Fastest way to solve these kinds of fractions (equal to zero and have two variables)? I'm trying to solve this:
$$0=\frac{4x-2y}{-5x+y}$$
I can't use partial fractions since there are two variables...
Also, given that there are 2 unknowns and only one equation then I assume one of the variables is "free" and the other is "forced" to be a solution constrained to whatever the free variable is.
What's the fastest way to solve this without inspection or trial-by-error and is it possible to get a solution in terms of $x$ and $y$ only?
EDIT: Too stupid to realize the obvious. As Git Gud hinted, this fraction is only equal to zero if the numerator is zero (and the denominator is non-zero). Therefore, this can be completed by solving for the numerator alone:
$$4x-2y=0$$
$$4x=2y$$
$$2x=y$$

The correct answer to this problem is that the solution set is
$$
\{ (x, 2x) :  x \ne 0 \}
$$
i.e., all pairs of the form $(x, 2x)$ where $x$ is any real number except $0$. 
What's wrong with $(0, 0)$? Plug it into your original equation, and you get
$$
0 = \frac{4x - 2y}{-5x + y} = \frac{4\cdot 0 - 2\cdot 0}{-5\cdot 0 + 1\cdot 0} = \frac{0}{0}.
$$
But the fraction $0/0$ does not equal zero -- it's in fact undefined. So the pair $(0,0)$ does not satisfy the original equation, but all other $(x, 2x)$ pairs do.   
