1) Consider the family of planes (x + y +cz=1) where c ranges all possible real values. Find the straight line on the xy-plane that is contained by all of these planes.

Haven't done any geometry for a while but here's how I would do it:
- Find two different points that are in the intersection of all the plans of equation x + y + cz = 1
- Find an equation of the line containing these two points.

The points (0, 1, 0) and (1, 0, 0) are in the intersection of all the plans because 0 + 1 + c*0 = 1 and 1 + 0 + c*0 = 1 for any real c.
Thus the vector (1, -1, 0) is a vector of the line we're looking for. So a parametric equation of this line could be {x = t, y = -t + 1, z = 0}, which gives you a cartesian equation in the xy plane: y = 1 -x

xy-plane has z=0 as equation.

Replace z by zero in the equation of a general plane of your family :

(*) x + y + cz = 1
(*) z = 0

(*) x + y = 1
(*) z = 0

Here is a system of equations for your straight-line.

Thanks guys. Really appreciate it.