Consider the system of linear equations $\linearsystem{A}{\vect{b}}$, and suppose that every element of the vector of constants $\vect{b}$ is a common multiple of the corresponding element of a certain column of $A$. More precisely, there is a complex number $\alpha$, and a column index $j$, such that $\vectorentry{\vect{b}}{i}=\alpha\matrixentry{A}{ij}$ for all $i$. Prove that the system is consistent.