Derive y=mx+b

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Derive y=mx+b · Grade 8 · Worksheet 2

1. Find the slope (m) and y-intercept (b) of the line that passes through points (2, 5) and (4, 9), then write the equation in y = mx + b form. Answer: y = 2x + 1 Solution: Calculate the slope using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (2, 5) and (4, 9): m = (9 - 5)/(4 - 2) = 4/2 = 2 Use the slope and one point to find the y-intercept Using point (2, 5) and m = 2 in y = mx + b: 5 = 2(2) + b 5 = 4 + b b = 5 - 4 = 1 Write the equation with m = 2 and b = 1…
   Full step-by-step solution

   Step 1: Calculate the slope using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (2, 5) and (4, 9): m = (9 - 5)/(4 - 2) = 4/2 = 2 Step 2: Use the slope and one point to find the y-intercept Using point (2, 5) and m = 2 in y = mx + b: 5 = 2(2) + b 5 = 4 + b b = 5 - 4 = 1 Step 3: Write the equation with m = 2 and b = 1 y = 2x + 1 The equation of the line is y = 2x + 1.

2. A quadrilateral is drawn on a coordinate plane with vertices at A(1, 2), B(5, 8), C(11, 8), and D(7, 2). A line is drawn connecting the midpoints of sides AB and CD. Find the equation of this line in slope-intercept form (y = mx + b). Answer: y = 1.5x + 1.5 Solution: Find the midpoint of AB, where A(1, 2) and B(5, 8) Midpoint = ((1+5)/2, (2+8)/2) = (6/2, 10/2) = (3, 5) Find the midpoint of CD, where C(11, 8) and D(7, 2) Midpoint = ((11+7)/2, (8+2)/2) = (18/2, 10/2) = (9, 5) Calculate the slope between the midpoints (3, 5) and (9, 5) Slope = (5 - 5)/(9 - 3) =…
   Full step-by-step solution

   Step 1: Find the midpoint of AB, where A(1, 2) and B(5, 8) Midpoint = ((1+5)/2, (2+8)/2) = (6/2, 10/2) = (3, 5) Step 2: Find the midpoint of CD, where C(11, 8) and D(7, 2) Midpoint = ((11+7)/2, (8+2)/2) = (18/2, 10/2) = (9, 5) Step 3: Calculate the slope between the midpoints (3, 5) and (9, 5) Slope = (5 - 5)/(9 - 3) = 0/6 = 0 Step 4: Use point-slope form with slope 0 and point (3, 5) y - 5 = 0(x - 3) y - 5 = 0 y = 5 The answer is y = 5.

3. Liam is tracking his savings for a new video game. He starts with $25 in his account and saves $8 each week from his allowance. After several weeks, he checks his account and finds he has $73. Write an equation in the form y = mx + b to represent this situation, then determine how many weeks Liam has been saving. Answer: 6 Solution: Identify the variables and the equation form. y = m x + b - y = total money in the account after x weeks - m = amount saved per week = $8 - b = starting money = $25 - x = number of weeks y = 8x + 25 Plug in the known total.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Identify the variables and the equation form.** We are told to use the form: y = m x + b Here: - y = total money in the account after x weeks - m = amount saved per week = $8 - b = starting money = $25 - x = number of weeks So the equation is: y = 8x + 25 --- **Step 2: Plug in the known total.** We know after x weeks, y = $73. So: 73 = 8x + 25 --- **Step 3: Solve for x.** Subtract 25 from both sides: 73 - 25 = 8x 48 = 8x Divide both sides by 8: 48 / 8 = x x = 6 --- **Step 4: Interpret the result.** x = 6 means Liam has been saving for 6 weeks. --- **Final answer:** Liam has been saving for 6 weeks.

4. Find the equation of the line in y = mx + b form that passes through points (3, 11) and (7, 27). Answer: y = 4x - 1 Solution: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (3, 11) and (7, 27): m = (27 - 11)/(7 - 3) = 16/4 = 4 y = mx + b Using point (3, 11): 11 = 4(3) + b 11 = 12 + b b = 11 - 12 b = -1 y = 4x - 1
   Full step-by-step solution

   Step 1: Calculate the slope (m) using the formula m = (y₂ - y₁)/(x₂ - x₁) Using points (3, 11) and (7, 27): m = (27 - 11)/(7 - 3) = 16/4 = 4 Step 2: Use point-slope form with one point to find b y = mx + b Using point (3, 11): 11 = 4(3) + b 11 = 12 + b b = 11 - 12 b = -1 Step 3: Write the final equation y = 4x - 1

5. Find the equation of the line in y = mx + b form that passes through points (9, 14) and (15, 32). Answer: y = 3x - 13 Solution: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Using points (9, 14) and (15, 32): m = (32 - 14) / (15 - 9) = 18 / 6 = 3. Use the slope and one point to find the y-intercept (b).
   Full step-by-step solution

   Step 1: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Using points (9, 14) and (15, 32): m = (32 - 14) / (15 - 9) = 18 / 6 = 3. Step 2: Use the slope and one point to find the y-intercept (b). Using 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 final equation: y = 3x - 13.

6. Find the slope of the line that passes through points (3, 7) and (8, 22). Answer: 3 Solution: Identify the coordinates: (x₁, y₁) = (3, 7) and (x₂, y₂) = (8, 22) Apply the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Substitute the values: m = (22 - 7)/(8 - 3) Calculate the numerator: 22 - 7 = 15 Calculate the denominator: 8 - 3 = 5 Divide: 15 ÷ 5 = 3 The slope is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (3, 7) and (x₂, y₂) = (8, 22) Step 2: Apply the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Step 3: Substitute the values: m = (22 - 7)/(8 - 3) Step 4: Calculate the numerator: 22 - 7 = 15 Step 5: Calculate the denominator: 8 - 3 = 5 Step 6: Divide: 15 ÷ 5 = 3 Step 7: The slope is 3.

7. Olivia is helping her family track the water level in their rain barrel. When they started monitoring at noon (time 0 hours), the barrel already had 9 inches of water. After 7 hours, the water level had risen to 23 inches. Assuming the water level increases at a constant rate, write a linear equation in the form y = mx + b that represents the water level (y) in inches after x hours. Answer: y = 2x + 9 Solution: Identify the two points from the problem. At time 0 hours, the water level is 9 inches, so the point is (0, 9). After 7 hours, the water level is 23 inches, so the point is (7, 23).
   Full step-by-step solution

   Step 1: Identify the two points from the problem. At time 0 hours, the water level is 9 inches, so the point is (0, 9). After 7 hours, the water level is 23 inches, so the point is (7, 23). Step 2: Calculate the slope (m) using the formula m = (y2 - y1) / (x2 - x1). Here, (x1, y1) = (0, 9) and (x2, y2) = (7, 23). So m = (23 - 9) / (7 - 0) = 14 / 7 = 2. Step 3: The y-intercept (b) is the water level at time 0, which is 9 inches. Step 4: Substitute m = 2 and b = 9 into the equation y = mx + b. Step 5: The equation is y = 2x + 9. The answer is y = 2x + 9.

8. Emma is tracking the temperature change during a science experiment. At the start of the experiment (time 0), the temperature was 68°F. After 5 minutes, the temperature had risen to 83°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 = 3x + 68 Solution: Identify the two points from the problem: (0, 68) and (5, 83) Calculate the slope (m) using the formula m = (y2 - y1)/(x2 - x1) m = (83 - 68)/(5 - 0) = 15/5 = 3 The y-intercept (b) is the temperature at time 0, which is 68 Substitute m and b into y = mx + b: y = 3x + 68 The equation is y = 3x + 68.
   Full step-by-step solution

   Step 1: Identify the two points from the problem: (0, 68) and (5, 83) Step 2: Calculate the slope (m) using the formula m = (y2 - y1)/(x2 - x1) Step 3: m = (83 - 68)/(5 - 0) = 15/5 = 3 Step 4: The y-intercept (b) is the temperature at time 0, which is 68 Step 5: Substitute m and b into y = mx + b: y = 3x + 68 The equation is y = 3x + 68.