Rearranging a formula containing a non-square matrix

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Given $4 \times 2$ matrix $V$ and $4 \times 1$ matrix $K$, solve the linear system $V A = K$ for $A$.

If V was a square matrix I could just multiply by the inverse, but I cannot since it is not. Apparently, the solution is of the form

$$A=(V'*V)\(V' K)$$

where $V'$ is the transpose of $V$.

What is this called? The algebra seems to be out of nowhere. I know that the transpose of $V$ theoretically is cancelled out, but it seems like dividing which is not allowed in matrix algebra.