Summary: System with three equations, four variables. Inconsistent. Null space of coefficient matrix has dimension 2. Coefficient matrix identical to that of Archetype D, constant vector is different.
Definition:\begin{align*} 2x_1 + x_2 + 7x_3 - 7x_4 &= 2 \\ -3x_1 + 4x_2 -5x_3 - 6x_4 &= 3 \\ x_1 +x_2 + 4x_3 - 5x_4 &= 2 \end{align*}