Constraint Systems

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Answer Key & Explanations

Constraint Systems · Grade 9 · Worksheet 1

1. Mason has a budget of $272 to buy two types of plants: ferns that cost $17 each and succulents that cost $12 each. He needs at least 7 ferns and at least 12 succulents. Write the system of inequalities that represents this situation, using x for ferns and y for succulents. Answer: 17x + 12y ≤ 272, x ≥ 7, y ≥ 12 Solution: Define the variables: Let x be the number of ferns, and y be the number of succulents. Write the cost constraint: Each fern costs $17, so the cost for ferns is 17x.
   Full step-by-step solution

   Step 1: Define the variables: Let x be the number of ferns, and y be the number of succulents. Step 2: Write the cost constraint: Each fern costs $17, so the cost for ferns is 17x. Each succulent costs $12, so the cost for succulents is 12y. The total cost must be less than or equal to the budget of $272. This gives: 17x + 12y ≤ 272. Step 3: Write the minimum fern constraint: Mason needs at least 7 ferns, so x ≥ 7. Step 4: Write the minimum succulent constraint: Mason needs at least 12 succulents, so y ≥ 12. Step 5: Combine all constraints: The system of inequalities is: 17x + 12y ≤ 272, x ≥ 7, y ≥ 12.

2. A right triangle is drawn on a coordinate plane with vertices at A(0,0), B(9,0), and C(0,12). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the area of this circumscribed circle? (Use π = 3.14) Answer: 176.625 Solution: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. The hypotenuse is between points B(9,0) and C(0,12).
   Full step-by-step solution

   Step 1: In a right triangle, the hypotenuse is the diameter of the circumscribed circle. Step 2: The hypotenuse is between points B(9,0) and C(0,12). Step 3: Calculate the length of BC using the distance formula: sqrt((9-0)^2 + (0-12)^2) = sqrt(81 + 144) = sqrt(225) = 15 units. Step 4: This means the diameter of the circle is 15 units, so the radius is 15/2 = 7.5 units. Step 5: Calculate the area of the circle using A = πr^2 = 3.14 × (7.5)^2 = 3.14 × 56.25 = 176.625 square units. The answer is 176.625.

3. Lena is designing a rectangular garden with a perimeter of 40 meters. She wants the length to be at least 3 meters more than the width, and the area must be at least 60 square meters. Write a system of inequalities that represents the constraints on the garden's dimensions, using l for length and w for width. Answer: 2l + 2w = 40, l ≥ w + 3, l * w ≥ 60 Solution: - Perimeter = 40 meters - Length = l, Width = w - Length at least 3 meters more than width → l ≥ w + 3 - Area at least 60 square meters → l × w ≥ 60 We need a system of inequalities (and one equation) for l and w.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the problem** We have a rectangular garden with: - Perimeter = 40 meters - Length = l, Width = w - Length at least 3 meters more than width → l ≥ w + 3 - Area at least 60 square meters → l × w ≥ 60 We need a system of inequalities (and one equation) for l and w. --- **Step 2: Perimeter equation** Perimeter of a rectangle = 2 × length + 2 × width So: 2l + 2w = 40 We can leave it as is, or simplify by dividing by 2: l + w = 20 But the given correct answer keeps it as 2l + 2w = 40, so we'll use that. --- **Step 3: Length and width relationship** "Length to be at least 3 meters more than width" means: l ≥ w + 3 --- **Step 4: Area condition** Area = l × w "Area must be at least 60" means: l × w ≥ 60 --- **Step 5: Combine all constraints** From above: 1. Perimeter: 2l + 2w = 40 2. Length condition: l ≥ w + 3 3. Area condition: l × w ≥ 60 --- **Final system of inequalities (and one equation):** 2l + 2w = 40 l ≥ w + 3 l × w ≥ 60 --- This matches the given correct answer.

4. Aisha is designing a rectangular garden with a perimeter of 40 meters. She wants the length to be at least 4 meters more than the width, and the area must be at least 96 square meters. Write a system of inequalities that represents the constraints on the length (L) and width (W) of Aisha's garden. Answer: 2L + 2W = 40, L ≥ W + 4, L * W ≥ 96 Solution: Perimeter = 2 × Length + 2 × Width We are told the perimeter is 40 meters.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Perimeter constraint** The perimeter of a rectangle is given by: Perimeter = 2 × Length + 2 × Width We are told the perimeter is 40 meters. So: 2L + 2W = 40 We can simplify this by dividing both sides by 2: L + W = 20 But the problem asks for a system of inequalities, so we keep the perimeter as an equation (which is allowed in a system with inequalities): 2L + 2W = 40 --- **Step 2: Length and width relationship** We are told: "the length to be at least 4 meters more than the width." "At least" means greater than or equal to. So: L ≥ W + 4 --- **Step 3: Area constraint** The area of a rectangle is Length × Width. We are told: "the area must be at least 96 square meters." So: L × W ≥ 96 --- **Step 4: Write the system** The system of inequalities (and one equation) is: 1. 2L + 2W = 40 2. L ≥ W + 4 3. L × W ≥ 96 --- **Final answer:** 2L + 2W = 40, L ≥ W + 4, L * W ≥ 96

5. Matiu is buying notebooks and pens for his study group. Each notebook costs $8 and each pen costs $2. He has a budget of $60 and needs at least 4 notebooks and at least 6 pens. Write the system of inequalities that represents these constraints, where x is the number of notebooks and y is the number of pens. Answer: 8x + 2y ≤ 60, x ≥ 4, y ≥ 6 Solution: Write the cost constraint inequality. Each notebook costs $8, so total notebook cost is 8x. Each pen costs $2, so total pen cost is 2y.
   Full step-by-step solution

   Step 1: Write the cost constraint inequality. Each notebook costs $8, so total notebook cost is 8x. Each pen costs $2, so total pen cost is 2y. The total cost must be ≤ $60: 8x + 2y ≤ 60 Step 2: Write the minimum notebook requirement. Matiu needs at least 4 notebooks: x ≥ 4 Step 3: Write the minimum pen requirement. Matiu needs at least 6 pens: y ≥ 6 Step 4: Combine all inequalities into the system: 8x + 2y ≤ 60 x ≥ 4 y ≥ 6 The system of inequalities is: 8x + 2y ≤ 60, x ≥ 4, y ≥ 6

6. Noah is planning a school event with a budget of $240. He needs to buy two types of items: decorations that cost $16 each and snacks that cost $9 each. He needs at least 8 decorations and at least 10 snacks. Write the system of inequalities that represents these constraints, where x is the number of decorations and y is the number of snacks. Answer: 16x + 9y ≤ 240, x ≥ 8, y ≥ 10 Solution: Write the cost constraint. Decorations cost $16 each, so the total cost for decorations is 16x. Snacks cost $9 each, so the total cost for snacks is 9y.
   Full step-by-step solution

   Step 1: Write the cost constraint. Decorations cost $16 each, so the total cost for decorations is 16x. Snacks cost $9 each, so the total cost for snacks is 9y. The total cost is 16x + 9y. Noah's budget is $240, and he cannot spend more than that. So the inequality is 16x + 9y ≤ 240. Step 2: Write the minimum decorations constraint. Noah needs at least 8 decorations, so x must be greater than or equal to 8. The inequality is x ≥ 8. Step 3: Write the minimum snacks constraint. Noah needs at least 10 snacks, so y must be greater than or equal to 10. The inequality is y ≥ 10. Step 4: Combine all constraints into a system. The system of inequalities is: 16x + 9y ≤ 240 x ≥ 8 y ≥ 10 Note: While not explicitly stated, it is also implied that x and y are non-negative (x ≥ 0, y ≥ 0), but the given minimum constraints already ensure this.

7. 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