How do you eliminate a variable to solve an equation

You can't remove a variable but you can separate the variable from the rest of the equation. If you are adding for example 64+x=88 you would subtract the 64 from 88. You will get the value of x. For anything else, you would do the opposite of whatever you are doing. Multiply if you are dividing, add if you are subtracting and hence forth. I hope this helped.

I saw your comment on the previous answer and I believe I solved the equation...
3x+y=21/-x+y=-7

If you take the 3x+y=21 and divide the 3 on each side you get x+y=7

If you take the -x+y=-7 and divide the -1 that is attached to the x you get x+y=7

The the equation really ends up being
x+y=7/x+y=7 which are equal, so when the divide each other the equation is just equal to 1