Create Systems

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Answer Key & Explanations

Create Systems · Grade 9 · Worksheet 1

1. Liam is managing a small business that produces handmade crafts. He finds that his weekly profit P (in dollars) can be modeled by the quadratic function P(x) = -2x² + 80x - 600, where x represents the number of crafts sold. His friend Emma suggests that a linear model P(x) = 20x - 200 might be more accurate for their current production scale. At what production levels do these two profit models predict the same profit, and what is that profit amount? Answer: x = 10 and x = 20 crafts; P = $0 and P = $200 Solution: To find where the two profit models predict the same profit, we set the quadratic model equal to the linear model and solve for x. Set the two profit functions equal to each other. Quadratic model: P(x) = -2x² + 80x - 600 Linear model: P(x) = 20x - 200 -2x² + 80x - 600 = 20x - 200 Bring all…
   Full step-by-step solution

   To find where the two profit models predict the same profit, we set the quadratic model equal to the linear model and solve for x. Step 1: Set the two profit functions equal to each other. Quadratic model: P(x) = -2x² + 80x - 600 Linear model: P(x) = 20x - 200 So we write: -2x² + 80x - 600 = 20x - 200 Step 2: Bring all terms to one side of the equation to set it equal to zero. Subtract (20x - 200) from both sides: -2x² + 80x - 600 - 20x + 200 = 0 Step 3: Combine like terms. -2x² + (80x - 20x) + (-600 + 200) = 0 -2x² + 60x - 400 = 0 Step 4: Simplify the equation by dividing all terms by -2 to make the leading coefficient positive. (-2x²)/(-2) + (60x)/(-2) + (-400)/(-2) = 0 x² - 30x + 200 = 0 Step 5: Solve the quadratic equation by factoring. We need two numbers that multiply to 200 and add to -30. These numbers are -10 and -20 because (-10) × (-20) = 200 and (-10) + (-20) = -30. So we can factor as: (x - 10)(x - 20) = 0 Step 6: Find the values of x that satisfy the equation. Set each factor equal to zero: x - 10 = 0 → x = 10 x - 20 = 0 → x = 20 So the two models predict the same profit when x = 10 crafts and x = 20 crafts. Step 7: Find the profit amount at each production level. We can use either model to calculate the profit. The linear model is simpler: P(x) = 20x - 200 For x = 10: P(10) = 20(10) - 200 = 200 - 200 = 0 For x = 20: P(20) = 20(20) - 200 = 400 - 200 = 200 Therefore, the two profit models predict the same profit at: - 10 crafts sold, with a profit of $0 - 20 crafts sold, with a profit of $200

2. Emma is selling two types of lemonade at her stand. Small cups sell for $1.50 and large cups sell for $2.50. On Saturday, she sold 35 cups total and collected $67.50. Write a system of equations to represent this situation, using s for small cups and l for large cups. Answer: s + l = 35, 1.5s + 2.5l = 67.5 Solution: Define variables: Let s = number of small cups, l = number of large cups Create equation for total cups: s + l = 35 Create equation for total money: (price of small × quantity) + (price of large × quantity) = total Substitute prices: 1.5s + 2.5l = 67.5 The system of equations is: s + l = 35 and…
   Full step-by-step solution

   Step 1: Define variables: Let s = number of small cups, l = number of large cups Step 2: Create equation for total cups: s + l = 35 Step 3: Create equation for total money: (price of small × quantity) + (price of large × quantity) = total Step 4: Substitute prices: 1.5s + 2.5l = 67.5 Step 5: The system of equations is: s + l = 35 and 1.5s + 2.5l = 67.5

3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). A diagonal path is drawn from (2, 1) to (8, 5), and a circular fountain is centered at the intersection point of this diagonal with another line segment from (2, 5) to (8, 1). Find the coordinates of the fountain's center by solving the system of equations formed by these two diagonals. Answer: (5, 3) Solution: A = (2, 1), B = (8, 1), C = (8, 5), D = (2, 5). First diagonal: from A(2, 1) to C(8, 5). Second diagonal: from D(2, 5) to B(8, 1).
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Identify the two diagonals** The rectangular garden has vertices: A = (2, 1), B = (8, 1), C = (8, 5), D = (2, 5). First diagonal: from A(2, 1) to C(8, 5). Second diagonal: from D(2, 5) to B(8, 1). The fountain is at the intersection of these two diagonals. --- **Step 2: Find the equation of diagonal AC** Points A(2, 1) and C(8, 5): Slope m1 = (5 - 1) / (8 - 2) = 4 / 6 = 2/3. Equation through A(2, 1): y - 1 = (2/3)(x - 2) y = (2/3)(x - 2) + 1 y = (2/3)x - 4/3 + 1 y = (2/3)x - 4/3 + 3/3 y = (2/3)x - 1/3. So AC: y = (2/3)x - 1/3. --- **Step 3: Find the equation of diagonal DB** Points D(2, 5) and B(8, 1): Slope m2 = (1 - 5) / (8 - 2) = (-4) / 6 = -2/3. Equation through D(2, 5): y - 5 = (-2/3)(x - 2) y = (-2/3)(x - 2) + 5 y = (-2/3)x + 4/3 + 5 y = (-2/3)x + 4/3 + 15/3 y = (-2/3)x + 19/3. So DB: y = (-2/3)x + 19/3. --- **Step 4: Solve the system of equations** We have: (1) y = (2/3)x - 1/3 (2) y = (-2/3)x + 19/3 Set them equal: (2/3)x - 1/3 = (-2/3)x + 19/3 Multiply through by 3 to eliminate denominators: 2x - 1 = -2x + 19 2x + 2x = 19 + 1 4x = 20 x = 5. --- **Step 5: Find y** Substitute x = 5 into (1): y = (2/3)(5) - 1/3 y = 10/3 - 1/3 y = 9/3 y = 3. --- **Step 6: Conclusion** The intersection point is (5, 3). This is the center of the fountain. --- **Final answer:** (5, 3)

4. Matiu is selling two types of handmade crafts at a market. He sold 8 of type A and 6 of type B for a total of $124. Type A costs $4 more than type B. Write a system of equations to find the price of each type. Answer: a = 8, b = 4 Solution: When creating systems from word problems, identify what the variables represent, then translate each piece of information into an equation.
   Full step-by-step solution

   When creating systems from word problems, identify what the variables represent, then translate each piece of information into an equation. For example, if someone sold x of item P and y of item Q for total revenue R, that gives one equation. If one item costs C more than another, that gives a second equation relating the prices.

5. 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.

6. Mere is selling two types of handmade candles. She sold 8 large candles and 12 small candles for a total of $186. The price of a large candle is $5 more than twice the price of a 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: 8L + 12S = 186, L = 2S + 5 Solution: Identify the first equation from the total revenue. Mere sold 8 large candles and 12 small candles for $186. This gives: 8L + 12S = 186.
   Full step-by-step solution

   Step 1: Identify the first equation from the total revenue. Mere sold 8 large candles and 12 small candles for $186. This gives: 8L + 12S = 186. Step 2: Identify the second equation from the price relationship. A large candle costs $5 more than twice a small candle. This gives: L = 2S + 5. Step 3: The system of equations is: 8L + 12S = 186 and L = 2S + 5.

7. Solve the system: 2x + 3y = 12 and x - y = 1 Answer: x = 3, y = 2 Solution: Use substitution method. From the second equation x - y = 1, we get x = y + 1 Substitute x = y + 1 into the first equation: 2(y + 1) + 3y = 12 Simplify: 2y + 2 + 3y = 12 Combine like terms: 5y + 2 = 12 Subtract 2 from both sides: 5y = 10 Divide by 5: y = 2 Substitute y = 2 into x = y + 1: x = 2…
   Full step-by-step solution

   Step 1: Use substitution method. From the second equation x - y = 1, we get x = y + 1 Step 2: Substitute x = y + 1 into the first equation: 2(y + 1) + 3y = 12 Step 3: Simplify: 2y + 2 + 3y = 12 Step 4: Combine like terms: 5y + 2 = 12 Step 5: Subtract 2 from both sides: 5y = 10 Step 6: Divide by 5: y = 2 Step 7: Substitute y = 2 into x = y + 1: x = 2 + 1 = 3 Step 8: Check: 2(3) + 3(2) = 6 + 6 = 12 ✓ and 3 - 2 = 1 ✓ The solution is x = 3, y = 2.