Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero $e$.
What does $e$ represent?
What information does this give us about $(x_0, y_0, z_0)$ in relation to P?
I believe that the set of all points such that $f(x,y,z)=e$ form a plane with equation $ax+by+cz=d+e$. Is this all that's useful about the extra information? The corresponding parallel plane that the point $(x_0, y_0, z_0)$ belongs to?