Archetype D (and identically, Archetype E) can be converted to the homogeneous system, \begin{align*} 2x_1 + x_2 + 7x_3 - 7x_4 &= 0 \\ -3x_1 + 4x_2 -5x_3 - 6x_4 &= 0 \\ x_1 +x_2 + 4x_3 - 5x_4 &= 0 \end{align*} whose augmented matrix row-reduces to \begin{equation*} \begin{bmatrix} \leading{1} & 0 & 3 & -2 & 0 \\ 0 & \leading{1} & 1 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} \end{equation*}
By Theorem HSC, the system is consistent, and so the computation $n-r=4-2=2$ means the solution set contains two free variables by Theorem FVCS, and hence has infinitely many solutions. We can describe this solution set using the free variables $x_3$ and $x_4$, \begin{align*} S&=\setparts{\colvector{x_1\\x_2\\x_3\\x_4}}{x_1=-3x_3+2x_4,\,x_2=-x_3+3x_4}\\ &=\setparts{\colvector{-3x_3+2x_4\\-x_3+3x_4\\x_3\\x_4}}{ x_3,\,x_4\in\complex{\null}}\\ \end{align*}