One Equals Two

Consider the following explanation for, why 1=2 :
1. Start out,                                                   Let y = x

2. Multiply through by x                            xy = x^2

3. Subtract y^2 from each side                xy – y^2 = x^2 – y^2

4. Factor each side                                      y(x-y) = (x+y)(x-y)

5. Divide both sides by (x-y)                     y = x+y

6. Divide both sides by y                            y/y = x/y + y/y

7. And so…                                                    1 = x/y + 1

8. Since x=y,       x/y = 1                             1 = 1 + 1

9. And so…                                                  1   =   2

How is this possible? Which step is invalid in the above calculation?