Fix a matrix A and a vector b. Suppose that y is any solution of the homogeneous system Ax=0 and that z is any solution of the system Ax=b. Show that y+z is also a solution of the system Ax=b.

Since y is a solution of the homogeneus system then satisfies Ay=0.

Since z is a solution of the system Ax=b then satisfies Az=b.

Now, we will show that A (y+z) = b.

Observe that A (y+z) = Ay+Az by properties of the product of matrices.

By hypotesis Ay=0 and Az=b.

Then A (y+z) = Ay+Az=0+b=b.

Then A (y+z) = b, this show that y+z is a solution of the system Ax=b.