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Solutions: Systems of 3 variable Equations

Where Planes Intersect

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).
  • picture of three variable plane and graph
Systems of Linear Equations ( 2 variables)
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.
  • no solution picture
    • 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
Before attempting to solve systems of three variable equations using elimination, you should probably be comfortable solving 2 variable systems of linear equations using elimination
Example of how to solve a system of three variable equations using elimination.
  • Steps to solve system of 3 equations
    • 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
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