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4-8 Augmented Matrices & Systems

4-8 Augmented Matrices & Systems . Objectives. Solving Systems Using Cramer’s Rule Solving Systems Using Augmented Matrices. Vocabulary. Cramer’s Rule. ax + by = m cx + dy = n. System. Replace the y-coefficients with the constants. Use the x- and y-coefficients.

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4-8 Augmented Matrices & Systems

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  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 Replace the y-coefficients with the constants Use the x- and y-coefficients. Replace the x-coefficients with the constants Then, &

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

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

  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. –7x + 4y = –3 x + 8y = 9 System of equations x-coefficients y-coefficients constants –74 –3 189 Augmented matrix Draw a vertical bar to separate the coefficients from constants. Writing an Augmented Matrix Write an augmented matrix to represent the system –7x + 4y = –3 x + 8y = 9

  8. 9x – 7y = –1 2x + 5y = –6 9–7–1 25–6 System of equations Augmented matrix x-coefficients y-coefficients constants Writing a System From an Augmented Matrix Write a system of equations for the augmented matrix . 9 –7 –1 2 5 –6

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

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

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

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

  14. Homework 4-8 pg 224 & 225 # 1, 4, 6, 7, 9, 12, 13, 18, 19 You must do by hand and show your work for all the steps.

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