Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Systems of Equations in Three Variables Identifying Solutions Solving Systems in Three Variables Dependency, Inconsistency, and Geometric Considerations 8.4

Slide 3- 2 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identifying Solutions A solution of a system of three equations in three variables is an ordered triple (x, y, z) that makes all three equations true. A linear equation in three variables is an equation equivalent to one in the form Ax + By + Cz = D, where A, B, C, and D are real numbers. We refer to the form Ax + By + Cz = D as standard form for a linear equation in three variables.

Slide 3- 3 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine whether (2, –1, 3) is a solution of the system Example

Slide 3- 4 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems in Three Variables The elimination method allows us to manipulate a system of three equations in three variables so that a simpler system of two equations in two variables is formed. Once that simpler system has been solved, we can substitute into one of the three original equations and solve for the third variable.

Slide 3- 5 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the following system of equations: (1) (2) (3) Example

Slide 3- 6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems of Three Linear Equations To use the elimination method to solve systems of three linear equations: 1. Write all the equations in standard form Ax + By+ Cz = D. 2. Clear any decimals or fractions. 3.Choose a variable to eliminate. Then select two of the three equations and work to get one equation in which the selected variable is eliminated.

Slide 3- 7 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems of Three Linear Equations (continued) 4. Next, use a different pair of equations and eliminate the same variable that you did in step (3). 5. Solve the system of equations that resulted from steps (3) and (4). 6.Substitute the solution from step (5) into one of the original three equations and solve for the third variable. Then check.

Slide 3- 8 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dependency, Inconsistency, and Geometric Considerations The graph of a linear equation in three variables is a plane. Solutions are points common to the planes of each system. Since three planes can have an infinite number of points in common or no points at all in common, we need to generalize the concept of consistency in three dimensions.

Slide 3- 9 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Consistency A system of equations that has at least one solution is said to be consistent. A system of equations that has no solution is said to be inconsistent.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the following system of equations: (1) (2) (3) Example

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Solving Applications: Systems of Three Equations Applications of Three Equations in Three Unknowns 8.5

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The sum of three numbers is 6. The first number plus twice the second, minus the third is 2. The first minus the second, plus three times the third is 8. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In triangle ABC, the measure of angle B is three times the measure of angle A. The measure of angle C is 60 o greater than twice the measure of angle A. Find the measure of each angle. Example

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Elimination Using Matrices Matrices and Systems Row-Equivalent Operations 8.6

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley In solving systems of equations, we perform computations with the constants. The variables play no important role until the end. simplifies to –3 7 if we do not write the variables, the operation of addition, and the equals signs. For example, the system

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Matrices and Systems In the previous slide, we have written a rectangular array of numbers. Such an array is called a matrix (plural, matrices). We ordinarily write brackets around matrices. The following are examples of matrices: The individual numbers are called elements or entries.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The rows of a matrix are horizontal, and the columns are vertical. column 1column 2column 3 row 1 row 2 row 3

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use matrices to solve the system. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Use matrices to solve the system. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Row-Equivalent Operations Each of the following row-equivalent operations produces a row-equivalent matrix: a) Interchanging any two rows. b) Multiplying all elements of a row by a nonzero constant. c) Replacing a row with the sum of that row and a multiple of another row.

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Determinants and Cramer’s Rule Determinants of 2 x 2 Matrices Cramer’s Rule: 2 x 2 Systems Cramer’s Rule: 3 x 3 Systems 8.7

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determinants of 2 x 2 Matrices When a matrix has m rows and n columns, it is called an “m by n” matrix. Thus its dimensions are denoted by m x n. If a matrix has the same number of rows and columns, it is called a square matrix. Associated with every square matrix is a number called its determinant, defined as follows for 2 x 2 matrices.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 x 2 Determinants The determinant of a two-by-two matrix and is defined as follows:

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate: Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule: 2 x 2 Matrices Using the elimination method, we can show that the solution to the system is These fractions can be rewritten using determinants.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule: 2 x 2 Systems The solution of the system if it is unique, is given by

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule: 2 x 2 Systems (continued) These formulas apply only if the denominator is not 0. If the denominator is 0, then one of two things happens: 1.If the denominator is 0 and the numerators are also 0, then the equations in the system are dependent. 2.If the denominator is 0 and at least one numerator is not 0, then the system is inconsistent.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve using Cramer’s rule: Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 x 3 Determinants The determinant of a three-by-three matrix is defined as follows: Subtract.Add.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Evaluate: Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule: 3 x 3 Systems The solution of the system can be found by using the following determinants:

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Cramer’s Rule: 3 x 3 Systems (continued) If a unique solution exists, it is given by

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve using Cramer’s rule: Example