A system of equations is consistent if it has a solution. The equations -6x-12y=3 and kx-24y=9 represent two straight lines. The solution is the coordinates of the point where the lines intersect.
Two lines do not intersect only of they are parallel to each other. This is the case if they have an equal slope.
The slope of -6x - 12y = 3 is -6/12 = -1/2. If the line kx-24y=9 has a slope -1/2, the value of k is -12.
The system of equations is consistent for all values of k except k = -12.