Final answer:
To ensure the system of equations has no solutions, lines must be parallel, meaning they have the same slope but different y-intercepts. The calculations show that the values h = -6 and k = 8 meet these conditions, making the two equations parallel with no points of intersection.
Step-by-step explanation:
To find the values of h and k such that the system of equations has no solutions, we need to ensure that the two equations are parallel. The equations are given as 7x - 8y = h and -4x + ky = -6. Parallel lines have the same slope; thus, we need to manipulate both equations into slope-intercept form (y = mx + b), where m is the slope. For the first equation, the slope is 7/8 (from 7x - 8y = h, rearranged to y = (7/8)x - h/8). The second equation should have the same slope for the lines to be parallel. So we need to rearrange the second equation like this: ky = 4x - 6 → y = (4/k)x - 6/k. Equating the slopes gives us the fractions 7/8 and 4/k to be equal. By setting up the proportion 7/8 = 4/k, we can find that k = 8. Once we know k, we need to ensure that the y-intercepts (h/8 and -6/k) are different because parallel lines cannot have the same y-intercept. However, for the system to have no solutions, the two equations must have the same left-hand expressions but different right-hand constants. Therefore, h should not equal -6. The only option that confirms both conditions is h = -6 and k = 8, corresponding to option D.