How to Solve a System of Three Linear Equations
In Section 8.3 Educo teaches the Gauss method for solving a system of three linear equations and three unknowns. At the beginning of that section the textbook says
However Educo neglects to show you how to apply "these methods" to solve a system of three equations but goes on to show the Gauss method, a rather long and laborious 8-step procedure to work out with pencil and paper. The Gauss method is important as it lends itself well to using matrices or a computer program that can easily swap equations around. Since you are using pencil and paper, we'll use a more traditional solving by elimination method typically taught before teaching the Gauss method. This method is simpler and builds on your knowledge of the "Elimination by Addition" method you learned for solving a system with two equations and two unknowns.
Procedure for Solving a Linear Three-Equation System by Elimination
1. Select a pair of equations and eliminate one variable from this pair.
2. Select a different pair of equations and eliminate the same variable as in step 1.
3. Solve the system of two equations resulting from steps 1 and 2. (Use procedures you learned in Section 8.1.)
4. Substitute the values found in step 3 into the simplest of the original equations, and then solve for the third variable.
Before you begin steps 1 and 2, look at the three equations and decide which variable would be easiest to eliminate. Then you proceed to eliminate that variable.
Example 1.
Solve the system of equations.
Step 1. It appears that eliminating
y
would require the least amount of arithmetic because adding equations (1) and
(3) will eliminate
y
without changing either equation.
So we'll first add those equations.
Step 2. Now we need to pick another pair of equations and eliminate y and we'll select equations (2) and (3). We'll eliminate y by adding equation (2) to 3 times equation (3).
Step 3. Now equations (5) and (6)
give us a system of two equations and two unknowns.
We solve those using the techniques previously
learned.
Add equations (5) and 3 times equation (6)
We now have the value for z.
Back substitute z into either equation (5) or (6)
to solve for x. Equation (5) looks the simplest so we’ll use it.
Step 4. We now have the values for
x and
z.
Back substitute into any one of the original
equations, we'll choose equation (1)
The solution is
Now you should substitute those values back into the original equations to
validate that the solution is correct.
Example 2.
Solve the system of equations.
Step 1.
We’ll choose to eliminate z. Add
equations (1) and -1 times equation (3).
Step 2. Add 2 times equation (2) to equation (3)
Step 3.
Solve the system of two equations, (5) and (6)
Substitute y=5 into equation(6)
Step 4.
Substitute y=5 and x=8 into equation (3)
The solution is
Example 3. In this examle, I'll show how I do it with pencil and paper. I'm given a system of three equations to solve. I see the third & fourth equations can be added to eliminate y. Then 4 times the first equation added to the third equation can be added to eliminate y. I'm then left with with two equations in x and z that I can solve using methods previously learned. The technique I've used here allows me to easily go back and double check my calculations.
Example 4. (from "Intermediate Algebra" by Baratto/Kohlmetz/Bergman, McGraw-Hill 2008)
Video Lecture.
See also video with solution of three equations. An equation of three variables defines a surface in 3-dimensional space. Three equations of three variables define three surfaces (planes) which might or might not all intersect. The first two minutes of the video illustrates how the three surfaces might intersect and provides a visual illustration of a solution. Beginning about two minutes into the video, the instructor tells how to solve an equation of three unknowns using the procedure we used here.