A shopper mixes two packs, A and B. A pack of A has 2 units of protein, 3 units of calcium and 1…

Step 1. The model. Minimise $Z = 8x + 5y$ pounds subject to $2x + y \ge 13$, $3x + y \ge 15$, $x + 3y \ge 14$, $x \ge 0$ and $y \ge 0$.

Step 2. The corners. The line $3x + y = 15$ meets the $y$ axis at $(0, 15)$. It meets $2x + y = 13$ at $x = 2$, $y = 9$. The lines $2x + y = 13$ and $x + 3y = 14$ meet at $(5, 3)$. The line $x + 3y = 14$ meets the $x$ axis at $(14, 0)$. At $(5, 3)$ the calcium is $3(5) + 3 = 18 \ge 15$, so the rule is met.

Step 3. The costs. At $(0, 15)$, $Z = 75$. At $(2, 9)$, $Z = 16 + 45 = 61$. At $(5, 3)$, $Z = 40 + 15 = 55$. At $(14, 0)$, $Z = 112$. The smallest is $55$, and the region is unbounded with positive prices, so it is the minimum.

Step 4. Say it. Buy $5$ packs of A and $3$ packs of B. The cost is $55$ pounds.