6.6 Matrix Row Operations 2017 ink.notebook Page 35 Page 36 6.6 Row operations (Solve Systems with Matrices) Lesson Objectives Page 37 Standards Lesson Notes Page 38 6.6 Matrix Row Operations Press the tabs to view details. 1
Lesson Objectives Standards Lesson Notes After this lesson, you should be able to solve linear systems by using matrices on a calculator. Press the tabs to view details. Lesson Objectives Standards Lesson Notes N VM.7. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N VM.8. Add, subtract, and multiply matrices of appropriate dimensions. A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Press the tabs to view details. Matrix = numbers arranged in rows and columns The plural is matrices. They are named by a capital letter. Dimensions = how many numbers going down by how many numbers going across 1. Give the dimensions of each matrix. A = B = C = 2 x 3 element = each individual number in the matrix A = x B = x C = x M = M 21 = 4 4 E = E 32 = - 4-4 2
6.6 Matrix Row Operations 2017 ink.notebook Augmented Matrix - Consists of the coefficients and the constant terms of a system of equations. The coefficients and the constants are separated by a dashed line. Linear system: 9 Augmented matrix: 3 8 1 2 4 9 3 8 2 4 Find the augmented form of each matrix. 2. 3. Row Reduction is a process of performing elementary row operations on an augmented matrix to get the coefficient portion of the matrix to have the form of the identity matrix. We are going to use the calculator to put the augmented matrix into reduced row echelon form. (RREF) 3
For both operating systems, you must see the identity matrix to get the correct answers. If there is a line as shown below, then the solution is no solution or many solutions. The numbers on the right after the identity matrix are the x and y values. Equations must be in standard form to be entered into the calculator. 2 nd x 1 fl edit enter Enter the dimensions: hitting enter after each number Hit enter after each number in the matrix mode 2nd To get the answer to the system: 2 nd x 1 fl math rref( enter 2 nd x 1 A enter Use an augmented matrix to solve each system of equations. If the system is inconsistent or dependent, say so. 4. 5. = 1-4 - 2 1 : : - 2-3 2 nd x 1 fl edit enter Hit enter after each number 2nd mode 2 nd x 1 fl math rref( enter 2 nd x 1 A enter 2 nd x 1 fl math rref( alpha zoom. = 1 0 0 1 : : 2 1 Answer: (2, 1) 4
6. 7. 8. 9. Write a system of equations for each augmented matrix. 10. 11. 5
6.6 Matrix Row Operations 2017 ink.notebook 12. Mel is having a few friends over and she is buying subs and cans of soda for them. Mel bought 28 items. If Mel spent $56.70, how many subs and how many sodas did she buy? a) If the reduced row echelon form of the augmented matrix for a system of equations is On Your Whiteboards a) many solutions b) no solution c) (0, 0, 1) d) none of these whiteboard 6
Use an augmented matrix to solve each system of equations. b) c) e) On a hot day, a public swimming pool sold 248 tickets. Tickets cost $0.25 for children, $2.00 for adults under 65, and $1.00 for senior citizens. Three times as many children s tickets as senior citizen s tickets were sold. The total amount of money collected was $221.00. How many tickets for adults were sold? a) 33 b) 144 c) 72 d) 36 d) On the Worksheet Homework 7
Find the augmented form of each matrix. 1. 2. Find the augmented form of each matrix. 3. Use an augmented matrix to solve each system of equations. 4. 5. Use an augmented matrix to solve each system of equations. 6. 7. 8
Use an augmented matrix to solve each system of equations. 8. 9. 11. 12. Write a system of equations for each augmented matrix. 13. 14. 9
Write a system of linear equations to model each situation. Then solve. 15. Darnell and Sandra went shopping for graphic novels. The store charges one price for all new books and another for all old books. Find the cost of one old book and one new book. 16. If the Franklin High School cheerleaders replace 8 uniforms and 6 poms, the cost is $378. If they replace 6 uniforms and 9 poms, the cost is $333. Find the cost of a uniform and a pom. 17. Tasha paid $25.75 for 3 games of miniature golf and 2 rides on go karts. Trevor paid $35.75 for 4 games of miniature golf and 3 rides on go karts. How much did each activity cost? Answers: 10