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

1 Answer
Oct 17, 2016

Answer:

#[ (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) ]#