For x ≥ 0 and y ≥ 0 satisfying x + y ≤ 8 and 3x + y ≤ 16, what is the greatest possible value of x + 2y?

The maximum of $x + 2y$ over the region occurs at a vertex. The region defined by $x, y \geq 0$, $x + y \leq 8$, and $3x + y \leq 16$ has vertices $(0, 0)$, $(\frac{16}{3}, 0)$, $(0, 8)$, and $(4, 4)$, where the two lines meet (subtracting $x + y = 8$ from $3x + y = 16$ gives $2x = 8$). Evaluate $x + 2y$: 0, $\frac{16}{3}$, 16, and $4 + 8 = 12$. The maximum is 16, choice C, at $(0, 8)$. Choice A (12) is the value at the intersection, which is tempting because it is the only non-axis corner. Choice B (8) and choice D ($\frac{16}{3}$) come from other points.