1. 4-8 Augmented Matrices & Systems

2. Objectives Solving Systems Using Cramer’s Rule Solving Systems Using Augmented Matrices

3. Vocabulary Cramer’s Rule ax + by = m cx + dy = n System Use the x- and y-coefficients. Replace the x- coefficients with the constants Replace the y- coefficients with the constants Then, &

4. Use Cramer’s rule to solve the system. Evaluate three determinants. Then find x and y. 7x – 4y = 15 3x + 6y = 8 D = = 54 7 –4 3 6 D x = = 122 15 –4 8 6 D y = = 11 7 15 3 8 x = = DxDDxD 61 27 11 54 DyDDyD y = = The solution of the system is,. 61 27 11 54 Using Cramer’s Rule

5. Find the y-coordinate of the solution of the system. –2x + 8y + 2z = –3 –6x + 2z = 1 –7x – 5y + z = 2 D = = –24Evaluate the determinant. –2 8 2 –6 0 2 –7 –5 1 D y = = 20Replace the y-coefficients with the constants and evaluate again. –2 –3 2 –6 1 2 –7 2 1 y = = – = –Find y. 20 24 DyDDyD 5656 The y-coordinate of the solution is –. 5656 Using Cramer’s Rule with

6. Vocabulary An augmented matrix contains the coefficients and the constants from a system of equations. Each row represents an equation. -6x + 2y = 10 4x = -20 System of Equations Augmented Matrix

7. Write an augmented matrix to represent the system –7x + 4y = –3 x + 8y = 9 System of equations –7x + 4y = –3 x + 8y = 9 x-coefficientsy-coefficientsconstants Augmented matrix –7 4 –3 1 8 9 Draw a vertical bar to separate the coefficients from constants. Writing an Augmented Matrix

8. Write a system of equations for the augmented matrix. 9 –7 –1 2 5 –6 Augmented matrix 9 –7 –1 2 5 –6 x-coefficientsy-coefficientsconstants System of equations 9x – 7y = –1 2x + 5y = –6 Writing a System From an Augmented Matrix

9. Vocabulary Row Operations To solve a system of equations using an augmented matrix, you can use one or more of the following row operations. Switch any two rows Multiply a row by a constant Add one row to another Combine one or more of these steps The goal is to get the matrix to the left of the line into the identity matrix. The values to the right of the line will be your solutions. Number here will be x-value Number here will be y-value

10. Use an augmented matrix to solve the system x – 3y = –17 4x + 2y = 2 1 –3 –17 4 2 2 Write an augmented matrix. Multiply Row 1 by –4 and add it to Row 2. Write the new augmented matrix. 1 –3 –17 0 14 70 –4(1 –3 –17) 4 2 2 0 14 70 1 14 1 –3 –17 0 1 5 Multiply Row 2 by. Write the new augmented matrix. (0 14 70) 0 1 5 1 14 Using an Augmented Matrix

11. (continued) 1 14 1 –3 –17 0 1 5 (0 14 70) 0 1 5 1 0 –2 0 1 5 1 –3 –17 3(0 1 5) 1 0 –2 Multiply Row 2 by 3 and add it to Row 1. Write the final augmented matrix. The solution to the system is (–2, 5). Check: x – 3y = –17 4x + 2y = 2Use the original equations. (–2) – 3(5) –17 4(–2) + 2(5) 2 Substitute. –2 – 15 –17 –8 + 10 2Multiply. –17 = –17 2 = 2 Continued

13. Use the rref feature on a graphing calculator to solve the system 4x + 3y + z = –1 –2x – 2y + 7z = –10. 3x + y + 5z = 2 Step 1:Enter the augmented matrix as matrix A. Step 2:Use the rref feature of your graphing calculator. The solution is (7, –9, –2). Using a Graphing Calculator

14. (continued) Partial Check: 4x + 3y + z = –1Use the original equation. 4(7) + 3(–9) + (–2) –1Substitute. 28 – 27 – 2 –1Multiply. –1 = –1Simplify. Continued

15. Homework 4-8 pg 220 # 2-20 even