Is there a law that you can add or multiply to both sides of an equation? It seems that given a statement $a = b$, that $a + c = b + c$ is assumed also to be true.
Why isn't this an axiom of arithmetic, like the commutative law or associative law?
Or is it a consequence of some other axiom of arithmetic?
Thanks!
Edit: I understand the intuitive meaning of equality. Answers that stated that $a = b$ means they are the same number or object make sense but what I'm asking is if there is an explicit law of replacement that allows us to make this intuitive truth a valid mathematical deduction. For example is there an axiom of Peano's Axioms or some other axiomatic system that allows for adding or multiplying both sides of an equation by the same number? 
In all the texts I've come across I've never seen an axiom that states if $a = b$ then $a + c = b + c$. I have however seen if $a < b$ then $a + c < b + c$. In my view $<$ and $=$ are similar so the absence of a definition for equality is strange.

You're perhaps missing the point of people saying 'That's what equality means.'  This is the natural result of equality. The basic property of equality is that one equal thing can be substituted in for another.  So if A = B, then you can substitute A in for B, or B in for A, anywhere it is found.  If you want the longer proof:
A = B  (initial proposition)
A = A  (equality substitution)
A + C = A + C  (all things are equal to themselves)  
A + C = B + C  (equality substitution)

