Theorem 4.16

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If $G$ is a group with binary operation $*$, and if $a$ and $b$ are any elements of $G$, then the linear equations
$a * x = b$ and $y * a = b$ have unique solutions.


First we show the existence of at least one solution by just computing that $a' * b$ is a solution of $a* x = b$.
Note that:
$a * (a' * b) = (a * a') * b$ by associative law
$= e * b$ definition of $a'$
$=b$ property of $e$

Thus $x = a' * b$ is a solution of $a * x = b$. In a similar fashion, $y = b * a'$ is a solution of $y * a = b$.

To show uniqueness of $y$, we use the standard method of assuming that we have two solutions, $y_{1}$ and $y_{2}$ so that $y_{1} * a = b$ and $y_{2} * a = b$. Then $y_{1} * a = y_{2} * a$ and by theorem 4.15 $y_{1} = y_{2}$. The uniqueness of x follows similarly.

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