Systems of equations that have three variables are systems of planes. Since all three variables equations such as 2x + 3y + 4z = 6 describe a a three dimensional plane.
Pictured below is an example of 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).
Like systems of linear equations, the solution of a system of planes can be no solution, one solution or infinite solutions.
No Solution of three variable systems
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.
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.
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.
Example of how to solve a system of three variable equations using elimination.
Practice Problems
Problem 1)
Use elimination to solve the following system of three variable equations.
4x + 2y – 2z = 10
2x + 8y + 4z = 32
30x + 12y – 4z = 24
Solution
Although you can indeed solve 3 variable systems using elimination and substitution as shown on this page, you may have noticed that this method is quite tedious. The most efficient method is to use matrices or, of course, you can use this online system of equations solver .
System of Equation Solver Just Type your equations in and let this calculator do the rest!
Graphing Calc
