Uniqueness Proof, Discrete Math Help

To show an object is unique one approach (the one taken here) is to assume that there is a second object that satisfies the given conditions. Then if you can show that this second object is actually the first object then you've shown that all objects that satisfy the condition are identical. In particular, by supposing that $s$ and $r$ are both arbitrary solutions to $ax+b=0$ and showing that $s=r$, you've shown that every solution to $ax+b=0$ is identical, i.e. the solution is unique (if it exists). Note that you can show uniqueness without showing existence.