When do variables cancel out?

Question

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?

Answer 1

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 $(x, y) = (\frac{1}{2}, -\frac{1}{4})$

Answer 2

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.

Answer 3

As John pointed out, there are actually two solutions of $x$ in the first equation, one of which doesn't cancel out. However, your question was why, after substituting for $x$ in the second equation, $y$ cancelled and your equation became independent of it. There is a nice geometrical interpretation of this: When you take the positive square root of your first equation, your are left with $$x=2y \quad x=2y+1$$ Which are the equations of two distinct parallel lines - as a result, they never intersect.

In general, if you have an equation in which all your variables cancel out, then it tells you that your equation is either true or false, independently of the vaues your variables take. In your example, you arrive at the equation $0=1$, which is false. No matter what values you take for $x$ and $y$, it will never be true.