This is a true statement. A proof is:
(⇒) Suppose we have a homogeneous system $\homosystem{A}$. Then by substituting the scalar zero for each variable, we arrive at true statements for each equation. So the zero vector is a solution. This is the content of Theorem HSC.
(⇐) Suppose now that we have a generic (i.e. not necessarily homogeneous) system of equations, $\linearsystem{A}{\vect{b}}$ that has the zero vector as a solution. Upon substituting this solution into the system, we discover that each component of $\vect{b}$ must also be zero. So $\vect{b}=\zerovector$.