hi,
multiplying by infinity makes no sense at all.

You know, multiplying by zero makes no sense at all too.

Your example where:
y1 = INF and y2 = INF, and finally y1 = y2 is valid.
may be easy modifyed to "y1 = 0 and y2 = 0, and finally y1 = y2 is valid" if we try to decrease 'a' instead growing.

Finally, if somebody still does not understand my bad English: the mistake is not at elimination common terms but some lines before it, when these common terms are inserted into the equation. The mistake is neither that zero=zero nor INF is undefined; the mistake is that such manipulations are prohibited with Gauss' method for solving linear equations.
Really, what will you say about man who goes across a street at the red light signal and then asks why a car knocks down him? He will discuss about the colour of a car, about that it moves wrong direction etc... But the mistake is that he went across at the red light signal and no more.
Here we have the same: we broke the rules and then come very surprised that 1=2. (=

Example about nonsence/rules: 0/0=NaN (not a number) is wrong! As I learned, a/a is always =1, so 0/0 = 1. If you say "no" then it means that zero in not a number too ((=
So what I say: there are rules and we must follow them.

Regards,
Oleg