Consider systems of three linear equations with two unknowns $x$ and $y$,
\begin{align*} a_1 x + b_1 y & = 0 \\ a_2 x + b_2 y & = 0 \\ a_3 x + b_3 y & = 0 \end{align*}
where each of the coefficients is either $0$ or $1$ and the system has a solution other than $x=y=0$. 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.)