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).
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Related : 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.
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.
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.
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.
Practice Problems
Problem 1) Use elimination to solve the
following system of three variable
equations.
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4x + 2y – 2z = 10
2x + 8y + 4z = 32
30x + 12y – 4z = 24
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 .