For x ≥ 0 and y ≥ 0 satisfying 2x + y ≤ 14 and x + 3y ≤ 17, what is the greatest possible value of 3x + 2y?

The maximum of a linear expression over a bounded region occurs at a vertex. The constraints $x, y \geq 0$, $2x + y \leq 14$, and $x + 3y \leq 17$ give vertices $(0, 0)$, $(7, 0)$, $(0, \frac{17}{3})$, and the intersection of the lines. From $y = 14 - 2x$ into $x + 3y = 17$: $x + 42 - 6x = 17$, so $x = 5$ and $y = 4$. Evaluate $3x + 2y$: 0, 21, $\frac{34}{3} \approx 11.3$, and $15 + 8 = 23$. The maximum is 23, choice C. Choice B (21) is the value at $(7, 0)$ only. Choices A (28) and D (25) are not achieved by any vertex.