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?