Question on relationship between Ax = 0 and Ax = b

Question

This is a true or false question.

"If x1 and x2 are two solutions of the nonhomogeneous linear system Ax = b, then x1 - x2 is a solution of the corresponding homogeneous linear system."

Now I know that the general solution of Ax = b can be found by adding a specific solution (x1 or x2) to the general solution of Ax = 0 but this question seems to suggest that x1 - x2 gives you a specific solution to Ax = 0. Why is this? I think everything is here to make the link but I can't seem to make it. Thanks.

Answer 1

Since $x_1$ and $x_2$ are solutions of $Ax=b$, we get

$$A(x_1-x_2)=Ax_1-Ax_2=b-b=0.$$

We are done !