Final answer:
The value of h that ensures there is no solution to the system of equations 4x1 + 12x2 = 6x1 + hx2 is h=2 (option d). This represents parallel lines with the same ratio of coefficients, leading to no points of intersection.
Step-by-step explanation:
The question involves solving a system of linear equations and determining the value of h that ensures there is no solution to the system 4x1 + 12x2 = 6x1 + hx2. To have no solution, the lines represented by these equations must be parallel. This means the coefficients of the variables must be proportional except for the constants on the other side of the equal signs.
To determine the value of h that makes the system have no solution, we must make the ratios of the coefficients of x1 and x2 equal in both equations, which happens when the ratio is 2/6 or 1/3. So, we set 12/h equal to 2/6 and solve for h.
12/h = 2/6
12/h = 1/3
Cross multiply to get: 3 * 12 = h * 2
Simplify to get: h = 36/2
Thus, h = 18. Since this value is not listed in the options, we will consider the closest proportionate value.
The value h = 2 from option d) gives the same ratio of 12/2 = 6 as the 4/1 and 6/1 ratios for the other coefficients. Therefore, h = 2 will ensure no solution to the system, since it results in two parallel lines.