Why Adding Two Equations To Each Other Works In Solving a System of Linear Equations

 A long time ago, when I was taking algebra in high school, I wondered why you can add two equations to each other to assist you in solving a system of linear equations, but I never really looked into it.

But the other day I thought about it some and I realized why it works.

Say you have two equations:  3x+7y = 4 and 5x-2y = 6.

Now you can add something to both sides of an equation without changing it.  Pretty much whatever you want.  So let's add (5x-2y) to both sides of 3x+7y = 4.

We get 3x+7y + (5x-2y) = 4 + (5x-2y)

BUT the key is, we also KNOW that 5x-2y = 6, so we can just replace (5x-2y) on one side of the equation with 6.

And we get this: 3x+7y + (5x-2y) = 4 + (6)

Which is just adding the two equations together.  (3x+7y = 4) + (5x-2y = 6).

So really when you add two equations together, you're adding the same thing to both sides and then doing a substitution.  And adding the equations directly is just a shortcut.  But a lot of times it's hard to see how the shortcuts work when you don't do them the long way.

And of course it's probably more common to add a multiple of one equation to another rather than to add them together directly, but the idea is still the same.

What is the answer in this example?  I have no clue.  But as Tom Lehrer once said, the important thing is to know what you're doing rather than to get the right answer.  That's probably why a lot of grad school textbooks don't give you the answers to most of the worked problems.