Good question!

Yes, this same formula extends to prove the result for any number of equations. With N equations there are N terms added up. Each term follows the same pattern with a general $b_i$, $c_i$, and product of all the other mods. I think the general formula makes this more confusing than they actually are, but hopefully the idea makes sense.

Also, to be clear, other than understanding the proof, you do NOT want to use this formula for solving these systems (unless you're writing a computer program or something). It is much easier to solve them directly/recursively looking at two equations at a time.