Read: Systems of Three Equations in Three Variables
Learning Objectives
- Define the solution to a system of three equations in three variables
- Determine whether an ordered triple is a solution to a system
- Solve a systems of equations in three variables using elimination and back-substitution
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.![Screen Shot 2016-07-11 at 2.08.59 PM](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/07/11211012/Screen-Shot-2016-07-11-at-2.08.59-PM.png)
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 systemAnswer: The third equation states that , so we substitute this into the second equation to obtain a solution for . \begin{array}y-\dfrac{1}{2}(-1)=4\\y+\dfrac{1}{2}=4\\y=4-\dfrac{1}{2}\\y=\dfrac{8}{2}-\dfrac{1}{2}\\y=\dfrac{7}{2}\end{array} Now we have two of our solutions and we can substitute them both into the first equation to solve for . \begin{array}x-\dfrac{1}{3}\left(\dfrac{7}{2}\right)+\dfrac{1}{2}\left(-1\right)=1\\x-\dfrac{7}{6}-\dfrac{1}{2}=1\\x-\dfrac{7}{6}-\dfrac{3}{6}=1\\x-\dfrac{10}{6}=1\\x=1+\dfrac{10}{6}\\x=\dfrac{6}{6}+\dfrac{10}{6}\\x=\dfrac{16}{6}=\dfrac{8}{3}\end{array} Now we have our ordered triple, remember that where you place the solutions matters!
Answer
Analysis of the solution:
Each of the lines in this system 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.![Three Planes Intersecting.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/121/2016/07/11200619/Screen-Shot-2016-07-11-at-1.04.41-PM-300x237.png)
Example
Find a solution to the following system:Answer:
Be careful here not to get confused with a solution of and an inconsistent solution. It's ok for variables to equal . Now we can substitute and back into the original equation.
Answer
Summary
- The solution to a system of linear equations in three variables is an ordered triple in 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.