Avon High School
Section: 8.2
ACE COLLEGE ALGEBRA II - NOTES
Systems of Equations in Three Variables
Mr. Record: Room ALC-129
Semester 2 - Day 2
Systems of Linear Equations in Three
Variables and Their Solutions
An equation such as x  2 y  3z  9 is called a linear equation in
three variables.
The process of solving a system of three linear equations in three
variables is geometrically equivalent to finding the intersection
point (if it exists) between three planes in space.
Solving Systems of Linear Equations in Three Variables by Eliminating Variables
Solving Linear Systems in Three Variables by Eliminating Variables
1. Reduce the system to two equations in two variables. This is usually accomplished by
taking two different pairs of equations and using the addition method to eliminate the same
variable from both pairs.
2. Solve the resulting system of two equations in two variables using addition or substitution.
The result is an equation in one variable that gives the value of that variable.
3. Back-substitute the value of the variable found in Step 2 into either of the equation in two
variables to find the value of the second variable.
4. Use the values of the two variables from Step 2 and Step 3 to find the value of the third
variable by back-substituting into one of the original equations.
5. Check the proposed solution in each of the system’s given equations.
Example 1
Solving a System in Three Variables
Solve the system.
x  4 y  z  20
3x  2 y  z  8
2 x  3 y  2 z  16
Let
be positive real numbers with
, and let p be any real number.
The logarithm of a number with an exponent is the product of the exponent and the logarithm
of that number.
Example 2
Solving a System in Three Variables with a Missing Term
Solve the system.
2 y  z  20
x  2 y  z  17
2 x  3 y  2 z  1
Applications
Systems of equations may allow us to find models for data without using a graphing utility. Three data points
that do not lie on or near a line determine the graph of a quadratic function of the form y  ax 2  bx  c, a  0 .
Quadratic functions often model situations in which values of y are decreasing and then increasing, suggesting
the cuplike shape of a parabola.
Example 3
Modeling Data Relating Sleep and Death Rate
In a study relating sleep and death rate, the following data were obtained. Use the function
y  ax 2  bx  c to model the data.