Solving system of linear equations by Gaussian elimination?

(1....1....1....0)
(1....1....0....3)
(0....1....1....1) Row 2 – Row 1

(1....1....1....0)
(0....0..–1....3)
(0....1....1....1) Interchange rows 2 and 3

(1....1....1....0)
(0....1....1....1)
(0....0..–1....3)

z = –3

y – 3 = 1, y = 4

x + 4 – 3 = 0, x = –1

(–1, 4, –3)
 
Start out by making an augmented matrix with the coefficients of the x, y, and z-terms on the left and the constants on the right as follows:
[ 1 1 1 | 0 ]
[ 1 1 0 | 3 ]
[ 0 1 1 | 1 ]

Then, use elementary row operations until the left side is the identity matrix.
[ 1 1 1 | 0 ]
[ 1 1 0 | 3 ]
[ 0 1 1 | 1 ]

[ 1 0 0 | -1 ] <== R1 - R3 --> R1
[ 1 1 0 | 3 ]
[ 0 1 1 | 1 ]

[ 1 0 0 | -1 ]
[ 0 1 0 | 4 ] <== R2 - R1 --> R2
[ 0 1 1 | 1 ]

[ 1 0 0 | -1 ]
[ 0 1 0 | 4 ]
[ 0 0 1 | -3 ] <== R3 - R2 --> R3.

By reading off of the numbers on the right:
x = -1, y = 4, and z = -3.

I hope this helps!
 
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