Constraint Systems

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Answer Key & Explanations

Constraint Systems · Grade 9 · Worksheet 3

1. Emma is organizing a school fundraiser by selling handmade bracelets and keychains. Each bracelet requires 3 hours of labor and $5 in materials, while each keychain requires 1 hour of labor and $3 in materials. Emma has at most 45 hours of labor available and a budget of at most $105 for materials. She wants to make at least 3 keychains for every bracelet to meet demand, and she must make at least 5 bracelets. Write a system of inequalities that represents all possible numbers of bracelets (b) and keychains (k) Emma can produce. Answer: 3b + k ≤ 45, 5b + 3k ≤ 105, k ≥ 3b, b ≥ 5, b ≥ 0, k ≥ 0 Solution: Labor constraint. Each bracelet uses 3 hours, each keychain uses 1 hour, total hours ≤ 45. So: 3b + k ≤ 45.
   Full step-by-step solution

   Step 1: Labor constraint. Each bracelet uses 3 hours, each keychain uses 1 hour, total hours ≤ 45. So: 3b + k ≤ 45. Step 2: Materials constraint. Each bracelet uses $5, each keychain uses $3, total cost ≤ $105. So: 5b + 3k ≤ 105. Step 3: Ratio constraint. At least 3 keychains for every bracelet means the number of keychains is at least three times the number of bracelets. So: k ≥ 3b. Step 4: Minimum bracelets. She must make at least 5 bracelets: b ≥ 5. Step 5: Non-negativity. You cannot make a negative number of items: b ≥ 0, k ≥ 0. The system of inequalities is: 3b + k ≤ 45 5b + 3k ≤ 105 k ≥ 3b b ≥ 5 b ≥ 0 k ≥ 0

2. Emma is organizing a school event with a budget of $300. She needs to buy pizzas ($15 each) and drinks ($5 each). She needs at least 10 pizzas and at least 20 drinks. Write the system of inequalities that represents these constraints, where x is the number of pizzas and y is the number of drinks. Answer: 15x + 5y ≤ 300, x ≥ 10, y ≥ 20 Solution: Identify the cost constraint. Each pizza costs $15 and each drink costs $5, with a total budget of $300. This gives: 15x + 5y ≤ 300.
   Full step-by-step solution

   Step 1: Identify the cost constraint. Each pizza costs $15 and each drink costs $5, with a total budget of $300. This gives: 15x + 5y ≤ 300. Step 2: Identify the minimum pizza requirement. She needs at least 10 pizzas: x ≥ 10. Step 3: Identify the minimum drink requirement. She needs at least 20 drinks: y ≥ 20. Step 4: Combine all constraints into a system: 15x + 5y ≤ 300, x ≥ 10, y ≥ 20.

3. Aisha is designing a rectangular stage for a school play. The stage must have a perimeter of 50 meters. She needs the length to be at least 5 meters more than the width to accommodate the set design. Additionally, the area must be at least 126 square meters to fit all the performers. Write a system of inequalities that represents these constraints, using L for length and W for width. Answer: 2L + 2W = 50, L ≥ W + 5, L × W ≥ 126 Solution: When modeling constraints for rectangular dimensions, we use the perimeter formula to create an equation, then translate comparative statements ('at least') into inequalities, and finally express area requirements as product inequalities.
   Full step-by-step solution

   When modeling constraints for rectangular dimensions, we use the perimeter formula to create an equation, then translate comparative statements ('at least') into inequalities, and finally express area requirements as product inequalities. This approach helps define the feasible region of possible dimensions that satisfy all conditions simultaneously.

4. A rectangular garden has a length that is 3 meters more than twice its width. The area of the garden must be at least 54 square meters, and the perimeter cannot exceed 30 meters. Write a system of inequalities that represents these constraints, using w for width and l for length. Answer: l = 2w + 3, l × w ≥ 54, 2l + 2w ≤ 30 Solution: - \( w \) = width of the garden (in meters) - \( l \) = length of the garden (in meters) "Length is 3 meters more than twice its width" means: l = 2w + 3 "Area must be at least 54 square meters" means: l \times w \geq 54 Perimeter of a rectangle is \( 2l + 2w \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define variables** We are told: - \( w \) = width of the garden (in meters) - \( l \) = length of the garden (in meters) --- **Step 2: Translate the length condition** "Length is 3 meters more than twice its width" means: \[ l = 2w + 3 \] --- **Step 3: Translate the area condition** "Area must be at least 54 square meters" means: \[ l \times w \geq 54 \] --- **Step 4: Translate the perimeter condition** Perimeter of a rectangle is \( 2l + 2w \). "Perimeter cannot exceed 30 meters" means: \[ 2l + 2w \leq 30 \] We can simplify this by dividing through by 2: \[ l + w \leq 15 \] But the problem's given correct answer uses \( 2l + 2w \leq 30 \), so we can keep it in that form. --- **Step 5: Write the system of inequalities** The system is: \[ l = 2w + 3 \] \[ l \times w \geq 54 \] \[ 2l + 2w \leq 30 \] --- **Final answer:** l = 2w + 3, l × w ≥ 54, 2l + 2w ≤ 30

5. Sophia is buying plants for her garden. She has a budget of $200. Rose bushes cost $15 each and hydrangea bushes cost $25 each. She wants at least 7 plants total and at least 3 of each type. Write the system of inequalities that represents these constraints, where r is the number of rose bushes and h is the number of hydrangea bushes. Answer: 15r + 25h ≤ 200, r + h ≥ 7, r ≥ 3, h ≥ 3 Solution: Write the cost constraint. Each rose bush costs $15 and each hydrangea costs $25, with a total budget of $200: 15r + 25h ≤ 200 Write the total plants constraint.
   Full step-by-step solution

   Step 1: Write the cost constraint. Each rose bush costs $15 and each hydrangea costs $25, with a total budget of $200: 15r + 25h ≤ 200 Step 2: Write the total plants constraint. Sophia wants at least 7 plants total: r + h ≥ 7 Step 3: Write the individual plant constraints. She wants at least 3 of each type: r ≥ 3 and h ≥ 3 Step 4: Combine all constraints into the system: 15r + 25h ≤ 200 r + h ≥ 7 r ≥ 3 h ≥ 3 The complete system of inequalities is: 15r + 25h ≤ 200, r + h ≥ 7, r ≥ 3, h ≥ 3

6. Aroha is organizing a school fundraiser with two types of tickets: student tickets cost $12 and adult tickets cost $18. She needs to raise at least $900 and can sell no more than 65 tickets total. Write the system of inequalities that represents these constraints, where x is the number of student tickets and y is the number of adult tickets. Answer: 12x + 18y ≥ 900, x + y ≤ 65, x ≥ 0, y ≥ 0 Solution: Identify the revenue constraint - Aroha needs at least $900, so 12x + 18y ≥ 900 Identify the ticket quantity constraint - No more than 65 tickets total, so x + y ≤ 65 Add non-negativity constraints since you can't have negative tickets, so x ≥ 0 and y ≥ 0 The complete system is: 12x + 18y ≥ 900,…
   Full step-by-step solution

   Step 1: Identify the revenue constraint - Aroha needs at least $900, so 12x + 18y ≥ 900 Step 2: Identify the ticket quantity constraint - No more than 65 tickets total, so x + y ≤ 65 Step 3: Add non-negativity constraints since you can't have negative tickets, so x ≥ 0 and y ≥ 0 Step 4: The complete system is: 12x + 18y ≥ 900, x + y ≤ 65, x ≥ 0, y ≥ 0