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Systems by Graphing

Grade 8 · Algebra · Worksheet 1

  1. Isabella is organizing a school bake sale and needs to choose between two pricing strategies for her team. In Strategy A, they charge a $40 base fee for table rental plus $2.50 per baked item sold. In Strategy B, they charge a $10 base fee for table rental plus $3.75 per baked item sold. Isabella wants to graph both cost equations on the same coordinate plane to determine for how many baked items the total cost would be the same with both strategies. Let x represent the number of baked items sold and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Write your answer as an ordered pair (x, y). Answer: ______________
  2. A scientist is tracking the growth of two different bacteria cultures. Culture A starts with 800 cells and grows at a rate of 300 cells per hour. Culture B starts with 2,000 cells and grows at a rate of 100 cells per hour. After how many hours will both cultures have the same number of cells? Answer: ______________
  3. y = 3x - 4 and y = -2x + 6 Answer: ______________
  4. Liam is planning a school fundraiser and needs to decide between selling pizza slices and hot dogs. He calculates that each pizza slice costs $2 to make and will be sold for $5, while each hot dog costs $1 to make and will be sold for $3. To break even on the event, he needs the total profit from both items to be $120. If he sells p pizza slices and h hot dogs, the system of equations representing this situation is: 3p + 2h = 120 (profit equation) and p + h = 50 (total items sold). Graph these equations to find how many pizza slices and hot dogs Liam needs to sell to meet his goal. Answer: ______________
  5. Liam is planning a school fundraiser and needs to decide between two pricing options for tickets. Option A has a fixed cost of $20 plus $2 per person, while Option B has a fixed cost of $10 plus $3 per person. Liam wants to know for how many people the total cost of both options would be the same. Write your answer as an ordered pair (number of people, total cost). Answer: ______________
  6. A rectangle is drawn on a coordinate plane with vertices at (2, 1), (6, 1), (6, 4), and (2, 4). A second rectangle has vertices at (1, 2), (5, 2), (5, 5), and (1, 5). Describe the transformation that maps the first rectangle onto the second rectangle, including the direction and distance of any translation. Answer: ______________
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Answer Key & Explanations

Systems by Graphing · Grade 8 · Worksheet 1

  1. Isabella is organizing a school bake sale and needs to choose between two pricing strategies for her team. In Strategy A, they charge a $40 base fee for table rental plus $2.50 per baked item sold. In Strategy B, they charge a $10 base fee for table rental plus $3.75 per baked item sold. Isabella wants to graph both cost equations on the same coordinate plane to determine for how many baked items the total cost would be the same with both strategies. Let x represent the number of baked items sold and y represent the total cost in dollars. After graphing the system of equations, what is the point of intersection? Write your answer as an ordered pair (x, y). Answer: (24, 100) Solution: Write the system of equations. For Strategy A: y = 2.50x + 40 For Strategy B: y = 3.75x + 10 Set the equations equal to each other to find the x-coordinate of the intersection. 2.50x + 40 = 3.75x + 10 Solve for x.
    Full step-by-step solution

    Step 1: Write the system of equations. For Strategy A: y = 2.50x + 40 For Strategy B: y = 3.75x + 10 Step 2: Set the equations equal to each other to find the x-coordinate of the intersection. 2.50x + 40 = 3.75x + 10 Step 3: Solve for x. 40 - 10 = 3.75x - 2.50x 30 = 1.25x x = 30 / 1.25 x = 24 Step 4: Substitute x = 24 into either equation to find y. Using y = 2.50x + 40: y = 2.50(24) + 40 y = 60 + 40 y = 100 Step 5: Write the solution as an ordered pair. The point of intersection is (24, 100). This means when 24 baked items are sold, both strategies result in a total cost of $100.

  2. A scientist is tracking the growth of two different bacteria cultures. Culture A starts with 800 cells and grows at a rate of 300 cells per hour. Culture B starts with 2,000 cells and grows at a rate of 100 cells per hour. After how many hours will both cultures have the same number of cells? Answer: 6 Solution: Let \( t \) = number of hours after the start. Culture A starts with 800 cells and grows at 300 cells per hour.
    Full step-by-step solution

    Let's solve this step by step. --- **Step 1: Define variables** Let \( t \) = number of hours after the start. Culture A starts with 800 cells and grows at 300 cells per hour. So the number of cells in Culture A after \( t \) hours is: \[ A(t) = 800 + 300t \] Culture B starts with 2000 cells and grows at 100 cells per hour. So the number of cells in Culture B after \( t \) hours is: \[ B(t) = 2000 + 100t \] --- **Step 2: Set up the equation** We want the time \( t \) when both cultures have the same number of cells: \[ A(t) = B(t) \] \[ 800 + 300t = 2000 + 100t \] --- **Step 3: Solve for \( t \)** Subtract \( 100t \) from both sides: \[ 800 + 300t - 100t = 2000 \] \[ 800 + 200t = 2000 \] Subtract 800 from both sides: \[ 200t = 1200 \] Divide both sides by 200: \[ t = 1200 / 200 \] \[ t = 6 \] --- **Step 4: Interpret the result** After 6 hours, both cultures will have the same number of cells. Check: A: \( 800 + 300 \times 6 = 800 + 1800 = 2600 \) B: \( 2000 + 100 \times 6 = 2000 + 600 = 2600 \) Yes, both have 2600 cells. --- **Final answer:** 6 hours

  3. y = 3x - 4 and y = -2x + 6 Answer: (2, 2) Solution: Set the equations equal to find the x-coordinate of intersection: 3x - 4 = -2x + 6 Add 2x to both sides: 5x - 4 = 6 Add 4 to both sides: 5x = 10 Divide both sides by 5: x = 2 Substitute x = 2 into either original equation to find y: y = 3(2) - 4 = 6 - 4 = 2 The solution is the ordered pair (2,…
    Full step-by-step solution

    Step 1: Set the equations equal to find the x-coordinate of intersection: 3x - 4 = -2x + 6 Step 2: Add 2x to both sides: 5x - 4 = 6 Step 3: Add 4 to both sides: 5x = 10 Step 4: Divide both sides by 5: x = 2 Step 5: Substitute x = 2 into either original equation to find y: y = 3(2) - 4 = 6 - 4 = 2 Step 6: The solution is the ordered pair (2, 2), which represents the point where the two lines intersect. The answer is (2, 2).

  4. Liam is planning a school fundraiser and needs to decide between selling pizza slices and hot dogs. He calculates that each pizza slice costs $2 to make and will be sold for $5, while each hot dog costs $1 to make and will be sold for $3. To break even on the event, he needs the total profit from both items to be $120. If he sells p pizza slices and h hot dogs, the system of equations representing this situation is: 3p + 2h = 120 (profit equation) and p + h = 50 (total items sold). Graph these equations to find how many pizza slices and hot dogs Liam needs to sell to meet his goal. Answer: 20 pizza slices and 30 hot dogs Solution: (1) 3p + 2h = 120 (profit equation) (2) p + h = 50 (total items sold) From equation (2): p + h = 50 So, p = 50 - h Substitute p = 50 - h into equation (1): 3(50 - h) + 2h = 120 150 - 3h + 2h = 120 150 - h = 120 -h = 120 - 150 -h = -30 h = 30 From p = 50 - h: p = 50 - 30 = 20 p = 20 pizza…
    Full step-by-step solution

    Let's solve the system of equations step by step. We are given: (1) 3p + 2h = 120 (profit equation) (2) p + h = 50 (total items sold) --- **Step 1: Solve one equation for one variable** From equation (2): p + h = 50 So, p = 50 - h --- **Step 2: Substitute into the other equation** Substitute p = 50 - h into equation (1): 3(50 - h) + 2h = 120 --- **Step 3: Simplify and solve for h** 150 - 3h + 2h = 120 150 - h = 120 -h = 120 - 150 -h = -30 h = 30 --- **Step 4: Find p** From p = 50 - h: p = 50 - 30 = 20 --- **Step 5: Interpret the result** p = 20 pizza slices h = 30 hot dogs --- **Step 6: Verify the solution** Check profit: 3(20) + 2(30) = 60 + 60 = 120 ✓ Check total items: 20 + 30 = 50 ✓ --- **Final answer:** Liam needs to sell 20 pizza slices and 30 hot dogs.

  5. Liam is planning a school fundraiser and needs to decide between two pricing options for tickets. Option A has a fixed cost of $20 plus $2 per person, while Option B has a fixed cost of $10 plus $3 per person. Liam wants to know for how many people the total cost of both options would be the same. Write your answer as an ordered pair (number of people, total cost). Answer: (10, 40) Solution: Let \( x \) = number of people. Let \( C_A \) = total cost for Option A. Let \( C_B \) = total cost for Option B.
    Full step-by-step solution

    Let's define the variables first. Let \( x \) = number of people. Let \( C_A \) = total cost for Option A. Let \( C_B \) = total cost for Option B. --- **Step 1: Write the cost equations for both options.** Option A: fixed cost $20 plus $2 per person. So: \( C_A = 20 + 2x \) Option B: fixed cost $10 plus $3 per person. So: \( C_B = 10 + 3x \) --- **Step 2: Set the costs equal to find when they are the same.** We set \( C_A = C_B \): \( 20 + 2x = 10 + 3x \) --- **Step 3: Solve for \( x \).** Subtract \( 2x \) from both sides: \( 20 = 10 + x \) Subtract 10 from both sides: \( 20 - 10 = x \) \( 10 = x \) So \( x = 10 \) people. --- **Step 4: Find the total cost for \( x = 10 \).** Use either cost formula. \( C_A = 20 + 2(10) = 20 + 20 = 40 \) \( C_B = 10 + 3(10) = 10 + 30 = 40 \) Both give $40. --- **Step 5: Write the answer as an ordered pair.** Ordered pair is (number of people, total cost) = (10, 40). --- **Final answer:** (10, 40)

  6. A rectangle is drawn on a coordinate plane with vertices at (2, 1), (6, 1), (6, 4), and (2, 4). A second rectangle has vertices at (1, 2), (5, 2), (5, 5), and (1, 5). Describe the transformation that maps the first rectangle onto the second rectangle, including the direction and distance of any translation. Answer: Translation 1 unit left and 1 unit up Solution: A = (2, 1) B = (6, 1) C = (6, 4) D = (2, 4) A' = (1, 2) B' = (5, 2) C' = (5, 5) D' = (1, 5) We can see that the rectangles are the same size because: First rectangle: width = 6 - 2 = 4, height = 4 - 1 = 3 Second rectangle: width = 5 - 1 = 4, height = 5 - 2 = 3 So no scaling or rotation is involved.
    Full step-by-step solution

    Let's compare the vertices of the first rectangle to the corresponding vertices of the second rectangle. First rectangle vertices: A = (2, 1) B = (6, 1) C = (6, 4) D = (2, 4) Second rectangle vertices: A' = (1, 2) B' = (5, 2) C' = (5, 5) D' = (1, 5) We can see that the rectangles are the same size because: First rectangle: width = 6 - 2 = 4, height = 4 - 1 = 3 Second rectangle: width = 5 - 1 = 4, height = 5 - 2 = 3 So no scaling or rotation is involved. Now let's find the translation from A to A': A = (2, 1) → A' = (1, 2) Change in x: 1 - 2 = -1 Change in y: 2 - 1 = +1 So translation is 1 unit left, 1 unit up. Check with another vertex to confirm: B = (6, 1) → B' = (5, 2) Change in x: 5 - 6 = -1 Change in y: 2 - 1 = +1 Same translation. C = (6, 4) → C' = (5, 5) Change in x: 5 - 6 = -1 Change in y: 5 - 4 = +1 Same translation. D = (2, 4) → D' = (1, 5) Change in x: 1 - 2 = -1 Change in y: 5 - 4 = +1 Same translation. Therefore, the transformation is a translation 1 unit left and 1 unit up.