Geometrical interpretation not matching with Cramer's Rule while solving system of 3 equations

Question

Consider 3 equation, all in 3 variables: E1, E2, E3.

Now in 3D space these would represent 3 planes. If I take random coefficients for them, and plot them, most of the time there won't be a point where all 3 meet(geometrically that would be a very specific case).

On the other hand, Cramer's rule states that if the determinant of the coefficient matrix is non-zero, a solution exists.

Now if I were to take random values again, a zero determinant is again an anomaly.

So how come a non-zero determinant(something very likely) ensure a solution or coincidence of the 3 planes(something very unlikely)

Answer 1

Your statment

Now in 3D space these would represent 3 planes. If I take random coefficients for them, and plot them, most of the time there won't be a point where all 3 meet(geometrically that would be a very specific case).

is not correct. Two planes which are not parallel intersect in a line. If another plane is not parallel to that line, then they intersect at a point. Thus, in general position, three planes intersect at a point. This agrees with your statement

On the other hand, Cramer's rule states that if the determinant of the coefficient matrix is non-zero, a solution exists.

where the solution corresponds to the point of intersection of the three planes.