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Derive y=mx+b

Grade 8 · Algebra · Worksheet 3

  1. Find the equation of the line in y = mx + b form that passes through points (1, 3) and (4, 9). Answer: ______________
  2. Aisha is designing a wheelchair ramp for her school's new accessibility project. The ramp needs to rise 3 feet to reach the entrance. Building codes require that for every 1 foot of vertical rise, there must be at least 12 feet of horizontal run. If the ramp starts at ground level (0 feet) and follows a linear path, write the equation in the form y = mx + b that represents the height y of the ramp at any horizontal distance x from the starting point. Answer: ______________
  3. Aroha tracks the growth of a plant. At day 3, the plant is 11 cm tall. At day 7, it is 27 cm tall. Assuming linear growth, write the equation in y = mx + b form where y is height in cm and x is the day number. 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. A line passes through points (9, 14) and (15, 32). Derive the equation in y = mx + b form. Answer: ______________
  6. Find the equation of the line in y = mx + b form that passes through points (1, 2) and (3, 8). Answer: ______________
  7. Emma is tracking the temperature change during a science experiment. At the start of the experiment (time 0 minutes), the temperature was 68°F. After 8 minutes, the temperature had risen to 84°F. Assuming the temperature increases at a constant rate, write a linear equation in the form y = mx + b that represents the temperature (y) after x minutes. Answer: ______________
  8. 2(3x - 5) + 4 = 3x + 7 Answer: ______________
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Answer Key & Explanations

Derive y=mx+b · Grade 8 · Worksheet 3

  1. Find the equation of the line in y = mx + b form that passes through points (1, 3) and (4, 9). Answer: y = 2x + 1 Solution: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (1, 3) and (4, 9): m = (9 - 3)/(4 - 1) = 6/3 = 2 Using point (1, 3) and m = 2: y = mx + b → 3 = 2(1) + b → 3 = 2 + b 3 - 2 = b → b = 1 y = 2x + 1
    Full step-by-step solution

    Step 1: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (1, 3) and (4, 9): m = (9 - 3)/(4 - 1) = 6/3 = 2 Step 2: Use point-slope form with one point to find b Using point (1, 3) and m = 2: y = mx + b → 3 = 2(1) + b → 3 = 2 + b Step 3: Solve for b 3 - 2 = b → b = 1 Step 4: Write the final equation y = 2x + 1

  2. Aisha is designing a wheelchair ramp for her school's new accessibility project. The ramp needs to rise 3 feet to reach the entrance. Building codes require that for every 1 foot of vertical rise, there must be at least 12 feet of horizontal run. If the ramp starts at ground level (0 feet) and follows a linear path, write the equation in the form y = mx + b that represents the height y of the ramp at any horizontal distance x from the starting point. Answer: y = (1/12)x Solution: Identify the slope (m). The building code states that for every 1 foot of vertical rise, there must be 12 feet of horizontal run. This means the slope is rise over run = 1/12.
    Full step-by-step solution

    Step 1: Identify the slope (m). The building code states that for every 1 foot of vertical rise, there must be 12 feet of horizontal run. This means the slope is rise over run = 1/12. Step 2: Identify the y-intercept (b). The problem states the ramp starts at ground level (0 feet), so when x = 0, y = 0. Therefore, b = 0. Step 3: Write the equation in slope-intercept form y = mx + b. Substituting m = 1/12 and b = 0 gives y = (1/12)x. The equation is y = (1/12)x.

  3. Aroha tracks the growth of a plant. At day 3, the plant is 11 cm tall. At day 7, it is 27 cm tall. Assuming linear growth, write the equation in y = mx + b form where y is height in cm and x is the day number. Answer: y = 4x - 1 Solution: Identify the points: (3, 11) and (7, 27). Calculate the slope m = (27 - 11) / (7 - 3) = 16 / 4 = 4. Use point (3, 11) and m = 4 in y = mx + b: 11 = 4(3) + b.
    Full step-by-step solution

    Step 1: Identify the points: (3, 11) and (7, 27). Step 2: Calculate the slope m = (27 - 11) / (7 - 3) = 16 / 4 = 4. Step 3: Use point (3, 11) and m = 4 in y = mx + b: 11 = 4(3) + b. Step 4: 11 = 12 + b, so b = 11 - 12 = -1. Step 5: The equation is y = 4x - 1.

  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. A line passes through points (9, 14) and (15, 32). Derive the equation in y = mx + b form. Answer: y = 3x - 13 Solution: Find the slope m = (y2 - y1) / (x2 - x1) = (32 - 14) / (15 - 9) = 18 / 6 = 3. Use point (9, 14) and m = 3 in y = mx + b: 14 = 3(9) + b → 14 = 27 + b. Solve for b: b = 14 - 27 = -13.
    Full step-by-step solution

    Step 1: Find the slope m = (y2 - y1) / (x2 - x1) = (32 - 14) / (15 - 9) = 18 / 6 = 3. Step 2: Use point (9, 14) and m = 3 in y = mx + b: 14 = 3(9) + b → 14 = 27 + b. Step 3: Solve for b: b = 14 - 27 = -13. Step 4: Write the equation: y = 3x - 13.

  6. Find the equation of the line in y = mx + b form that passes through points (1, 2) and (3, 8). Answer: y = 3x - 1 Solution: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (1, 2) and (3, 8): m = (8 - 2)/(3 - 1) = 6/2 = 3 Use the slope and one point to find the y-intercept (b) Using point (1, 2) and m = 3 in y = mx + b: 2 = 3(1) + b 2 = 3 + b b = 2 - 3 = -1 Write the equation using m = 3…
    Full step-by-step solution

    Step 1: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (1, 2) and (3, 8): m = (8 - 2)/(3 - 1) = 6/2 = 3 Step 2: Use the slope and one point to find the y-intercept (b) Using point (1, 2) and m = 3 in y = mx + b: 2 = 3(1) + b 2 = 3 + b b = 2 - 3 = -1 Step 3: Write the equation using m = 3 and b = -1 y = 3x - 1

  7. Emma is tracking the temperature change during a science experiment. At the start of the experiment (time 0 minutes), the temperature was 68°F. After 8 minutes, the temperature had risen to 84°F. Assuming the temperature increases at a constant rate, write a linear equation in the form y = mx + b that represents the temperature (y) after x minutes. Answer: y = 2x + 68 Solution: Identify the two points from the problem: (0, 68) and (8, 84) Calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) m = (84 - 68) / (8 - 0) = 16 / 8 = 2 The y-intercept (b) is the temperature at time 0, which is 68 Substitute m and b into the equation y = mx + b y = 2x + 68 The…
    Full step-by-step solution

    Step 1: Identify the two points from the problem: (0, 68) and (8, 84) Step 2: Calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1) Step 3: m = (84 - 68) / (8 - 0) = 16 / 8 = 2 Step 4: The y-intercept (b) is the temperature at time 0, which is 68 Step 5: Substitute m and b into the equation y = mx + b Step 6: y = 2x + 68 The answer is y = 2x + 68.

  8. 2(3x - 5) + 4 = 3x + 7 Answer: x = 13/3 Solution: 2(3x - 5) + 4 = 3x + 7 Distribute the 2 Multiply 2 by each term inside the parentheses: 2 * 3x = 6x 2 * (-5) = -10 6x - 10 + 4 = 3x + 7 -10 + 4 = -6 6x - 6 = 3x + 7 Subtract 3x from both sides: 6x - 3x - 6 = 7 3x - 6 = 7 Add 6 to both sides: 3x = 7 + 6 3x = 13 Divide both sides by 3: x = 13/3…
    Full step-by-step solution

    Let's solve the equation step by step. We start with: 2(3x - 5) + 4 = 3x + 7 **Step 1: Distribute the 2** Multiply 2 by each term inside the parentheses: 2 * 3x = 6x 2 * (-5) = -10 So we have: 6x - 10 + 4 = 3x + 7 **Step 2: Combine like terms on the left** -10 + 4 = -6 So: 6x - 6 = 3x + 7 **Step 3: Move x terms to one side** Subtract 3x from both sides: 6x - 3x - 6 = 7 3x - 6 = 7 **Step 4: Move constant terms to the other side** Add 6 to both sides: 3x = 7 + 6 3x = 13 **Step 5: Solve for x** Divide both sides by 3: x = 13/3 **Final answer:** x = 13/3