algebralinear equationAMCAMC 102022 AMC2022 AMC 10Bsystem of equations
Problem Statement
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 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.)<spanclass=′latex−bold′>(A)</span>302<spanclass=′latex−bold′>(B)</span>338<spanclass=′latex−bold′>(C)</span>340<spanclass=′latex−bold′>(D)</span>343<spanclass=′latex−bold′>(E)</span>344