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?

You need to be careful about finding squareroots. For example, 
$x^2 = 4y^2$ gives $x = \pm 2y$ instead of just $x = 2y$. 
I would personally start with the linear equation and substitute that into the quadratic to get you the answers.
i.e.
substitute $x = 2y + 1$ into $x^2 = 4y^2$ which gives:
$(2y + 1)^2 = 4y^2$
This gives you a solution of $(x, y) = (\frac{1}{2}, -\frac{1}{4})$
