Suppose that $\vect{w}$ is one solution to the linear system of equations $\linearsystem{A}{\vect{b}}$. Then $\vect{y}$ is a solution to $\linearsystem{A}{\vect{b}}$ if and only if $\vect{y}=\vect{w}+\vect{z}$ for some vector $\vect{z}\in\nsp{A}$.