1. **State the problem:** A shop wants to buy Pens and Notebooks to maximize profit. - Each Pen costs 2 and yields a profit of 1. - Each Notebook costs 5 and yields a profit of 2. - The shop can spend at most 20. - The shop can buy at most 10 items in total. 2. **Define variables:** Let $x$ = number of Pens Let $y$ = number of Notebooks 3. **Write constraints:** - Cost constraint: $$2x + 5y \leq 20$$ - Quantity constraint: $$x + y \leq 10$$ - Non-negativity: $$x \geq 0, y \geq 0$$ 4. **Profit function to maximize:** $$P = 1 \cdot x + 2 \cdot y = x + 2y$$ 5. **Find feasible integer points satisfying constraints:** - From cost: $2x + 5y \leq 20$ - From quantity: $x + y \leq 10$ 6. **Check corner points of the feasible region:** - $(0,0)$: $P=0$ - $(0,4)$: cost $= 5\times4=20$, quantity $=4$, profit $=0+2\times4=8$ - $(5,0)$: cost $=2\times5=10$, quantity $=5$, profit $=5+0=5$ - $(5,2)$: cost $=2\times5+5\times2=10+10=20$, quantity $=7$, profit $=5+4=9$ - $(7,1)$: cost $=14+5=19$, quantity $=8$, profit $=7+2=9$ 7. **Evaluate profit at these points:** - Max profit is 9 at $(5,2)$ and $(7,1)$. 8. **Conclusion:** The shop should buy either 5 Pens and 2 Notebooks or 7 Pens and 1 Notebook to maximize profit.