Line Intersection Problem

The Problem is:

Find the equations of the two straight lines, each of which cuts all four of the lines

x = 1, y = 0; y = 1, z = 0; z = 1, x = 0; x = y = 6z

I parameterized the four lines as: a(q)=(1,0,q), b(r)=(r,1,0), c(s)=(0,s,1), and d(t)=(6t,6t,t).

I've constructed two solution lines and formed a system of equations for each coordinate, but Mathematica is telling me there is no solution.

Can someone provide me with some insights on how to solve such a problem and if I've started this correctly?