Prove why the 'elimination method' of solving simultaneous equations works. [This has been deleted due to me stressing out over something obvious that I didn't see; apologies if I seemed aggressive, I was merely stressed. Thanks.

I've been searching around the internet looking for an explanation for my Algebra students as to why one can solve a system by elimination. It's not immediately clear as to why you can just "combine" equations, resulting in a new equation (which if graphed, creates a new line, but happens to go through the same solution point!)
My students are comfortable with the idea of substitution (sure, if something is equivalent, then I can replace it somewhere else in the equation with what it's equivalent to!). The intuition is lost when we get to solving by elimination. Well, finally I've come across an accessible answer for these 7th and 8th graders! These are my notes:
Question: Why does the elimination method work?
My two equations are:


*

*$x-2y=1$

*$3x+2y=11$
We know that we can add something to both sides of an equation and maintain equality. For example I can take
$3x+2y=11$
and add 1 to both sides to get
$3x+2y+1=12$
And from the first equation, we know that $x-2y=1$, which we can substitute back in.
$3x+2y+1=12$ becomes $3x+2y+x-2y=12$
Combine like-terms to get $4x=12$. Solve for $x$, then solve for $y$. 
Then I guess I would show my students how doing it in columns saves a lot of time, and also makes it more obvious of multiplying the equations by some constant to ensure they will eventually eliminate a variable.
