Section 3.4 – Solve 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:
2x + y – z = 5
3x – 2y + z = 16
4x + 3y – 5z = 3
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
A solution of such a system is an ordered
triple (x, y, z) whose coordinates make each
equation true.
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.
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 1:
Solve the system:
4x + 2y + 3z = 1
2x – 3y + 5z = -14
6x – y + 4z = -1
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 2:
Solve the system:
2x – y + 6z = -4
6x + 4y – 5z = -7
-4x – 2y + 5z = 9
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 3:
Solve the system:
x+y+z=3
4x + 4y + 4z = 7
3x – y + 2z = 5
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 4:
Solve the system:
x+y+z=4
x+y–z=4
3x + 3y + z = 12
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 5:
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 ads and
newspaper ads combined. How many of each
type of ad should the department run each
month.
Section 3.4 – Solve Systems of Linear
Equations in Three Variables
Example 6:
At a carry-out pizza restaurant, an order of 3
slices of pizza, 4 breadsticks, and 2 juice drinks
costs $13.35. A second order of 5 slices of
pizza, 2 breadsticks, and 3 juice drinks costs
$19.50. If four breadsticks and a juice drink
cost $.30 more than a slice of pizza, what is the
cost of each item?