Summary: System with three equations, four variables. Consistent. Null space of coefficient matrix has dimension 2. Coefficient matrix identical to that of Archetype E, vector of constants is different.
Definition:\begin{align*} 2x_1 + x_2 + 7x_3 - 7x_4 &= 8 \\ -3x_1 + 4x_2 -5x_3 - 6x_4 &= -12 \\ x_1 +x_2 + 4x_3 - 5x_4 &= 4 \end{align*}