Writing A Program That Does Algebra For Me

I want to try and write a program that can solve algebra problems in C++, how would I go about doing this (I already know some C++, I just need a basic idea of how to go about using variables, constants, etc. to do algebra)?

It's called numerical programming.
Any decent 4 year college will offer a specialized CS degree in it.
Go buy the text book. Make sure it covers Gaussian Elimination.