Consider systems of three linear equations with unknowns $x$, $y$, and $z$,
\begin{align*} a_1 x + b_1 y + c_1 z & = 0 \\ a_2 x + b_2 y + c_2 z & = 0 \\ a_3 x + b_3 y + c_3 z & = 0 \end{align*}
where each of the coefficients is either $0$ or $1$. Additionally, each equation must have exactly two non-zero coefficients. The system has a solution other than $x=y=z=0$. How many such systems of equations are there? (The equations in a system need not be distinct, and two systems containing the same equations in a different order are considered different.)