HMM. I must be dense today. So, when you are adding two equations to find a solution that satisfies both, you are really saying that the equations are equivalent for that solution (or set of solutions) and it's OK to add equivalent things.

So if you have x + 2y = 35 and 3x + 14 y = 120 then you've decided they are equivalent so adding the left sides and the right sides makes them stay equivalent?