Systems of Three Equations in Three Variables
Learning Outcomes
- Solve systems of three equations in three variables
Solution Set, One Solution
The figure below illustrates how a system with three variables can have one solution. Systems that have a single solution are those which result in a solution set consisting of an ordered triple . Graphically, the ordered triple defines a point that is the intersection of three planes in space.
Example
Determine whether the ordered triple is a solution to the system.Answer: We will check each equation by substituting in the values of the ordered triple for , and . The ordered triple is indeed a solution to the system.
How To: Given a linear system of three equations, solve for three unknowns
- Pick any pair of equations and solve for one variable.
- Pick another pair of equations and solve for the same variable.
- You have created a system of two equations in two unknowns. Solve the resulting two-by-two system.
- Back-substitute known variables into any one of the original equations and solve for the missing variable.
Example
Solve the given system.Answer: The third equation states that , so we substitute this into the second equation to obtain a solution for . Now we have two of our solutions, and we can substitute them both into the first equation to solve for . Now we have our ordered triple; remember that where you place the solutions matters!
Analysis of the Solution:
Each of the lines in the system above represents a plane (think about a sheet of paper). If you imagine three sheets of notebook paper each representing a portion of these planes, you will start to see the complexities involved in how three such planes can intersect. Below is a sketch of the three planes. It turns out that any two of these planes intersect in a line, so our intersection point is where all three of these lines meet.
Example
Find a solution to the following system:Answer: We labeled the equations this time to be able to keep track of things a little more easily. The most obvious first step here is to eliminate by adding equations (2) and (3).
Now we can substitute the value for that we obtained into equation .
Be careful here not to get confused with a solution of and an inconsistent solution. It is ok for variables to equal . Now we can substitute and back into the first equation.
Try It
[ohm_question]38331[/ohm_question]Summary
- The solution to a system of linear equations in three variables is an ordered triple of the form .
- Solutions can be verified using substitution and the order of operations.
- Systems of three variables can be solved using the same techniques as we used to solve systems with two variables, including elimination and substitution.
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