When Two Equals One

Given that, a = b
It ensues that, a² = a b
Hence also, a² - b² = ab - b²
Clearly, (a + b) (a - b) = 2b (a - b)
And, (a + b) (a - b) = b (a - b)
Divide both sides by (a - b), they cancel each other out...
Then, (a + b) = b
And since a = b, then logically (a + a) = a
Restated, 2a = a
Dividing by "a", 2 = a/a
Therefore 2 = 1

Where is the error? Is there an error? What is the error?