# Group Product/Examples/a x b = c

## Examples of Operations on Group Product

Solve for $x$ in:

$a x b = c$

## Solution

 $\displaystyle a x b$ $=$ $\displaystyle c$ $\quad$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle a^{-1} a x b$ $=$ $\displaystyle a^{-1} c$ $\quad$ Group Product of both sides with $a^{-1}$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle a^{-1} a x b b^{-1}$ $=$ $\displaystyle a^{-1} c b^{-1}$ $\quad$ Group Product of both sides with $b^{-1}$ $\quad$ $\displaystyle \leadsto \ \$ $\displaystyle x$ $=$ $\displaystyle a^{-1} c b^{-1}$ $\quad$ Group Axiom $G \, 3$: Inverses $\quad$

$\blacksquare$