There 3 types of solutions of systems with three variables. A system can have 1 solution, infinitely many solutions, or no solutions. You can use graphs in three dimensions to represent systems of equations with three variables.
The solutions of a system of equations with three variables can be shown graphically as the intersections of planes.
When a system with three variables has only one solution, the planes intersect at one common point. The common point is shown in black in the figure below.
When a system with three variables has an infinite number of solutions, the planes intersect at all the points along a common line. The common line is shown in black above.
When systems with three variables have no solution, we call these systems inconsistent systems. A couple of cases could present itself.
Case #1:
The three planes are parallel, so they never intersect.
Case #2:
The planes intersect with each other. However, they do not intersect at a common point.
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