Application to Solution of Simultaneous Equations

Find the condition that, the equation $lx+my+nz=0$ may be satisfied by the three sets of values $\left(a_1, b_1, c_1\right),\left(a_2, b_2, c_2\right),\left(a_3, b_3, c_3\right)$ and show that, it is the same as the condition, that the three equations $a_{1}x+b_{1}y+c_{1}z=0, a_{2}x+b_{2}y+c_{2}z=0, a_{3}x+b_{3}y+c_{3}z=0$ may be simultaneously satisfied by $l, m, n$.