How to find disjoint set of two planes in form $ax+by+cz=d$?

Question

How to find disjoint set of two planes in form $ax+by+cz=d$?

How do I need to alter the equations?

Answer 1

Keeping $a,b,c$ fixed and taking different values of $d$ gives parallel planes... just because all these planes share the same normal vector $(a,b,c)$.

For example $x+y+z=1$ and $x+y+z=2$ are equations of parallel planes. A proof of this is that if there was a common point $(x,y,z)$, we would have a proof that $1=2$...