# How do you solve the system 5x-4y+2z=21, -x-5y+6z=-24, and -x-4y+5z=-21?

Oct 11, 2016

Please see the explanation for the matrix row operations.

$x = 5 , y = - 1$ and $z = - 4$

#### Explanation:

Write the row for −x − 5y + 6z = −24 into an augmented matrix:

[ (-1, -5, 6, |, -24) ]

Add the row for −x − 4y + 5z = −21:

[ (-1, -5, 6, |, -24), (-1, -4, 5, |, -21) ]

Add the row for 5x − 4y + 2z = 21:

[ (-1, -5, 6, |, -24), (-1, -4, 5, |, -21), (5, -4, 2, |, 21) ]

Multiply row 1 by -1 (and leave it that way) then add to row 2:

[ (1, 5, -6, |, 24), (0, 1, -1, |, 3), (5, -4, 2, |, 21) ]

Multiply row 1 by -5 and add to row 3:

[ (1, 5, -6, |, 24), (0, 1, -1, |, 3), (0, -29, 32, |, -99) ]

Multiply row 2 by 29 and add to row 3:

[ (1, 5, -6, |, 24), (0, 1, -1, |, 3), (0, 0, 3, |, -12) ]

Divide row 3 by 3:

[ (1, 5, -6, |, 24), (0, 1, -1, |, 3), (0, 0, 1, |, -4) ]

Add row 3 to row 2:

[ (1, 5, -6, |, 24), (0, 1, 0, |, -1), (0, 0, 1, |, -4) ]

Multiply row 3 by 6 and add to row 1:

[ (1, 5, 0, |, 0), (0, 1, 0, |, -1), (0, 0, 1, |, -4) ]

Multiply row 2 by -5 and add to row 1:

[ (1, 0, 0, |, 5), (0, 1, 0, |, -1), (0, 0, 1, |, -4) ]

This means that $x = 5 , y = - 1$ and $z = - 4$