For each archetype that is a system of equations, we have formulated a similar, yet different, homogeneous system of equations by replacing each equation's constant term with a zero. To wit, for Archetype C, we can convert the original system of equations into the homogeneous system, \begin{align*} 2x_1 - 3x_2 + x_3 - 6x_4 &= 0 \\ 4x_1 +x_2 +2x_3 + 9x_4 &= 0 \\ 3x_1 +x_2 +x_3 + 8x_4 &= 0 \end{align*}
Can you quickly find a solution to this system without row-reducing the augmented matrix?