both sides of equation

Hi,

Looking at the basic algebraic rules, I always see commutativity,
associativity, distrib., inverse and identity properties for mult. and
addition. One of the most common things I do in algebra is add,
subtract, multiply and divide *both sides of an equation* by the same
thing, and yet, I don't see how that practice fits in with any of the
basic rules. Where does it say that whatever you do to one side of an
equation you have to do to the other? Can anyone help?

Thanks

.