How do you solve #x+4y-z=7#, #x+y+z=2# and #-2x-2y+2z=-8# using matrices?

1 Answer
Oct 14, 2016

Answer:

Please see the explanation for the process.
#x = 2, y = 1, z = -1#

Explanation:

Write #x + 4y - z = 7# into the first row of an augmented matrix:

#[ (1,4,-1,|,7) ]#

Write #x + y + z = 2# into the second row:

#[ (1,4,-1,|,7), (1,1,1,|,2) ]#

Write #-2x - 2y + 2z = -8# into the third row:

#[ (1,4,-1,|,7), (1,1,1,|,2), (-2, -2, 2, |, -8) ]#

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

#[ (1,4,-1,|,7), (0,-3,2,|,-5), (-2, -2, 2, |, -8) ]#

Multiply row 1 by 2 and add to row 3:

#[ (1,4,-1,|,7), (0,-3,2,|,-5), (0, 6, 0, |, 6) ]#

Divide row 3 by 6 and swap with row 2:

#[ (1,4,-1,|,7), (0, 1, 0, |, 1), (0,-3,2,|,-5) ]#

Multiply row 2 by 3 and add to row 3:

#[ (1,4,-1,|,7), (0, 1, 0, |, 1), (0,0,2,|,-2) ]#

Divide row 3 by 2:

#[ (1,4,-1,|,7), (0, 1, 0, |, 1), (0,0,1,|,-1) ]#

Add row 3 to row 1:

#[ (1,4,0,|,6), (0, 1, 0, |, 1), (0,0,1,|,-1) ]#

Multiply row 2 by -4 and add to row 1:

#[ (1,0,0,|,2), (0, 1, 0, |, 1), (0,0,1,|,-1) ]#

#x = 2, y = 1, z = -1#