# Help! Use Gauss-Jordan elimination (row reduction) to find all solutions to the following system of linear equations? 2x + 3y - z = 3 - x + 4y = 9 4x - y - z = -7 How? Help!

Nov 8, 2016

$x = - 1 , y = 2 \mathmr{and} z = 1$

#### Explanation:

Given: 2x + 3y - z = 3, - x + 4y = 9, 4x - y - z = -7

I am going to put the second equation into the first row of an augmented matrix:

[ (-1, 4, 0,|, 9) ]

Add the first equation as the second row in the matrix:

[ (-1, 4, 0,|, 9), (2,3,-1,|,3) ]

Add the third equation as the third row in the matrix:

[ (-1, 4, 0,|, 9), (2,3,-1,|,3), (4,-1,-1,|,-7) ]

Multiply the row 1 by 2 and add to row 2:

[ (-1, 4, 0,|, 9), (0,11,-1,|,21), (4,-1,-1,|,-7) ]

Multiply the row 1 by 4 and add to row 3:

[ (-1, 4, 0,|, 9), (0,11,-1,|,21), (0,15,-1,|,29) ]

Subtract row 3 from row 2:

[ (-1, 4, 0,|, 9), (0,-4,0,|,-8), (0,15,-1,|,29) ]

Add row 2 to row 1:

[ (-1, 0, 0,|, 1), (0,-4,0,|,-8), (0,15,-1,|,29) ]

Multiple row 1 by -1:

[ (1, 0, 0,|, -1), (0,-4,0,|,-8), (0,15,-1,|,29) ]

Divide row 2 by -4:

[ (1, 0, 0,|, -1), (0,1,0,|,2), (0,15,-1,|,29) ]

Multiply row 2 by -15 and add to row 3:

[ (1, 0, 0,|, -1), (0,1,0,|,2), (0,0,-1,|,-1) ]

Multiply row 3 by -1:

[ (1, 0, 0,|, -1), (0,1,0,|,2), (0,0,1,|,1) ]

$x = - 1 , y = 2 \mathmr{and} z = 1$