Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 1

1. A technology startup is analyzing two different revenue models for their new app. Model X follows a linear growth pattern: R(x) = 25x + 500, where x represents months after launch and R(x) is revenue in dollars. Model Y follows a quadratic growth pattern: R(x) = -x² + 50x + 400. After how many months will both models predict the same revenue, and what will that revenue be? Answer: After 5 months, the revenue will be $625 Solution: In business and economics, companies often compare different growth models to make strategic decisions.
   Full step-by-step solution

   In business and economics, companies often compare different growth models to make strategic decisions. Linear models represent steady growth, while quadratic models can represent growth that eventually slows down or peaks. Setting the equations equal allows you to find when these different predictions align, which is useful for planning and decision-making.

2. Matiu is analyzing two different investment plans for his savings. Plan A offers a fixed annual return of $120 plus 4% simple interest on the principal amount invested. Plan B offers a fixed annual return of $80 plus 6% simple interest on the same principal amount. Matiu wants to know what principal amount would make both plans yield the same total annual return, and what that total annual return would be. Write a system of equations to represent this situation and solve for the principal amount and the total annual return. Answer: $2000 principal, $200 total annual return Solution: Define variables. Let x = principal amount invested (in dollars). Let y = total annual return (in dollars).
   Full step-by-step solution

   Step 1: Define variables. Let x = principal amount invested (in dollars). Let y = total annual return (in dollars). Step 2: Write equations for each plan. Plan A: y = 0.04x + 120 Plan B: y = 0.06x + 80 Step 3: Set the equations equal to each other to find when the returns are the same. 0.04x + 120 = 0.06x + 80 Step 4: Solve for x. 0.04x + 120 - 0.04x = 0.06x + 80 - 0.04x 120 = 0.02x + 80 120 - 80 = 0.02x + 80 - 80 40 = 0.02x x = 40 / 0.02 x = 2000 Step 5: Substitute x = 2000 into either equation to find y. y = 0.04(2000) + 120 y = 80 + 120 y = 200 Step 6: Interpret the solution. With a principal of $2000, both plans yield a total annual return of $200. The answer is $2000 principal, $200 total annual return.

3. A technology startup is analyzing two different revenue models for their new app. Model A generates revenue according to the function R(x) = 2x² + 50x, where x is the number of premium subscribers in hundreds. Model B follows R(x) = -x² + 120x + 800. The company wants to know at what subscriber count both models predict the same revenue, and what that revenue will be. Write and solve a system of equations to answer this question. Answer: 1200 Solution: Set the two revenue functions equal to find when they predict the same revenue: 2x² + 50x = -x² + 120x + 800 2x² + 50x + x² - 120x - 800 = 0 3x² - 70x - 800 = 0 Solve the quadratic equation using the quadratic formula: x = [70 ± sqrt(70² - 4×3×(-800))] / (2×3) x = [70 ± sqrt(4900 + 9600)] / 6 x…
   Full step-by-step solution

   Step 1: Set the two revenue functions equal to find when they predict the same revenue: 2x² + 50x = -x² + 120x + 800 Step 2: Bring all terms to one side: 2x² + 50x + x² - 120x - 800 = 0 3x² - 70x - 800 = 0 Step 3: Solve the quadratic equation using the quadratic formula: x = [70 ± sqrt(70² - 4×3×(-800))] / (2×3) x = [70 ± sqrt(4900 + 9600)] / 6 x = [70 ± sqrt(14500)] / 6 x = [70 ± 50√5.8] / 6 Step 4: Calculate the two possible solutions: x = [70 + 120.416] / 6 ≈ 190.416/6 ≈ 31.74 x = [70 - 120.416] / 6 ≈ -50.416/6 ≈ -8.40 Step 5: Since x represents number of subscribers in hundreds, we discard the negative solution. x ≈ 31.74 means approximately 3,174 subscribers. Step 6: Calculate the revenue using either function: R(31.74) = 2(31.74)² + 50(31.74) R(31.74) = 2(1007.4) + 1587 R(31.74) = 2014.8 + 1587 ≈ 1201.8 The revenue is approximately $1,200 (in hundreds of dollars, so $120,000).

4. Emma is selling two types of concert tickets. Premium tickets cost $15 each and standard tickets cost $10 each. She sold 40 tickets total and collected $500. Write a system of equations to represent this situation, using p for premium tickets and s for standard tickets. Answer: p + s = 40, 15p + 10s = 500 Solution: Identify the variables: p = number of premium tickets, s = number of standard tickets Create the first equation for total tickets: p + s = 40 Create the second equation for total revenue: 15p + 10s = 500 The system of equations is: p + s = 40 and 15p + 10s = 500
   Full step-by-step solution

   Step 1: Identify the variables: p = number of premium tickets, s = number of standard tickets Step 2: Create the first equation for total tickets: p + s = 40 Step 3: Create the second equation for total revenue: 15p + 10s = 500 Step 4: The system of equations is: p + s = 40 and 15p + 10s = 500

5. Ava is comparing two different cell phone plans. Plan A charges a base fee of $36 per month plus $0.06 per text message. Plan B charges a base fee of $21 per month plus $0.11 per text message. Write a system of equations to represent this situation and determine the number of text messages for which both plans cost the same total amount per month, and what that total cost is. Answer: 300 text messages, $54 Solution: Define variables. Let x = number of text messages, and y = total monthly cost in dollars. Write equations for each plan.
   Full step-by-step solution

   Step 1: Define variables. Let x = number of text messages, and y = total monthly cost in dollars. Step 2: Write equations for each plan. Plan A: y = 0.06x + 36 Plan B: y = 0.11x + 21 Step 3: Set the equations equal to find when the costs are the same. 0.06x + 36 = 0.11x + 21 Step 4: Solve for x. 0.06x + 36 - 0.06x = 0.11x + 21 - 0.06x 36 = 0.05x + 21 36 - 21 = 0.05x + 21 - 21 15 = 0.05x x = 15 / 0.05 x = 300 Step 5: Substitute x = 300 into either equation to find y. y = 0.06(300) + 36 y = 18 + 36 y = 54 Step 6: Check with the other equation. y = 0.11(300) + 21 y = 33 + 21 y = 54 Both plans cost $54 when 300 text messages are sent. The answer is 300 text messages, $54.

6. Olivia is comparing two different pricing plans for a photography studio rental. Plan A charges a flat booking fee of $80 plus $25 per hour of use. Plan B charges a flat booking fee of $50 plus $35 per hour of use. Olivia wants to know after how many hours both plans would cost the same total amount, and what that total cost would be. Write a system of equations to represent this situation and solve for the number of hours and total cost. Answer: 3 hours, $155 Solution: Define variables. Let x = number of hours Let y = total cost in dollars Write equations for each plan. Plan A: y = 25x + 80 Plan B: y = 35x + 50 Since both equal y, set the expressions equal to each other.
   Full step-by-step solution

   Step 1: Define variables. Let x = number of hours Let y = total cost in dollars Step 2: Write equations for each plan. Plan A: y = 25x + 80 Plan B: y = 35x + 50 Step 3: Since both equal y, set the expressions equal to each other. 25x + 80 = 35x + 50 Step 4: Solve for x. 25x + 80 - 25x = 35x + 50 - 25x 80 = 10x + 50 80 - 50 = 10x + 50 - 50 30 = 10x x = 30/10 x = 3 Step 5: Substitute x = 3 into either equation to find y. Using Plan A: y = 25(3) + 80 = 75 + 80 = 155 Using Plan B: y = 35(3) + 50 = 105 + 50 = 155 Step 6: Interpret the solution. After 3 hours, both plans cost $155. The answer is 3 hours, $155.