898 CHAPTER 9 Systems and Matrices The Gaussian Elimination Method With the Gauss-Jordan method for solving a system of linear equations, a matrix is transformed into reduced rowechelon form. At that point the solution can be read directly from the matrix. A similar form, called row-echelon form, is the same as reduced rowechelon form with one exception—the numbers above the leading 1s are not intentionally converted to 0s. With the Gaussian elimination method, the matrix is first transformed to row-echelon form. Then the system of linear equations corresponding to the matrix is solved using back-substitution. EXAMPLE 5 Using the Gaussian Elimination Method Solve the system. 3x - 6y + 9z = 0 4x - 6y + 8z = -4 -2x - y + z = 7 SOLUTION £ 3 -6 9 4 -6 8 -2 -1 1 3 0 -4 7§ Write the augmented matrix. £ 1 -2 3 4 -6 8 -2 -1 1 3 0 -4 7§ 1 3R1 £ 1 -2 3 0 2 -4 -2 -1 1 3 0 -4 7§ -4R1 + R2 £1 -2 3 0 2 -4 0 -5 7 3 0 -4 7§ 2R1 + R3 £1 -2 3 0 1 -2 0 -5 7 3 0 -2 7§ 1 2R2 £1 -2 3 0 1 -2 0 0 -3 3 0 -2 -3§ 5R2 + R3 £1 -2 3 0 1 -2 0 0 1 3 0 -2 1§ -1 3R3 Row-echelon form The corresponding equations are x - 2y + 3z = 0 y - 2z = -2 z = 1 The value of z is given. The values of y and x can now be found in reverse order by back-substitution.
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