298 Chapter 3 Systems of Linear Equations and Inequalities It is particularly easy to recognize a system of dependent equations or an inconsistent system of equations from the reduced row echelon form of an augmented matrix. This is demonstrated in Examples 6 and 7. Solving a System of Dependent Equations by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. Solution: x 3y 4 Set up the augmented matrix. 12 Multiply row 1 by and add the result to row 2. 1 2 The second row of the augmented matrix represents the equation 0 0. The equations are dependent. The solution set is Skill Practice Solve by using the Gauss-Jordan method. 12. 4x 6y 16 6x 9y 24 51x, y2 0 x 3y 46. 12 R1 R21 R2 c 1 3 0 0 ` 4 0 d c 1 3 12 32 ` 4 2 d x 3 2 y 2 Example 6 Answers 12. Infinitely many solutions; 51x, y2 0 4x 6y 166; dependent equations 13. No solution; { }; inconsistent system Solving an Inconsistent System by Using the Gauss-Jordan Method Solve by using the Gauss-Jordan method. Solution: x 3y 2 3x 9y 1 Set up the augmented matrix. Multiply row 1 by 3 and add the result to row 2. c 1 3 3 9 ` 2 1 The second row of the augmented matrix represents the contradiction 0 7. The system is inconsistent. There is no solution, { }. Skill Practice Solve by using the Gauss-Jordan method. 13. 6x 10y 1 15x 25y 3 c1 3 ` 0 0 2 7 3R1 R2 1 R2 d d Example 7
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