Division Laws for Groups

Theorem
Let $G$ be a group.

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

Then:
 * $(1): \quad a x = b \iff x = a^{-1} b$
 * $(2): \quad x a = b \iff x = b a^{-1}$

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

Proof of $(1)$
and the converse:

Proof of $(2)$
and the converse: