Greetings!
On a test recently I ended up having to solve this for y:
$$ x = \frac{2y}{y + 1} $$
But I kept getting stuck in cirlces...
$$ \begin{aligned} x(y + 1 ) = 2y \\ xy + x = 2y \\ \frac{xy + x}{2} = y \\ \end{aligned} $$
That didn't get me anywhere, so then I started over and tried multiplying both sides by the reciprocal:
$$ \begin{aligned} (\frac{y+1}{2y})(x) = 1 \\ \frac{xy+x}{2y} = 1 \\ \end{aligned} $$
And still I can't see a way to isolate y.
The worst part of this is that I remember being specifically taught a trick for this particular conundrum, but I can't remember the trick!
Wolfram gives the answer as
$$ y = - \frac{x}{x-2} $$
but it doesn't show the steps.