systems in 3 variables

BUC Beginning

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.

system of three linear equations in three variables:
2x + y – z = 5
3x – 2y + z = 16
4x + 3y – 5z = 3

• A solution of such a system is an ordered triple
(x, y, z) whose coordinates make each equation
true.

Solving Steps:

1. Eliminate a variable in the first two equations.

2. Eliminate the same variable in the last two
equations.
3. Use the elimination method again with the two
new equations.
4. Solve for the remaining variable.
5. Substitute to find the other two variables.
6. Write the solution as an ordered triple.

Example 1:
Solve the system:
4x + 2y + 3z = 1
2x – 3y + 5z = -14
6x – y + 4z = -1

Example 2:
Solve the system:
2x – y + 6z = -4
6x + 4y – 5z = -7
-4x – 2y + 5z = 9

Example 3:
Solve the system:
x+y+z=3
4x + 4y + 4z = 7
3x – y + 2z = 5

Example 4:
Solve the system:
x+y+z=4
x+y–z=4
3x + 3y + z = 12

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.

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?