From your last line simply divide through by k^2-9:
[ 0 0 1 | (3k+9)/k^2-9]
This tells you that z = (3k+9)/(k^2-9). As long as this value is defined you get a unique set of solutions for x, y, and z. For example if k=0 you get x=-1, y=1, and z = -1. If k= 4 you get x= -17, y= -11, z = 3. But notice that the (3k+0)/(k^2-9) is not defined for k= 3 or k=-3, so we need to see what happens for these two cases. For k=3 your last row becomes [0 0 0 | 15] which has no solution and for k = -3 it's [0 0 0| 0] which means the three original equations are not linearly indpendent and hence there are an infininite number of solutions.