It's simple.I would say that it is mathmatical proof that there is something wrong with claiming a=b. I never seen any algebra books that claimed that $$ a^{2} = ab $$ from then on it is just nonsense.

When you substitute equations into other equations, then it makes the variable have to agree with all of those equations. So then by saying that a=b, and then factoring out a zero from both sides, the solution to "a" and "b" would then have to be zero. It is the only number that you could factor out a zero and not change its value. Everything multiplied by zero is zero, so if it had a zero multiplied in it and you factored that out, then that number would have to be zero.

I think this is why they say "a" does not equal "b", and they cannot be equal to zero. The algebra just wouldn't work out anymore, as shown. I think this is a good example of that. It is never said that "a" can equal "b" in any fundemental math principals.

A = B (FINE)

Then multiply both sides by A, so we get A^2 = AB (FINE)

Subtract B^2 from both sides, so we A^2 - B^2 = AB - B^2 (FINE)

Divide both sides by (A-B), so we get A + B = B (NOT FINE!!)