When do variables cancel out? Sometimes if I randomly combine different equation and try to solve for a variable, one of them will cancel out. Why?
For example:
$\displaystyle x^2 = 4y^2$ and
$\displaystyle x = 2y + 1$
And solve for $x$. One of the things I would start doing is taking the square root of the first equation, which gets me $x = 2y$. But then plugging this into the second equation to solve for $x$, $x$ will then cancel out. Why is this so?

The reason that the "$x$ cancels out" is because you chose the positive branch and ignored the negative branch of $x^2 = 4y^2$. This can be seen from the graph below. The green line is $x=2y+1$, the blue line is the positive branch of $x^2 = 4y^2$ and the red line is the negative branch of $x^2 = 4y^2$. As you can see the negative branch intersects with the green line whilst the positive branch does not intersect with the green line ever. (It's parallel) $y = \frac{x}{2} - \frac{1}{2}$ and $y = \frac{x}{2}$, one is a vertical translation of the other by $\frac{1}{2}$ units. So if you choose the positive branch, the $x$'s will "cancel" because there are no solutions/intersections. 

