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 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

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

Example 3

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 $ 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: x y z3x - 4 3x + 4