4-8 Augmented Matrices and Systems

Cramer’s Rule System Use the x- and y- coefficients. Replace the x-coefficient with the constants Replace the y-coefficient with the constants The solution of the system is :

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

–2x + 8y + 2z = –3 –6x + 2z = 1 –7x – 5y + z = 2 Find the y-coordinate of the solution of the system . –2x + 8y + 2z = –3 –6x + 2z = 1 –7x – 5y + z = 2

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

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

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

(continued) 1 –3 –17 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. 1 0 –2 0 1 5 The solution to the system is (–2, 5).

4x + 3y + z = –1 –2x – 2y + 7z = –10. 3x + y + 5z = 2 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).

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