Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 3

1. Matiu is selling two types of handmade candles. The large candles sell for $18 each and the small candles sell for $12 each. In one day, he sold 25 candles and collected $390. Write a system of equations to represent this situation, using L for large candles and S for small candles. Answer: L + S = 25, 18L + 12S = 390 Solution: The first equation comes from the total number of candles sold: L + S = 25 The second equation comes from the total money collected: 18L + 12S = 390 This system of equations represents the situation: L + S = 25 18L + 12S = 390
   Full step-by-step solution

   Step 1: The first equation comes from the total number of candles sold: L + S = 25 Step 2: The second equation comes from the total money collected: 18L + 12S = 390 Step 3: This system of equations represents the situation: L + S = 25 18L + 12S = 390

2. Olivia runs a small business that produces custom notebooks. She has two production methods. Method A uses a machine with a fixed monthly cost of $75 and a material cost of $3 per notebook. Method B uses handcrafting with a fixed monthly cost of $45 and a material cost of $5 per notebook. Write a system of equations to represent the total monthly cost for each method in terms of the number of notebooks produced. Determine the number of notebooks for which both methods have the same total monthly cost, and what that cost is. Answer: 15 notebooks, $120 Solution: Define variables. Let x = number of notebooks produced per month. Let y = total monthly cost in dollars.
   Full step-by-step solution

   Step 1: Define variables. Let x = number of notebooks produced per month. Let y = total monthly cost in dollars. Step 2: Write equations for each method. Method A: y = 3x + 75 Method B: y = 5x + 45 Step 3: Since we want the number of notebooks where the costs are equal, set the equations equal to each other. 3x + 75 = 5x + 45 Step 4: Solve for x. 3x + 75 - 3x = 5x + 45 - 3x 75 = 2x + 45 75 - 45 = 2x + 45 - 45 30 = 2x x = 15 Step 5: Substitute x = 15 into either equation to find y. Using Method A: y = 3(15) + 75 = 45 + 75 = 120 Using Method B: y = 5(15) + 45 = 75 + 45 = 120 Step 6: Interpret the solution. When 15 notebooks are produced, both methods cost $120. The answer is 15 notebooks, $120.

3. Sophia is selling two types of handmade candles. The small candles sell for $6 each and the large candles sell for $11 each. Yesterday she sold 16 candles total and made $156. Write a system of equations to represent this situation, using s for small candles and l for large candles. Answer: s + l = 16, 6s + 11l = 156 Solution: The first equation comes from the total number of candles sold: s + l = 16 The second equation comes from the total money earned: (price of small candle × number sold) + (price of large candle × number sold) = total money Substitute the prices: 6s + 11l = 156 The system of equations is: s + l =…
   Full step-by-step solution

   Step 1: The first equation comes from the total number of candles sold: s + l = 16 Step 2: The second equation comes from the total money earned: (price of small candle × number sold) + (price of large candle × number sold) = total money Step 3: Substitute the prices: 6s + 11l = 156 Step 4: The system of equations is: s + l = 16 and 6s + 11l = 156

4. Isabella is selling two types of handmade candles. The large candles sell for $12 each and the small candles sell for $8 each. In one day, she sold 25 candles and collected $260. Write a system of equations to represent this situation, using L for large candles and S for small candles. Answer: L + S = 25, 12L + 8S = 260 Solution: When creating systems of equations from word problems, identify what each variable represents and what relationships are given.
   Full step-by-step solution

   When creating systems of equations from word problems, identify what each variable represents and what relationships are given. The total number of items gives one equation, while the total value or cost gives another equation. Make sure both equations use the same variables consistently.

5. Aroha is selling two types of handmade candles. She sold 9 large candles and 11 small candles for a total of $285. Each large candle costs $5 more than each small candle. Write a system of equations to represent this situation, using L for the price of a large candle and S for the price of a small candle. Answer: 9L + 11S = 285, L = S + 5 Solution: When creating systems of equations from word problems, look for two distinct mathematical relationships between the variables.
   Full step-by-step solution

   When creating systems of equations from word problems, look for two distinct mathematical relationships between the variables. One equation often comes from a total quantity (like total money earned or total items), while the other comes from a comparative relationship between the variables (like one being a certain amount more than the other).

6. Mason and Charlotte are comparing two different subscription plans for a streaming service. Plan A has a monthly fee of $12 plus a cost of $2 per movie rental. Plan B has a monthly fee of $7 plus a cost of $3 per movie rental. At what number of movie rentals will the total monthly cost of both plans be the same, and what will that total cost be? Write and solve a system of equations to find the answer. Answer: 5 movie rentals, $22 Solution: Define the variables. Let x = number of movie rentals, and let y = total monthly cost in dollars. Write an equation for Plan A.
   Full step-by-step solution

   Step 1: Define the variables. Let x = number of movie rentals, and let y = total monthly cost in dollars. Step 2: Write an equation for Plan A. The monthly fee is $12, and each rental costs $2, so y = 2x + 12. Step 3: Write an equation for Plan B. The monthly fee is $7, and each rental costs $3, so y = 3x + 7. Step 4: Since both equations equal y, set them equal to each other: 2x + 12 = 3x + 7. Step 5: Solve for x. Subtract 2x from both sides: 12 = x + 7. Subtract 7 from both sides: x = 5. Step 6: Substitute x = 5 into either original equation to find y. Using Plan A: y = 2(5) + 12 = 10 + 12 = 22. Using Plan B: y = 3(5) + 7 = 15 + 7 = 22. Both give y = 22. The answer is 5 movie rentals, $22.

7. Solve the system: 3x² - 2x + y = 7 and 2x + y = 3 Answer: x = 2, y = -1 Solution: From the second equation 2x + y = 3, solve for y: y = 3 - 2x Substitute y = 3 - 2x into the first equation: 3x² - 2x + (3 - 2x) = 7 Simplify: 3x² - 2x + 3 - 2x = 7 → 3x² - 4x + 3 = 7 Subtract 7 from both sides: 3x² - 4x - 4 = 0 Solve the quadratic equation using factoring: (3x + 2)(x - 2) = 0…
   Full step-by-step solution

   Step 1: From the second equation 2x + y = 3, solve for y: y = 3 - 2x Step 2: Substitute y = 3 - 2x into the first equation: 3x² - 2x + (3 - 2x) = 7 Step 3: Simplify: 3x² - 2x + 3 - 2x = 7 → 3x² - 4x + 3 = 7 Step 4: Subtract 7 from both sides: 3x² - 4x - 4 = 0 Step 5: Solve the quadratic equation using factoring: (3x + 2)(x - 2) = 0 Step 6: Find x values: x = -2/3 or x = 2 Step 7: Substitute back to find y: When x = -2/3: y = 3 - 2(-2/3) = 3 + 4/3 = 13/3 When x = 2: y = 3 - 2(2) = 3 - 4 = -1 Step 8: The solution is x = 2, y = -1