Write the equations as an augmented matrix:
#|
(2,1,1,|,0),
(3, -2,-3,|,-21),
(4,5,3,|,-2)
|#
Multiply row 2 by -2:
#|
(2,1,1,|,0),
(-6, 4,6,|,42),
(4,5,3,|,-2)
|#
Multiply row 1 by 3 and add to row 2:
#|
(2,1,1,|,0),
(0, 7,9,|,42),
(4,5,3,|,-2)
|#
Multiply row 1 by -2 and add to row 3:
#|
(2,1,1,|,0),
(0, 7,9,|,42),
(0,3,1,|,-2)
|#
Multiply row 3 by -2 and add to row 2:
#|
(2,1,1,|,0),
(0, 1,7,|,46),
(0,3,1,|,-2)
|#
Multiply row 2 by -3 and add to row 3:
#|
(2,1,1,|,0),
(0, 1,7,|,46),
(0,0,-20,|,-140)
|#
Multiply row 2 by 7/20
#|
(2,1,1,|,0),
(0, 1,7,|,46),
(0,0,-7,|,-49)
|#
Add row 3 to row 2:
#|
(2,1,1,|,0),
(0, 1,0,|,-3),
(0,0,-7,|,-49)
|#
Divide row 3 by -7
#|
(2,1,1,|,0),
(0, 1,0,|,-3),
(0,0,1,|,7)
|#
Subtract row 3 from row 1:
#|
(2,1,0,|,-7),
(0, 1,0,|,-3),
(0,0,1,|,7)
|#
Subtract row 2 from row 1:
#|
(2,0,0,|,-4),
(0, 1,0,|,-3),
(0,0,1,|,7)
|#
Divide row 1 by 2:
#|
(1,0,0,|,-2),
(0, 1,0,|,-3),
(0,0,1,|,7)
|#
#x = -2, y = -3, and z = 7#
check:
#2(-2) + (-3) + 7 = 0#
#3(-2) - 2(-3) -3(7) = -21#
#4(-2) + 5(-3) + 3(7) = -2#
#0 = 0#
#-21 = -21#
#-2 = -2#
This checks.
