Okay so here's what I tried: $ a + x = b\\\implies (-a) + (a + x) = (-a) +b \\\implies (-a+a)+x=b+(-a) \\\implies 0 + x = b + (-a) \\\implies x = b+(-a)$
But I'm not sure whether this shows that the solution is unique. I've only shown that $b+(-a)$ is one solution. Maybe there is some other combination of $b$ and $a$ that'll work. But I don't know how to prove that this is the only one.