# Solutions of systems with three variables

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.

## Systems with three variables with one solution

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.

### Systems with three variables with an infinite number of solutions

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.

### Systems with three variables with no solution

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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