Give a geometric explanation, Linear Algebra.?

Question

Give a geometric explanation of why a homogeneous linear system consisting of two equations in three unknowns must have infinitely many solutions? What are the possible numbers of solutions for a nonhomogeneous 2x3 linear system? Give a geometric explanation of your answer.

Answer 1

The geometric representation of a linear equation in three unknowns is a plane. If the equation is homogeneous (i.e., has the form ax + by + cz = 0), then the plane passes through the origin. If the system of two equations is homogeneous, then both planes pass through the origin and hence are not parallel. Therefore they must intersect; either they intersect in a line, or else the planes are coincident. Both of these cases result in infinitely many solutions.

A nonhomogeneous system of two equations in three unknowns may have infinitely many solutions as well -- again, the planes may intersect in a line, or be coincident. Unlike the homogeneous case, the planes may be parallel, in which case the system has no solution