How to prove the uniqueness of the solution of $ax+b=0$? I have no background in mathematical analysis or the like, but I am interested to know how to prove the uniqueness of the solution of $ax+b=0$? Perhaps your answers will help me to prove other uniqueness problems.

Suppose $x$ and $y$ are both solutions to your equation. Then we have:
$ax + b = 0$
and
$ay + b = 0$
Subtracting the two equations gives
$a(x - y) = 0$
So if $a$ is non-zero, then $x-y$ must be zero, i.e. $x = y$.
