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.
