The equations can be written in matrix form as
[3, 8, -2; 3, 1, 4; -6, -9, -2]*[x; y; z] = [-8; 12; 6]
Make a new second row by subtracting it from the first row. The augmented coefficient matrix becomes
[3, 8, -2, -8; 0, 7, -6, -20; -6, -9, -2, 6]
Now add twice the first row to the third row to get a new third row.
[3, 8, -2, -8; 0, 7, -6, -20; 0, 7, -6, 10]
Subtract the third row from the second to get
[3, 8, -2, -8; 0, 7, -6, -20; 0, 0, 0, -30]
The third equation has been transformed to
0*z + 0*y + 0*z = -30
or
0 = -30
This tells us there is no solution to this set of equations. They are inconsistent.
[3, 8, -2; 3, 1, 4; -6, -9, -2]*[x; y; z] = [-8; 12; 6]
Make a new second row by subtracting it from the first row. The augmented coefficient matrix becomes
[3, 8, -2, -8; 0, 7, -6, -20; -6, -9, -2, 6]
Now add twice the first row to the third row to get a new third row.
[3, 8, -2, -8; 0, 7, -6, -20; 0, 7, -6, 10]
Subtract the third row from the second to get
[3, 8, -2, -8; 0, 7, -6, -20; 0, 0, 0, -30]
The third equation has been transformed to
0*z + 0*y + 0*z = -30
or
0 = -30
This tells us there is no solution to this set of equations. They are inconsistent.
