You begin by writing each equation in the form #ax+by+cz = d#:
#13x-43y-4z=8#
#3x-33y-13z=-10#
#343x-121y-64z=0#
Next, you use the 3 equations to create an augmented matrix :
#[
(13,-43,-4,|,8),
(3,-33,-13,|,-10),
(343,-121,-64,|,0)
]#
Perform elementary row operations on the augmented matrix, until its left side is an identity matrix .
Multiply row 2 by -4 and add it to row 1:
#[
(1,89,48,|,48),
(3,-33,-13,|,-10),
(343,-121,-64,|,0)
]#
Multiply row 1 by -3 and add it to row 2:
#[
(1,89,48,|,48),
(0,-300,-157,|,-154),
(343,-121,-64,|,0)
]#
Multiply row 1 by -343 and add it to row 3:
#[
(1,89,48,|,48),
(0,-300,-157,|,-154),
(0,-30648,-16528,|,-16464)
]#
Multiply row 2 by -102.16 and add it row 3:
#[
(1,89,48,|,48),
(0,-300,-157,|,-154),
(0,0,-488.88,|,-731.36)
]#
Divide row 3 by -488.88:
#[
(1,89,48,|,48),
(0,-300,-157,|,-154),
(0,0,1,|,1306/873)
]#
Multiply row 3 by 157 and add to row 2:
#[
(1,89,48,|,48),
(0,-300,0,|,211800/2619),
(0,0,1,|,1306/873)
]#
Divide row 2 by -300:
#[
(1,89,48,|,48),
(0,1,0,|,-706/2619),
(0,0,1,|,1306/873)
]#
Multiply row 3 by -48 and add to row 1:
#[
(1,89,0,|,-62352/2619),
(0,1,0,|,-706/2619),
(0,0,1,|,1306/873)
]#
Multiply row 2 by -89 and add to row 1:
#[
(1,0,0,|,482/2619),
(0,1,0,|,-706/2619),
(0,0,1,|,1306/873)
]#
We have an identity matrix on the left, therefore, the values of x,y,z are in the column vector on the right:
#x = 482/2619, y = -706/2619, z = 1306/873#
