Question

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!

Answer 1

Please see the explanation for steps leading to the answer:
`x = -1, y = 2 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 and z = 1`