Archetype B can be converted to the homogeneous system, \begin{align*} -7x_1 -6 x_2 - 12x_3 &=0\\ 5x_1 + 5x_2 + 7x_3 &=0\\ x_1 +4x_3 &=0 \end{align*} whose augmented matrix row-reduces to \begin{equation*} \begin{bmatrix} \leading{1} & 0 & 0 & 0\\ 0 & \leading{1} & 0 & 0\\ 0 & 0 & \leading{1} & 0 \end{bmatrix} \end{equation*}
By Theorem HSC, the system is consistent, and so the computation $n-r=3-3=0$ means the solution set contains just a single solution. Then, this lone solution must be the trivial solution.