determine the equation of the line through the point (a,0,0)and (0,a,0)
Neither of these points have a z-component. In both cases, the z-component = 0.
This means we have a line that lies completely, and only, in the xy-plane.
The two points reduce to: (a,0) amd (0,a), for which the equation is,
(y - 0)/(x - a) = (a - 0)/(0 - a)
y/(x - a) = a/(-a) = -1
y = a - x