The system ax + 3y = 12 and 4x + by = 24 has infinitely many solutions. If a = 2, what is the value of b?

With $a = 2$, the first equation is $2x + 3y = 12$. For infinitely many solutions, the second equation, $4x + by = 24$, must be a multiple of the first. The $x$-coefficients go from 2 to 4, a factor of 2, and the constants go from 12 to 24, also a factor of 2, so it is consistent. Then $b = 2 \times 3 = 6$, choice C. Choice A (3) copies the $y$-coefficient without scaling. Choice B (8) and choice D (12) do not satisfy the proportion; $b = 12$ would be the constant of the first equation. Every coefficient must be scaled by the same factor.