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week04re - Math 43, Fall 2007 B. Dodson Week 5: Problem 1....

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Math 43, Fall 2007 B. Dodson Week 5: Problem 1. For the system x + y + z - w = 3 , 2 x + 4 y - 3 z + 7 w = 2 determine the coef. matrix A, and the augmented matrix £ A | ± b / . Solve the system using elemenary row operations, and write the solution in vector form. Solution: A = 1 1 1 - 1 2 4 - 3 7 , £ A | ± b / = 1 1 1 - 1 | 3 2 4 - 3 7 | 2 . £ A | ± b / 1 1 1 - 1 | 3 0 2 - 5 9 | - 4 ( R 2 - 2 R 1 ) 1 1 1 - 1 | 3 0 1 - 5 2 9 2 | - 2 ( 1 2 R 2 ) 1 0 7 2 - 11 2 | 5 0 1 - 5 2 9 2 | - 2 ( R 1 - R 2 ) The augmented matrix has been “reduced” to Reduced Row Echelon Form. For the last step, we notice “leading 1’s” in column 1 and 2, so x and y are leading variables. The other variables are free variables, and we use each row to solve for one leading variable. x + 7 2 z - 11 2 w = 5 , so x = 5 - 7 2 z + 11 2 w and y - 5 2 z + 9 2 w = - 2 , so y = - 2 + 5 2 z - 9 2 w. Finally, we replace the free variables by parameters, z = r, w = s, then [ x, y, z, w ] = [5 - 7 2 r + 11 2 s, -
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week04re - Math 43, Fall 2007 B. Dodson Week 5: Problem 1....

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