# Geometric Interpretations of the Cramer's Rule

In Cramer's rule when $Δ_x=Δ_y=Δ_z=Δ=0$ what are the possibilities for the three planes (three simultaneous equations in $x,y,z$) to look like? My book says that they can have either infinite solutions or no solutions in such cases. One possibility I can think of is all the planes intersect in a line (i.e. a family of planes). Another is when any two planes are parallel.

• Is there any third possibility in such a case?
• Also, what are the geometric possibilities (in terms of three planes) when $\Delta=0$ but any one or more of $Δ_x,Δ_y,Δ_z$ is non-zero?

P.S: I know that the $\Delta$ vanishing implies that the normal vectors of the three planes are coplanar.

• – Yashas May 4 '17 at 9:36