Solution set of a consistent linear system with given rank

Let $A$ be a $5\times3$ real matrix of rank $2$. Let $b\in\mathbb R^5$ be a non-zero vector that is in the column space of $A$. Let $S=\{x\in\mathbb R^3:Ax=b\}$. Define the translation of a subspace $V$ of $\mathbb R^3$ by $x_0\in\mathbb R^3$ as the set $x_0+V=\{x_0+v:v\in V\}$. Then