Group Product/Examples/a x b = c

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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 $\quad$

$\blacksquare$


Sources