How can I prove that in a linear system can be there are three solution sets without use matrices?

Question

I would like to know how to prove that in a linear system can be there are three solution sets (empty, a set with one solution or a set with infinitely many solutions) without use matrices? Thanks in advance!

EDIT: sorry guys, maybe I have expressed myself wrong, I know the geometric interpretation and that I need to do the Gaussian elimination, but my doubt is how to write this formally considering a generic linear system, observing just if the system is homogeneous or not, the number of equations and the number of variables.

Answer 1

Two lines either intersect (in one place), don't, or are collinear ( the same line)

Answer 2

You can use Gaussian elimination. If you find same number of independent equations as variables the system has only one solution. If the number of independent equations is smaller than the number of variables, the system has infinity solutions. And finally, if you end up with something like $0=\mbox{something}$, then the system has no solution

Answer 3

You can solve the system using substitution. Find an unknown from the first equation and substitute in all other equations, than find a second unknown from the second equation and substitute in all other... and so on. You finish this process with a single equation in one unknown.

If this equation has one solution than the system has one solution,

if this equation is ''impossible'' (that is it reduce to $0=a$ for $a \ne 0$) this means that the system has no solutions,

if this equation is an identity ($0=0$), this means that the system has infinitely many solutions.