True, but it is still valid algebra. There are other examples where one might turn a false statement into a true statement without doing something as pointless

True, but it is still valid algebra. There are other examples where one might turn a false statement into a true statement without doing something as pointless as multipling by zero. A common example is squaring an equation. Consider a system of equation in which it can be shown that x=3. Within this system, the statement x=-3 is false, however the statement x^2=9 is true. Again, this specific example is contrived, but this does come up annoyingly often.