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

[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.