3.5 Algebra 2 Notes
Solving Systems of Equations in Three Variables
Learning Target: Solve a system of linear equations in three variables. Solve real-world problems using systems of linear
equations in 3 variables.
Review:
Solve, use the calculator:
9 x  5 y  12
5 x  6 y  26
System of 3 equations with 3 variables has either one solution, infinitely many solutions, or no
solution. If a solution, the 3 values represent a point in space that is contained in all three
planes, the x-plane, the y-plane, and the z-plane- Ordered Triple(x,y,z)
1. Solve the system of equations
5x  3 y  2 z  2
2x  y  z  5
x  4 y  2 z  16
On the more difficult problems, eliminate the same variable in both equations. (the last
variable). For this, z would be a good choice.
5x  3 y  2 z  2
2x  y  z  5
Now do elimination of these 2 equations! Solve:
Now solve for the other variable(s).
2x  y  z  5
x  4 y  2 z  16
2. Solve the system of equations.
2x  y  2
3z  21
4 x  z  19
Look and analyze the problem first!
3. Solve the system of equations.
x  2 y  12
3 y  4 z  25
x  6 y  z  20
For this, z would be a good choice.
(Why?)
3 y  4 z  25
x  6 y  z  20
x  2 y  12
Special case #1:
Solve
r  3s  t  4
3r  6 s  9t  5
4r  9s  10t  9
Eliminate t
9(r  3s  t  4)
3r  6s  9t  5
10(r  3s  t  4)
4r  9s  10t  9
New equations:
9r  27 s  9t  36
3r  6s  9t  5
10r  30s  10t  40
4r  9s  10t  9
6r  21s  31
6r  21s  31
Think! What happens?
Special case #2:
Solve
2x  y  z  1
x  2 y  4z  3
4 x  3 y  7 z  8
Eliminate z
4(2 x  y  z  1)
x  2 y  4z  3
7(2 x  y  z  1)
4 x  3 y  7 z  8
8x  4 y  4z  4
x  2 y  4z  3
14 x  7 y  7 z  7
4 x  3 y  7 z  8
9x  2 y  7
18 x  4 y  1
Solve:
The planes for the equations in a system of three linear equations in three variables determine
the number of solutions. Match each graph description below with the description of the
number of solutions of the system. (Some of the items on the right may be used more than
once, and not all possible types of graphs are listed.)
a. three parallel planes
I. one solution
b. three planes that intersect in a line
II. no solutions
c. three planes that intersect in one point
III. infinite solutions
Application: There are 49,000 seats in a sports stadium. Tickets for the seats in the upper level sell for
$25, the ones in the middle level cost $30, and the ones in the bottom level are $35 each. The number
of seats in the middle and bottom levels together equal the number of seats in the upper level. When
all of the seats are sold for an event, the total revenues are $1,419,500. How many seats are there in
each level?
Suppose that three classmates, Monique, Josh, and Lilly, are studying for a quiz on this lesson. They
work together on solving a system of equations in three variables, x, y, and z, following the algebraic
method shown in your textbook. They first find that z = 3, then that y =-2, and finally that x =-1. The
students agree on these values, but disagree on how to write the solution. Here are their answers:
Monique: (3, -2, -1)
Which student is correct?
Josh: (-2, -1, 3)
Lilly: (-1, -2, 3)