Write the coefficients of the equations into an augmented matrix:
x - 2z = 1 to [(1, 0, -2,|,1)]
x + y - z = 2 to [(1, 0, -2,|,1),(1, 1, -1,|,2)]
2x - 3y - z = 8 to [(1, 0, -2,|,1),(1, 1, -1,|,2),(2, -3, -1,|,8)]
Perform Guass-Jordan Elimination using row operations on the augmented matrix:
[
(1, 0, -2,|,1),
(1, 1, -1,|,2),
(2, -3, -1,|,8)
]
Elementary Row Operations Notation
R_2 - R_1 to R_2
[
(1, 0, -2,|,1),
(0, 1, 1,|,1),
(2, -3, -1,|,8)
]
R_3 - 2R_1 to R_3
[
(1, 0, -2,|,1),
(0, 1, 1,|,1),
(0, -3, 3,|,6)
]
3R_2 + R_3 to R_3
[
(1, 0, -2,|,1),
(0, 1, 1,|,1),
(0, 0, 6,|,9)
]
R_3/6
[
(1, 0, -2,|,1),
(0, 1, 1,|,1),
(0, 0, 1,|,3/2)
]
R_2 - R_3 to R_2
[
(1, 0, -2,|,1),
(0, 1, 0,|,-1/2),
(0, 0, 1,|,3/2)
]
2R_3 + R_1 to R_1
[
(1, 0, 0,|,4),
(0, 1, 0,|,-1/2),
(0, 0, 1,|,3/2)
]
x = 4, y = -1/2, and z = 3/2
check:
4 - 2(3/2) = 1
4 + (-1/2) - 3/2 = 2
2(4) - 3(-1/2) - 3/2 = 8
1 = 1
2 = 2
8 = 8
This checks.