TITLE: Algebra: What allows us to do the same thing to both sides of an equation?
QUESTION [50 upvotes]: I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. What I do not understand is why making a new equation (adding or taking away from an expression) allows one to know what an unknown represents. What about equations lets us do this?

REPLY [3 votes]: It really comes down to the basic properties of equality: the reflexive property and substitution. 
Equality Property of Addition: Say $a = b$. Obviously $a+c = a+c$ (reflexive property). So $a+c = b+c$ (substituting $b$ for $a$ on the right side). 
Equality Property of Multiplication: Say $a = b$. Obviously $ac = ac$ (reflexive property). So $ac = bc$ (substituting $b$ for $a$ on the right side). 
And so forth.