I want to fit a plane which has the following equation: $ax+by+cz+d=0$ Having a overdetermined system (more than 3 points), most people solve that equation using the normal equations (Least square). Because most people minimize over $z$-Direction, they set $c=1$ and therefore obtain the following equation $ax+by+d=-z$ which can be solved using the normal equations.
Now my question: Why can they choose $c=1$? Can I also choose $b=1$ and obtain $ax+cz+d=-y$ ? Does the selection depend on anything?