While studying Cramer's Rule for non-homogeneous equations in three unknowns one comes across the statement that for the system to possess infinite solutions the following conditions are required:
(i)$\triangle=0$
(ii)$\triangle_{x}=\triangle_{y}=\triangle_{z}=0$
Geometrically what do (i) and (ii) imply individually and together
Can we construct a system of equations where $\triangle=\triangle_{x}=\triangle_{y}=0$ but $\triangle_{z}\neq 0$ and then what would be its geometrical implications.
I have read lot of text on this but everywhere authors want students to simply memorize these statements and get it over with. I am not able to find any reasonable explanation as why this should be the case and what would happen otherwise. Nobody gives a geometrical viewpoint. For equations in two unknowns things are quite obvious but in three unknowns I am having difficulty in comprehending these situations.
(1) implies that the equations have either many solutions or no solution.
(2) alone implies all variables are zero if $\Delta\ne 0$
(1) and (2) imply many solutions: the three equations may be identical (three identical planes) or two identical planes and one intersecting them in a line or three nonidentical planes meeting in a line like three pages of a book.