Question

Let $\mathit{a}\mathbf{,}\mathit{b}\in \mathit{G}$. Prove that the equations$\mathit{a}\mathit{x}\mathbf{=}\mathit{b}$ and$\mathit{y}\mathit{a}\mathbf{=}\mathit{b}$each have a unique solution in G. [Hint: Two things must be done for each equation: First find a solution and then show that it is the only solution.]

Short answer

It is proved that, $ax=b$and$ya=b$ have unique solutions.

Step-by-step solution

Determine solution

Consider the given functions, $ax=b$and $ya=b$.

The solution for $ax=b$and $ya=b$are$x=a^{-1}b$ and $y=a^{-1}b$ respectively.

Determine solution are unique

Now consider the solution $x=c$for $ax=b$then,

Multiplying by$a^{-1}$on left-hand side then the solution is,$c=a^{-1}b$.

Similarly,

the solution $y=d$for $ya=b$then,

Multiplying by$a^{-1}$on right-hand side then the solution is, $d=a^{-1}b$.

Hence the solution are unique.