Let's take the equation $2x = 1 + x$. We know that in the real numbers we can cancel out an $x$ on both sides (since the inverse of $x$ exists) and we get that $x=1$.
However, if we include infinity (take countable infinity as an example), then infinity would also be a valid solution to the equation. It would also be a valid solution for many other problems, such as $x^2 = x$.
Unfortunately, of course, allowing infinity into arithmetic would cause many "laws", such as $x - x = 0$ for all $x$, to break down. My question is - are there any situations where it might be advantageous to treat infinity as a number?