The last step, the division by (a-b).a=b for some a, some b

a+a=a+b (add a to both sides)

2a=a+b

2a-2b=a+b-2b (subtract 2b from oth sides)

2(a-b)=a+b-2b [2a-2b=2(a-b)]

2(a-b)=a-b

2 = 1 (divide by a-b)

One step here is wrong, or so the teacher says. Which one is it?

You said a=b at the beginning. If that is so, then a-b = 0.

Hence, since you cannot divide by zero, then the last step is wrong.