5m + 2n = 3
-5m + 3n = 7
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I don't get it at all... should I change the bottom -5m + 3n = 7 to -5m - 3n = -7?

No because it's not an equlivelent equation. This is though 5m - 3n = -7.

You don't have to do ANYTHING but add them.
5n=10; remember 5+-5 = 0

Solve for n; find m.

Well, you can change them, but that won't change the solution of the problem. What you need to do is make sure that you don't leave mistakes.

Because, the one you did, you multiplied only two of the three terms by -1, and what about the -5m? It should be 5m instead.