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

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Answer 1 · 2016-10-11T16:36:07

Please see the explanation for the matrix row operations.

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

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 $z = -4$

How do you solve the system 5x-4y+2z=215x−4y+2z=215x-4y+2z=21, -x-5y+6z=-24−x−5y+6z=−24-x-5y+6z=-24, and -x-4y+5z=-21−x−4y+5z=−21-x-4y+5z=-21?