What does the solution of a set of three linear equations in three variables looks like when the planes form the shape of triangular prism in $R^3$?

Question

We know that a linear equation in three variables represents a plane in $R^3$ and if we take three such equations then we can find the exact values of the 3 variables that support all the three equations i.e. we find the solution to this system of equations. But how does the 3 equations looks relative to each other when the 3 corresponding planes of them represents 3 faces of a triangular prism? Also, how does this situation differs from the case in which at least 2 pair of planes are parallel to each other?

Answer 1

The solution set of the linear system looks like this: $L = \emptyset$.

The reason is that $L$ is the common intersection of all planes. $$ L = E_1 \cap E_2 \cap E_3 $$ Two of them, $E_i$ and $E_j$ will intersect in a line $g_{ij}$. But the three lines do not intersect. So no common solution.

If we already know that two planes are parallel, let them be $E_1$ and $E_2$, then we do not have to look at $E_3$, because for whatever $E_3$ we get $$ L = E_1 \cap E_2 \cap E_3 = \emptyset \cap E_3 = \emptyset $$ How does this differ? The first case had all pairwise intersections non-empty, while the second case features at least one pairwise empty intersection.