3.6 Solving Systems of Linear Equations in Three Variables
A linear equation in three variables x, y, and z is an equation of the form
ax + by + cz = d where a, b, and c are not all zero.
The following is an example of a system of three linear equations in three variables.
x + 2y − 3z = −3
2x − 5y + 4z = 13
5x + 4y − z = 5
A solution of such a system is an ordered triple (x , y , z ) that is a solution of all three
equations. For instance, (2, −1, 1) is a solution of the system above.
The graph of a linear equation in three variables is a plane in three-dimensional space.
The graphs of three such equations that form a system are three planes whose
intersection determines the number of solutions of the system, as shown in the
The Linear Combination Method for Solving a Three-Variable System
1.) Use the linear combination method to rewrite the linear system in three
variables as a linear system in two variables.
2.) Solve the new linear system for both its variables.
3.) Substitute the values found in step 2 into one of the original equations and
solve for the remaining variable.
Ex: Solve the systems below.
4x + 2y + 3z = 1
a) 2x − 3y + 5z = −14
6x − y + 4z = −1
3x + 2y + 4z = 11
b) 2x − y + 3z = 4
5x − 3y + 5z = −1
x +y +z = 3
c) 4x + 4y + 4z = 7
3x − y + 2z = 5
x +y +z = 4
d) x + y − z = 4
3x + 3y + z = 12
Solving a Three-Variable System Using Substitution
The marketing department of a company has a budget of $30,000 for advertising.
A television ad costs $1000, a radio ad costs $200, and a newspaper ad costs
$500. The department wants to run 60 ads per month and have as many radio
ads as television and newspapers ads combined. How many of each type should
the department run each month?