Division Laws for Groups

Theorem
Let $G$ be a group.

Let $a, b, x \in G$.

Then:
 * $a x = b \iff x = a^{-1} b$
 * $x a = b \iff x = b a^{-1}$

Proof
All derivations can be achieved using applications of the group axioms.

Proof of First Result
and the converse:

Proof of Second Result
and the converse: