978 CHAPTER 9 Systems and Matrices Concepts Examples Elimination Method Use multiplication and addition to eliminate a variable from one equation. To eliminate a variable, the coefficients of that variable in the equations must be additive inverses. Solving a Linear System with Three Unknowns Step 1 Eliminate a variable from any two of the equations. Step 2 Eliminate the same variable from a different pair of equations. Step 3 Eliminate a second variable using the resulting two equations in two variables to obtain an equation with just one variable whose value we can now determine. Step 4 Find the values of the remaining variables by substitution. Write the solution of the system as an ordered triple. Solve the system. x + 2y - z = 6 (1) x + y + z = 6 (2) 2x + y - z = 7 (3) Add equations (1) and (2). The variable z is eliminated, and the result is 2x + 3y = 12. Eliminate z again by adding equations (2) and (3) to obtain 3x + 2y = 13. Solve the resulting system. 2x + 3y = 12 (4) 3x + 2y = 13 (5) -6x - 9y = -36 Multiply (4) by -3. 6x + 4y = 26 Multiply (5) by 2. -5y = -10 Add. y = 2 Divide by -5. Substitute 2 for y in equation (4). 2x + 3122 = 12 (4) with y = 2 2x + 6 = 12 Multiply. 2x = 6 Subtract 6. x = 3 Divide by 2. Let y = 2 and x = 3 in any of the original equations to find z = 1. The solution set is 513, 2, 126. 9.2 Matrix Solution of Linear Systems Matrix Row Transformations For any augmented matrix of a system of linear equations, the following row transformations will result in the matrix of an equivalent system. 1. Interchange any two rows. 2. Multiply or divide the elements of any row by a nonzero real number. 3. Replace any row of the matrix by the sum of the elements of that row and a multiple of the elements of another row. Gauss-Jordan Method The Gauss-Jordan method is a systematic technique for applying matrix row transformations in an attempt to reduce a matrix to diagonal (reduced row-echelon) form, with 1s along the diagonal. Gaussian Elimination Method The Gaussian elimination method is a systematic technique for applying matrix row transformations in an attempt to reduce a matrix to row-echelon form. Solve the system. x + 3y = 7 2 x + y = 4 c 1 2 3 1 2 7 4d Augmented matrix c 1 0 3 -5 2 7 -10d -2R1 + R2 c 1 0 3 1 2 7 2d - 1 5 R2 c 1 0 0 1 2 1 2d -3R2 + R1 This leads to the system x = 1 y = 2. The solution set is 511, 226.
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