Fallacy or Paradox?

Start with a simple identity,

a = b

Multiply each side by a:

a^2 = a b

Subtract b^2 (b squared) from each side of the equality:

a^2 – b^2 = a b – b^2

Using high school algebra, factor out the term (a – b) from each side:

(a + b) (a – b) = b (a – b)

Divide each side by (a – b), effectively canceling out the term from each side, leaving:

(a + b) = b

Per our opening equality, substitute a for b:

a + a = a

Combine terms:

2 a = a

Factor out the a’s, and you’re left with:

2 = 1

Where did we go wrong???