3.3 Systems of Linear Equations in Three Variables

[Pages:3]Sec 3.3 Systems of Linear Equations in Three Variables

Learning Objectives:

1. Solve systems of three linear equations containing three variables. 2. Model and solve problems involving three linear equations containing three variables.

1. Solve systems of three linear equations containing three variables Definitions:

1. Linear Equations in Three Variables--Algebraic equation of the form Ax + By + Cz = D , where A, B, C and D are real numbers, with A, B and C are not all zero.

2. A Solution to a system of equation is any ordered triple (x, y, z) that give true statement to all

equations in the system.

Example 1. Determine whether the given ordered triple (- 1, 2, - 3) is solution of the system of

linear equation. x + y + z = -2 x + 2 y - 3z = 12 2x - 2 y + z = -9

Three types of the system 1. Consistent System with independent equations (independent system)-has exactly one

solution (x, y, z) .

2. Inconsistent System-has no solution, . 3. Consistent System with dependent equations (dependent system)--has infinitely many

solutions. Steps for Solving Systems of Linear Equations in Three Variables 1. Select two of the equations and eliminate one of the variables form one of the equations. Select

any two other equations and eliminate the same variable from one of the equations. 2. You will have two equations that have only two unknowns. Eliminate a second variable form the

two linear equations in two unknown. 3. Solve the remaining variable. Example 2. Solve each system of equations.

2x - y + 3z = 14 1. x + y - 2z = -5

3x + y - z = 2

1

x - y = 3 2. 2x + 2z = 7

y + z = 8

x + y - 2z = 3 3. - 2x - 3y + z = -7

x + 2 y + z = 4

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2. Model and solve problems involving three linear equations containing three variables Example 3. Curve Fitting

The function f (x) = ax2 + bx + c is a quadratic function, where a, b, and c are constant. a. If f (1) = 4 , then 4 = a(1)2 + b(1) + c or a + b + c = 4 . Find two additional linear equations if f (- 1) = -6 and f (2) = 3 .

b. Use the three linear equations found in part (a) to determine a, b and c. What is the

quadratic function that contains the points (- 1, - 6), (1, 4) and (2, 3)?

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