Solutions: Systems of 3 variable Equations

What is a linear equation with 3 variables?

It is a plane ! picture of three variable plane and graph

The picture on the left is the graph of the plane 2x + 3y + z = 6.

The red triangle is the portion of the plane when x, y, and z values are all positive. This plane actually continues off in the negative direction. All that is pictured is the part of the plane that is intersected by the positive axes (the negative axes have dashed lines).

What is a system of 3 variables equations?

Just like a system of linear equations with 2 variables is more than 1 line, a system of 3 variable equations is just more than plane.

Video Tutorial

on Systems of 3 variable equations

No Solutions, 1 Solution or Infinite Solutions

Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions.

No Solution

Case I

Below is a picture of three planes that have no solution. There is no single point at which all three planes intersect, therefore this system has no solution.

no solution picture

Case II

The other common example of systems of three variables equations that have no solution is pictured below. In the case below, each plane intersects the other two planes. However, there is no single point at which all three planes meet. Therefore, the system of 3 variable equations below has no solution.

picture of no solution 3 variable equation of planes

One Solution

of three variable systems

If the three planes intersect as pictured below then the three variable system has 1 point in common, and a single solution represented by the black point below.

Infinite Solutions

of three variable systems

If the three planes intersect as pictured below then the three variable system has a line of intersection and therefore an infinite number of solutions.

Picture of infinite solutions 3 variable system
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