The easiest way to approach this problem is to study the coefficient matrix M.
The first row of M is 2 4 -k
The second row is 4 k 2
The third row is k 2 2
In order for there to be an infinite number of solutions, the determinant of M must be zero.
This leads to the equation k3 + 4 k - 40 = 0
A graphical calculator can be used to find that this equation has just one solution k = 3.03.
This means that, ultimately, the answer to the question is just one value.
Once the value of k has been determined, the analysis can be carried a bit further to show that
the three direction vectors ((2, 4, -3.03 ) ; (4, 3.03, 2) ; (3.03 , 2 ,2) all lie in the same plane.
These direction vectors give the directions normal to three planes. These three planes will have a common line of intersection. The direction of this line can be obtained by taking the the cross product of any two of these. The cross product of the first two of these is C = (17.18 , -16.12 , -9.94) . Thus any (x ,y, z) of the form (x, y, z) = ∧ C (for any real number ∧) will solve the three original equations ( with k = 3.03)
This whole analysis is made relatively easy by the fact the the original equations are all equal to zero (homogeneous case)
