# How do you solve the system -5x+3y+6z=4, -3x+y+5z=-5, and -4x+2y+z=13?

Oct 17, 2016

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

$x = - 2 , y = 4 , z = - 3$

#### Explanation:

Write −5x + 3y + 6z =4 into the first row of an augmented matrix:

[ (−5, 3, 6, |, 4) ]

Add −3x + y + 5z = −5 to the second row of the augmented matrix:

[ (−5, 3, 6, |, 4), (−3, 1, 5,|, −5) ]

Add −4x + 2y + z = 13 to the third row of the augmented matrix:

[ (−5, 3, 6, |, 4), (−3, 1, 5,|, −5), (−4, 2, 1, |, 13) ]

Subtract row 2 from row 1:

[ (−2, 2, 1, |, 9), (−3, 1, 5,|, −5), (−4, 2, 1, |, 13) ]

Subtract row 3 from row 1:

[ (2, 0, 0, |, -4), (−3, 1, 5,|, −5), (−4, 2, 1, |, 13) ]

Divide row 1 by 2:

[ (1, 0, 0, |, -2), (−3, 1, 5,|, −5), (−4, 2, 1, |, 13) ]

Multiply row 1 by 3 and add to row 2:

[ (1, 0, 0, |, -2), (0, 1, 5,|, -11), (−4, 2, 1, |, 13) ]

Multiply row 1 by 4 and add to row 3:

[ (1, 0, 0, |, -2), (0, 1, 5,|, -11), (0, 2, 1, |, 5) ]

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

[ (1, 0, 0, |, -2), (0, 1, 5,|, -11), (0, 0, -9, |, 27) ]

Divide row 3 by -9:

[ (1, 0, 0, |, -2), (0, 1, 5,|, -11), (0, 0, 1, |, -3) ]

Multiply row 3 by -5 and add to row 2:

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