What is a proof, derivation or explanation of the process of solving equations?

Question

This probably sounds a little silly coming from some one taking calculus (maybe I'm confused because I skipped 2 years), but most things (theorems, methods, etc.) that I read in math textbooks are formally proven, however I was thinking about the process of solving equations, and realized I didn't really know any proof or derivation of it, and could find none - it was simply something I was taught to do in elementary school. Also, I have a related question of whether when you perform an operation to both sides of an equation is that the same equation, a different expression of the original equation, a statement that is also true? Thanks to any answers!

Answer 1

I would definitely recommend taking a mathematical foundation class, or something like it.  I took this course and it explained the very logic on why we do things a certain way, this class also helped me prove why we use certain equations and where they came from!  Calculus really is just a more hard form algebra!