which is inverse variation), or $y = x$, which is direct variation. Indeed you can check that both of these satisfy the equation. So the answer is both! If you plot this equation, you get something like this: