Why you can add equations using the Addition Method to solve a system of equations

For the longest time, I never understood how the Addition Method solved a system of simultaneous equations. What I mean to say is that I could carry out the method’s steps, but I didn’t understand why adding two equations was legal. It seemed unintuitive to me.

For example, take this system of two linear equations:


Here is how I would solve it using the Addition Method:


Which is great, but why Step 3? I could understand the motivation for adding two equations to eliminate a variable, but what mathematical principle said it was okay to do that in the first place? It seemed to just about as rational to me as doing a head-stand while holding my nose to get the answer!

So I thought about it obsessively for a couple of hours and eventually realized the following:

The initial system basically says that y-x is 2 and y+x is 4. Conversely, 2 is y-x and 4 is y+x.

Nothing to write home about, but let’s say I want to add 2 and 4. Yes, the answer is 6, but suppose I want the answer to be in terms of x and y.





That is, Step 3! Effectively, we can add up the equations because a side of an equation can stand-in for its equivalent on the other side. The adding of equations is effectively the addition of numbers on the righthand sides of the two equations as the sum of those same numbers in terms of x and y.

For some, perhaps, this entry is a useless explanation of the obvious. Maybe it is, but no matter: now I understand something which I didn’t before, which I think is better than blindly accepting something as true, even if it’s small thing such as adding equations.