Prove the part of the theorem which lets w be any solution of ax = b, and defines vh = w - p. show that v_h is a solution of ax = 0. this shows that every solution of ax = b has the form w = p + v_h, with p a particular solution of ax = b and v_h a solution of ax = 0. let w and p be solutions of ax = b. substitute for v_h from the equation w = p+ v_h.