Section 3.6 Systems of Equations in Three Variables
If A, B, C, and D are real numbers, with A, B, and C not all zero, then
Ax + By + Cz = D
is called a linear equation in three variables.
Definition: when the solution of a system of equations is represented by one point, you can write it as
an ordered triple, (x, y, z).
Steps to solving a system in three variables:
1. Simplify and write all three equations in the form Ax+By+Cz = D , if needed.
2. Use substitution or elimination to eliminate any one of the variables from a pair of equations of the
system. Look for the easiest variable to eliminate.
3. Eliminate the same variable from another pair of equations of the system.
4. Solve the resulting system of two equations in two unknowns.
5. After you have found the values of two of the variables, substitute into one of the original equations
to find the value of the third variable.
6. Check the three values in all of the original equations.
The graph of a system of equations in three variables is a plane. Depending on the type of system, one of
three possibilities will occur:
All three planes intersect at one point. The system
will have one solution. Therefore, it is consistent
and independent
The three systems intersect in a line. The system is
consistent and dependent. Therefore, it will have an
infinite number of solutions.
Each of the diagrams below shows three planes that have no points in common. The
system is inconsistent. Therefore, there is no solution.
Example 1 – Solve the system by elimination:
x – 3y + 3z = - 4
2x + 3y – z = 15
4x – 3y – z = 19
Example 2 – Solve the system by elimination:
2x + y – z = 5
x + 4y + 2z = 16
15x + 6y – 2z = 12
Example 3 – Solve the system by substitution:
x – 2y + z = - 4
-4x + y – 2z = 1
2x + 2y – z = 10
Example 4 – John inherited $25,000 and invested part of it in a money market account, part in municipal
bonds, and part in a mutual fund. After one year, he received a total of $1,620 in simple interest from the three
investments. The money market paid 6% annually, the bonds paid 7% annually, and the mutually fund paid 8%
annually. There was $6,000 more invested in the bonds than the mutual funds. Find the amount John invested in
each category.
Homework – Page 153 #’s 2, 4, 7, 8, 10, 12, 20, 22, 23, 24, and 29.