Example of solving a 3-by-3 system of linear equations by row-reducing the augmented matrix

Example: Solving a 3-by-3 System by Row Reduction

Solve:

2x + y - z =  8
-3x - y + 2z = -11
-2x + y + 2z = -3

Step 1: Write augmented matrix

[ 2  1 -1 |  8]
[-3 -1  2 |-11]
[-2  1  2 | -3]

Step 2: R₁ → R₁/2

[ 1  0.5 -0.5 |  4]
[-3 -1    2    |-11]
[-2  1    2    | -3]

Step 3: R₂ → R₂ + 3R₁, R₃ → R₃ + 2R₁

[ 1  0.5 -0.5 |  4   ]
[ 0  0.5  0.5 |  1   ]
[ 0  2    1    |  5   ]

Step 4: R₂ → 2R₂

[ 1  0.5 -0.5 |  4 ]
[ 0  1    1    |  2 ]
[ 0  2    1    |  5 ]

Step 5: R₃ → R₃ - 2R₂

[ 1  0.5 -0.5 |  4 ]
[ 0  1    1    |  2 ]
[ 0  0   -1    |  1 ]

Step 6: R₃ → -R₃, then back-substitute: z = -1, y = 2 - (-1) = 3, x = 4 - 0.5(3) + 0.5(-1) = 2

Solution: x = 2, y = 3, z = -1