Can 2 = 1?

Here’s our "proof":

1. Let x = y
2. so
   x² = xy
3. adding x² to both sides of the equation we get
   x² + x² = x² + xy
4. simplifying we get
   2 x² = x² + xy
5. subtract 2xy from both sides and we get
   2 x² – 2xy = x² + xy – 2xy
6. simplifying we get
   2 x² – 2xy = x² – xy
7. factoring for (x² – xy) we get
   2 (x² – xy) = 1 (x² – xy)
8. divide both sides by (x² – xy) we get
   2 = 1

Since 2 can’t equal, 1 there must be something wrong here. What’s wrong with our proof?