How to implement parametric form in system of equations

I need to implement lines in 3D in a system of equations. Assume a system of equation like this:

   F(1) = x(1)-x(3);
   F(2) = -x(5)+x(6);
   F(3) = x(7)+x(9)-4;
   F(4) = x(11)+x(12)-4;
   F(5)= sqrt((x(1)-x(4))^2+(x(2)-x(5))^2+(x(3)-x(6))^2)-2; 
   ...
   F(12)=...

where F(1)-F(4) are equations of planes in coordinate form. Instead of equations for planes, equations of lines should be implemented. To be more specific, these lines shall be the intersecting lines of the four planes F(1)-F(4). How to implement this, since there is no possibility to have lines in 3D in coordinate form.

complete system of equation attached.