Download Section 7.2 Systems of Linear Equations in Three Variables

Section 7.2
Systems of Linear Equations in Three
Variables
Overview
• When solving systems of linear equations in
two variables, we utilized the following
techniques:
1. Substitution
2. Elimination
3. Graphing
• In this section we will develop techniques
that can be used for larger systems.
Three-variable Systems
• A system of linear equations in three variables
is in the form
Ax  By  Cz  D
Ex  Fy  Gz  H
Jx  Ky  Lz  M
• The solution to a three-variable system is an
ordered triple (x,y,z).
Example 1
• Determine if (2, -4, 4) is a solution to
x  9 z  34
4 x  4 y  24
8 y  2 z  40
Remember This?
• Solve the following system by elimination:
3x – 4y = 25
5x + y = 11
• We will use a similar strategy to solve a system
that has three equations and three unknown
variables.
Steps In The Process
1. Pick any two equations and use elimination to
cancel one variable (x, y, or z).
2. Pick another set of two equations and use
elimination to cancel the same variable.
3. Now solve this new, smaller system for the
remaining variables.
4. Take those values and substitute them into any
of the three original equations to find the third
and final variable.
Example 2
5 x  10 y  3z  21
x  5y  z  0
10 x  y  z  12
Example 3
 3x  y  3
y  2 z  1
x  4 y  z  12
Example 4
• The sum of three numbers is 8. The sum of
twice the first number, 3 times the second
number, and 4 times the third number is 22.
The difference between 5 times the first
number and the second number is 11. Find
the three numbers.
Let’s Go Shopping!
• On a recent trip to the convenience store, you
picked up 4 gallons of milk, 5 bottles of water,
and 6 snack-size bags of chips. Your total bill
(before tax) was $26.40. If a bottle of water
costs twice as much as a bag of chips, and a
gallon of milk costs $1.80 more than a bottle
of water, how much does each item cost?
A Little Geometry Never Hurt
Anyone…
• Find the values of x, y, and z in the triangle
below:
y
x
3x + 4
z
3x - 4