(a) find parametric equations for the line of intersection of the planes x + y + z = 2 and x + 7y + 7z = 2

Direction vector of line of intersection of two planes is the cross product of the normal vectors of the planes, namely
p1: x+y+z=2
p2: x+7y+7z=2

and the corresponding normal vectors are: (equiv. to coeff. of the plane)
n1:<1,1,1>
n2:<1,7,7>

The cross product n1 x n2
vl=
 i  j  l
1 1 1
1 7 7
=<7-7, 1-7, 7-1>
=<0,-6,6>
Simplify by reducing length by a factor of 6
vl=<0,-1,1>

By observing the equations of the two planes, we see that (2,0,0) is a point on the intersection, because this points satisfies both plane equations.

Thus the parametric equation of the line is
L: (2,0,0)+t(0,-1,1)
or
L: x=2, y=-t, z=t