1. Warm-Up Write each system as a matrix equation. Then solve the system, if possible, by using the matrix equation. 6 minutes

2. 4.5.1 Using Matrix Row Operations 4.5.1 Using Matrix Row Operations Objectives: Represent a system of equations as an augmented matrix Perform elementary row operations on matrices

3. Matrix Row Operations The row-reduction method of solving a system allows you to determine whether the system is independent, dependent, or inconsistent. The row-reduction method of solving a system is performed on an augmented matrix. An augmented matrix consists of the coefficients and constant terms in the system of equations. SystemAugmented Matrix

4. Matrix Row Operations The goal of the row-reduction method is to transform, if possible, the coefficient columns into columns that form an identity matrix. This is called the reduced row-echelon form of an augmented matrix if the matrix represents an independent system. The resulting constants will represent the unique solution to the system.

5. Elementary Row Operations The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form. -Interchange two rows -Multiply all entries in one row by a nonzero number -Add a multiple of one row to another row

6. Row Operations and their Notations -Interchange rows 1 and 2 -Multiply each entry in row 3 by -2 -Replace row 1 with the sum of row 1 and 4 times each entry in row 2

7. Example 1 Perform the indicated row operations on matrix A. 11192014 -116-75

8. Practice Perform the indicated row operations on matrix A.

9. Homework p.256 #8-12

10. Warm-Up 6 minutes Perform the indicated row operations on matrix A.

11. 4.5.2 Using Matrix Row Operations 4.5.2 Using Matrix Row Operations Objectives: Solve a system of linear equations by using elementary row operations

12. Example 1 Solve the system of equations by using the row- reduction method. Then classify the system. 1216 0-3-21 x = 2; y = 7 independent

13. Practice Solve the system of equations by using the row- reduction method. Then classify the system.

14. Example 2 Solve the system of equations by using the row- reduction method. Then classify the system. x – 1.4z = 0 y – 0.2z = 0 0 = 1 no solution, inconsistent

15. Example 3 Solve the system of equations by using the row- reduction method. Then classify the system. x – z = -1 y + 2z = 3 0 = 0 infinitely many solutions dependent

16. Practice Solve the system of equations by using the row- reduction method. Then classify the system.

17. Homework worksheet