If you were asked to solve a system of equations in which there is no linear equation to start with, can you sometimes begin by isolating and substituting a variable that is squared in both equations?

I believe you can. You would want to start by isolating a variable that way you can plug it in in the other equation. For example, let's say I have:

3x+y=15
And
6x+12y=0

Start by taking

3x+Y=15
You'd want to divide each side by 3x, making your equation 15/3x=Y

Now that you have a separated variable, you can plug it in like:

6x+12(15/3x)=0