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Derive y=mx+b

Grade 8 · Algebra · Worksheet 2

  1. Olivia is saving money to buy a new bicycle. She already has $45 saved from her birthday. She plans to mow lawns in her neighborhood to earn more money. For each lawn she mows, she charges $15. Olivia wants to know how much total money she will have after mowing a certain number of lawns. Write a linear equation in the form y = mx + b that represents the total amount of money y (in dollars) Olivia will have after mowing x lawns. Answer: ______________
  2. A spacecraft is traveling at a constant speed of 2.8 × 10^4 kilometers per hour. The distance it has traveled is given by the equation y = mx + b, where y is the total distance in kilometers and x is the time in hours. If the spacecraft had already traveled 1.5 × 10^5 kilometers before starting its current journey, how many kilometers will it have traveled after 8 hours? Answer: ______________
  3. A line passes through points (1, 11) and (6, 36). Write the equation in y = mx + b form. Answer: ______________
  4. A rectangle is drawn on a coordinate plane with vertices at (1, 2), (7, 2), (7, 8), and (1, 8). A line is drawn from the midpoint of the left side to the midpoint of the right side. What is the equation of this line in slope-intercept form (y = mx + b)? Answer: ______________
  5. Hana is monitoring the water level in a tank for her school's science project. At the start of her observation (day 0), the water level is 40 centimeters. After 6 days, the water level has dropped to 22 centimeters. If the water level decreases at a constant rate, write a linear equation in the form y = mx + b that represents the water level (y) after x days. Answer: ______________
  6. A scientist is studying bacterial growth. The number of bacteria in a petri dish follows the linear equation y = 2500x + 800, where y is the total number of bacteria and x is the time in hours. How many bacteria were initially present in the petri dish (at time x = 0)? Answer: ______________
  7. Liam is designing a ramp for his skateboard. He knows that for every 3 feet of horizontal distance, the ramp rises 1 foot. If the ramp needs to reach a platform that is 5 feet high, how many feet away from the platform's base should Liam start building the ramp? Write your answer as a whole number. Answer: ______________
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Answer Key & Explanations

Derive y=mx+b · Grade 8 · Worksheet 2

  1. Olivia is saving money to buy a new bicycle. She already has $45 saved from her birthday. She plans to mow lawns in her neighborhood to earn more money. For each lawn she mows, she charges $15. Olivia wants to know how much total money she will have after mowing a certain number of lawns. Write a linear equation in the form y = mx + b that represents the total amount of money y (in dollars) Olivia will have after mowing x lawns. Answer: y = 15x + 45 Solution: Identify the y-intercept (b). Olivia starts with $45 before mowing any lawns, so when x = 0, y = 45. Identify the slope (m).
    Full step-by-step solution

    Step 1: Identify the y-intercept (b). Olivia starts with $45 before mowing any lawns, so when x = 0, y = 45. Thus, b = 45. Step 2: Identify the slope (m). For each lawn mowed, Olivia earns $15. So the rate of change is 15 dollars per lawn. Thus, m = 15. Step 3: Write the equation in slope-intercept form y = mx + b. Substitute m = 15 and b = 45 to get y = 15x + 45. The equation is y = 15x + 45.

  2. A spacecraft is traveling at a constant speed of 2.8 × 10^4 kilometers per hour. The distance it has traveled is given by the equation y = mx + b, where y is the total distance in kilometers and x is the time in hours. If the spacecraft had already traveled 1.5 × 10^5 kilometers before starting its current journey, how many kilometers will it have traveled after 8 hours? Answer: 374000 Solution: - m (speed) = 2.8 × 10^4 = 28,000 km/h - b (initial distance) = 1.5 × 10^5 = 150,000 km - x (time) = 8 hours y = mx + b y = (2.8 × 10^4)x + (1.5 × 10^5) Substitute x = 8 into the equation y = (2.8 × 10^4)(8) + (1.5 × 10^5) (2.8 × 10^4)(8) = 2.8 × 8 × 10^4 = 22.4 × 10^4 = 224,000 y = 224,000 +…
    Full step-by-step solution

    Step 1: Identify the values from the problem - m (speed) = 2.8 × 10^4 = 28,000 km/h - b (initial distance) = 1.5 × 10^5 = 150,000 km - x (time) = 8 hours Step 2: Write the equation y = mx + b y = (2.8 × 10^4)x + (1.5 × 10^5) Step 3: Substitute x = 8 into the equation y = (2.8 × 10^4)(8) + (1.5 × 10^5) Step 4: Calculate the first term (2.8 × 10^4)(8) = 2.8 × 8 × 10^4 = 22.4 × 10^4 = 224,000 Step 5: Add the initial distance y = 224,000 + 150,000 = 374,000 Step 6: The total distance traveled after 8 hours is 374,000 kilometers.

  3. A line passes through points (1, 11) and (6, 36). Write the equation in y = mx + b form. Answer: y = 5x + 6 Solution: Find the slope (m). m = (36 - 11) / (6 - 1) = 25 / 5 = 5. Use point (1, 11) and m = 5 in y = mx + b: 11 = 5(1) + b → 11 = 5 + b.
    Full step-by-step solution

    Step 1: Find the slope (m). m = (36 - 11) / (6 - 1) = 25 / 5 = 5. Step 2: Use point (1, 11) and m = 5 in y = mx + b: 11 = 5(1) + b → 11 = 5 + b. Step 3: Solve for b: b = 11 - 5 = 6. Step 4: Write the equation: y = 5x + 6. The answer is y = 5x + 6.

  4. A rectangle is drawn on a coordinate plane with vertices at (1, 2), (7, 2), (7, 8), and (1, 8). A line is drawn from the midpoint of the left side to the midpoint of the right side. What is the equation of this line in slope-intercept form (y = mx + b)? Answer: y = 5 Solution: Identify the midpoints of the left and right sides. The left side has endpoints (1, 2) and (1, 8). Its midpoint is ((1+1)/2, (2+8)/2) = (1, 5).
    Full step-by-step solution

    Step 1: Identify the midpoints of the left and right sides. The left side has endpoints (1, 2) and (1, 8). Its midpoint is ((1+1)/2, (2+8)/2) = (1, 5). The right side has endpoints (7, 2) and (7, 8). Its midpoint is ((7+7)/2, (2+8)/2) = (7, 5). Step 2: Find the slope (m) using the two points (1, 5) and (7, 5). m = (5 - 5)/(7 - 1) = 0/6 = 0. Step 3: Use the slope-intercept form y = mx + b with m = 0 and one point, say (1, 5). 5 = 0*1 + b 5 = b Step 4: Write the equation. y = 0*x + 5 y = 5 The equation of the line is y = 5.

  5. Hana is monitoring the water level in a tank for her school's science project. At the start of her observation (day 0), the water level is 40 centimeters. After 6 days, the water level has dropped to 22 centimeters. If the water level decreases at a constant rate, write a linear equation in the form y = mx + b that represents the water level (y) after x days. Answer: y = -3x + 40 Solution: Identify the two points: (0, 40) and (6, 22). Calculate the slope m using m = (y2 - y1) / (x2 - x1). m = (22 - 40) / (6 - 0) m = (-18) / 6 m = -3 The y-intercept b is the water level at day 0, which is 40.
    Full step-by-step solution

    Step 1: Identify the two points: (0, 40) and (6, 22). Step 2: Calculate the slope m using m = (y2 - y1) / (x2 - x1). m = (22 - 40) / (6 - 0) m = (-18) / 6 m = -3 Step 3: The y-intercept b is the water level at day 0, which is 40. Step 4: Substitute m = -3 and b = 40 into y = mx + b. y = -3x + 40 The equation is y = -3x + 40.

  6. A scientist is studying bacterial growth. The number of bacteria in a petri dish follows the linear equation y = 2500x + 800, where y is the total number of bacteria and x is the time in hours. How many bacteria were initially present in the petri dish (at time x = 0)? Answer: 800 Solution: We are given the linear equation for bacterial growth: y = 2500x + 800 Here, y = total number of bacteria, x = time in hours. We want the number of bacteria initially present, which means at time x = 0.
    Full step-by-step solution

    Step 1: Understand the problem We are given the linear equation for bacterial growth: y = 2500x + 800 Here, y = total number of bacteria, x = time in hours. We want the number of bacteria initially present, which means at time x = 0. Step 2: Substitute x = 0 into the equation y = 2500 * (0) + 800 Step 3: Perform the multiplication 2500 * 0 = 0 So y = 0 + 800 Step 4: Perform the addition y = 800 Step 5: Interpret the result At x = 0 hours, the number of bacteria is 800. This is the initial number of bacteria in the petri dish. Final answer: 800

  7. Liam is designing a ramp for his skateboard. He knows that for every 3 feet of horizontal distance, the ramp rises 1 foot. If the ramp needs to reach a platform that is 5 feet high, how many feet away from the platform's base should Liam start building the ramp? Write your answer as a whole number. Answer: 15 Solution: We are told that for every 3 feet of horizontal distance, the ramp rises 1 foot. This means the ratio of horizontal distance to vertical rise is 3 : 1. We know the ramp must reach a height of 5 feet.
    Full step-by-step solution

    We are told that for every 3 feet of horizontal distance, the ramp rises 1 foot. This means the ratio of horizontal distance to vertical rise is 3 : 1. We know the ramp must reach a height of 5 feet. Let the horizontal distance be \( x \) feet. Set up the proportion using the given ratio: Horizontal distance / Vertical rise = 3 / 1 So, \( x / 5 = 3 / 1 \) Multiply both sides by 5: \( x = 5 \times 3 \) \( x = 15 \) So, Liam should start building the ramp 15 feet away from the platform's base. Answer: 15