TITLE: How to solve this rational equation for y
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: Another way to arrive at the conclusion, if you don't "see" factors jumping out at you is this: When you see a fraction like $\frac{2y}{y+1}$, I am sure you feel the urge to split into summands. Now, of course, you cannot do it with the sum in the denominator, so that suggests inverting the whole thing (assuming that everything that you want to be non-zero is non-zero):
$$
\frac{1}{x} = \frac{y+1}{2y} = \frac{y}{2y} + \frac{1}{2y} = \frac{1}{2} + \frac{1}{2y}.
$$
Now subtract 1/2 and invert back again to get the result.