2022 AMC 10B Problem 18

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 $1$ and the system has a solution other than $x=y=z=0$ . For example, one such system is $$\{ 1x + 1y + 0z = 0, 0x + 1y + 1z = 0, 0x + 0y + 0z = 0 \}$$ with a nonzero solution of $\{x,y,z\} = \{1, -1, 1\}$ . 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.)