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?
Equation 1:
x + 2y = 35
Equation 2:
3x + 14 y = 120
Let's say we multiply both sides of equation 1 by 3 so we can subtract it from equation 2:
3*(x + 2y) = 3*35
3x + 6y = 105
Let's alter equation 2:
3x + 14y - 105 = 120 - 105
Subtracting 105 from both sides didn't change anything, right? It's perfectly valid to do?
But we know that 105 is equal to (3x + 6y).
So we can re-write that as 3x + 14y - (3x + 6y) = 120 - 105
Which becomes 8y = 15.
This is the same as subtracting the equations directly
3x + 14 y = 120
-3x -6y -105
8y = 15
I hope that makes it clear.
