The first equation, #x - y + 4z = 6#, makes the following row in the augmented matrix:

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

The second equation, #2x + z = 1#, makes the following row in the augmented matrix:

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

The third equation, #x + 5y + z = -9#, makes the following row in the augmented matrix:

#[
(1,-1,4,|,6),
(2,0,1,|,1),
(1,5,1,|,-9)
]#

Now, perform elementary row operations until, you obtain an identity matrix on the left.

#R_2-2R_1toR_2#

#[
(1,-1,4,|,6),
(0,2,-7,|,-11),
(1,5,1,|,-9)
]#

#R_3-R_1toR_3#

#[
(1,-1,4,|,6),
(0,2,-7,|,-11),
(0,6,-3,|,-15)
]#

#R_3-3R_2toR_3#

#[
(1,-1,4,|,6),
(0,2,-7,|,-11),
(0,0,18,|,18)
]#

#R_3/18#

#[
(1,-1,4,|,6),
(0,2,-7,|,-11),
(0,0,1,|,1)
]#

#R_2+7R_3toR_2#

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

#R_1-4R_2toR_1#

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

#R_2/2#

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

#R_1+R_2toR_1#

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

We have an identity matrix on the left, therefore, the solution set is on the right:

#x = 0, y = -2, and z = 1#

Check:

#x - y + 4z = 6#

#2x + z = 1#

#x + 5y + z = -9#

#0 - -2 + 4(1) = 6#

#2(0) + 1 = 1#

#0 + 5(-2) + 1 = -9#

#6 = 6#

#1 = 1#

#-9 = -9#

This checks.