In this explanation, the teacher’s logic is sound up until the cancellation step. Rewrite the first part of the problem (I’ve numbered the steps for easy reference):
If a = b,
- a2 = ab
- a2 – b2 = ab – b2
- (a + b)(a-b) = b(a – b)
Up until this point, the proof is just basic algebra. Since a = b, we can say that a*a = a*b. The two sides of the equation remain equal if you subtract b2 from both terms, as he does in step 2. In step 3, he uses factoring to write the equation without exponents. This gives a common term (a – b) on both sides of the equation.
The next step is the error. He cancels the term on both sides of the equation, but to do this, he has to divide both sides by (a – b). Seems doable, right? Wrong! Because a = b, we can see that a – b = 0. There is no way to divide by zero in math—even if you were to attempt to solve this on your calculator using numbers in place of the variables, it would give you an error message.
After this step, the math is once again legitimate algebra—he substitutes an equivalent term and simplifies the equation to yield 2 = 1, which is, of course, incorrect, due to the error of dividing by zero.