Derive y=mx+b

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Derive y=mx+b · Grade 8 · Worksheet 1

1. Find the slope of the line passing through points (2, 5) and (6, 17). Answer: 3 Solution: Identify the coordinates: (x₁, y₁) = (2, 5) and (x₂, y₂) = (6, 17) Use the slope formula: m = (y₂ - y₁) ÷ (x₂ - x₁) Substitute the values: m = (17 - 5) ÷ (6 - 2) Calculate the numerator: 17 - 5 = 12 Calculate the denominator: 6 - 2 = 4 Divide: 12 ÷ 4 = 3 The slope of the line is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (2, 5) and (x₂, y₂) = (6, 17) Step 2: Use the slope formula: m = (y₂ - y₁) ÷ (x₂ - x₁) Step 3: Substitute the values: m = (17 - 5) ÷ (6 - 2) Step 4: Calculate the numerator: 17 - 5 = 12 Step 5: Calculate the denominator: 6 - 2 = 4 Step 6: Divide: 12 ÷ 4 = 3 Step 7: The slope of the line is 3.

2. Points (3, 17) and (9, 41). Find equation in y = mx + b form. Answer: y = 4x + 5 Solution: Find the slope m using m = (y2 - y1) / (x2 - x1). Using (3, 17) and (9, 41): m = (41 - 17) / (9 - 3) = 24 / 6 = 4. Use point (3, 17) and m = 4 in y = mx + b: 17 = 4(3) + b → 17 = 12 + b.
   Full step-by-step solution

   Step 1: Find the slope m using m = (y2 - y1) / (x2 - x1). Using (3, 17) and (9, 41): m = (41 - 17) / (9 - 3) = 24 / 6 = 4. Step 2: Use point (3, 17) and m = 4 in y = mx + b: 17 = 4(3) + b → 17 = 12 + b. Step 3: Solve for b: b = 17 - 12 = 5. Step 4: Write the equation: y = 4x + 5.

3. A line passes through points (1, 6) and (6, 31). Write the equation in y = mx + b form. Answer: y = 5x + 1 Solution: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (1, 6) and (6, 31): m = (31 - 6) / (6 - 1) = 25 / 5 = 5. Use point (1, 6) and m = 5 in y = mx + b: 6 = 5(1) + b → 6 = 5 + b.
   Full step-by-step solution

   Step 1: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (1, 6) and (6, 31): m = (31 - 6) / (6 - 1) = 25 / 5 = 5. Step 2: Use point (1, 6) and m = 5 in y = mx + b: 6 = 5(1) + b → 6 = 5 + b. Step 3: Solve for b: b = 6 - 5 = 1. Step 4: Write the equation: y = 5x + 1.

4. Liam is designing a rectangular garden. The length of the garden is 5 feet more than twice its width. If the perimeter of the garden is 82 feet, what is the length of the garden in feet? Answer: 29 Solution: Let the width of the garden be \( w \) feet. The length is 5 feet more than twice the width, so: length \( l = 2w + 5 \).
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Define variables** Let the width of the garden be \( w \) feet. The length is 5 feet more than twice the width, so: length \( l = 2w + 5 \). --- **Step 2: Write the perimeter formula** Perimeter of a rectangle is: \( P = 2 \times (\text{length} + \text{width}) \) Given \( P = 82 \), so: \( 2 \times (l + w) = 82 \). --- **Step 3: Substitute \( l \) in terms of \( w \)** Substitute \( l = 2w + 5 \) into the perimeter equation: \( 2 \times ( (2w + 5) + w ) = 82 \). --- **Step 4: Simplify inside the parentheses** \( (2w + 5) + w = 3w + 5 \). So: \( 2 \times (3w + 5) = 82 \). --- **Step 5: Solve for \( w \)** Divide both sides by 2: \( 3w + 5 = 41 \). Subtract 5 from both sides: \( 3w = 36 \). Divide by 3: \( w = 12 \). --- **Step 6: Find length** \( l = 2w + 5 = 2 \times 12 + 5 = 24 + 5 = 29 \). --- **Final answer:** The length is 29 feet.

5. Points (1, 6) and (6, 31). Find equation in y = mx + b form. Answer: y = 5x + 1 Solution: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (1, 6) and (6, 31): m = (31 - 6) / (6 - 1) = 25 / 5 = 5. Use point (1, 6) and m = 5 in y = mx + b: 6 = 5(1) + b.
   Full step-by-step solution

   Step 1: Find the slope m using the formula m = (y2 - y1) / (x2 - x1). Using (1, 6) and (6, 31): m = (31 - 6) / (6 - 1) = 25 / 5 = 5. Step 2: Use point (1, 6) and m = 5 in y = mx + b: 6 = 5(1) + b. Step 3: Solve for b: 6 = 5 + b, so b = 6 - 5 = 1. Step 4: Write the equation: y = 5x + 1. The answer is y = 5x + 1.

6. Find the slope of the line passing through (2, 5) and (6, 13). Answer: 2 Solution: Identify the coordinates: (x₁, y₁) = (2, 5) and (x₂, y₂) = (6, 13) Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Substitute the values: m = (13 - 5)/(6 - 2) Calculate the numerator: 13 - 5 = 8 Calculate the denominator: 6 - 2 = 4 Divide: 8 ÷ 4 = 2 The slope is 2.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (2, 5) and (x₂, y₂) = (6, 13) Step 2: Use the slope formula: m = (y₂ - y₁)/(x₂ - x₁) Step 3: Substitute the values: m = (13 - 5)/(6 - 2) Step 4: Calculate the numerator: 13 - 5 = 8 Step 5: Calculate the denominator: 6 - 2 = 4 Step 6: Divide: 8 ÷ 4 = 2 Step 7: The slope is 2.

7. 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

8. (-2, 5) and (4, -1) lie on a line. Write the equation in y = mx + b form. Answer: y = -x + 3 Solution: Find the slope (m) using the formula m = (y2 - y1)/(x2 - x1) Using points (-2, 5) and (4, -1): m = (-1 - 5)/(4 - (-2)) = (-6)/(6) = -1 Use the slope-intercept form y = mx + b and substitute one point to find b Using point (-2, 5): 5 = (-1)(-2) + b 5 = 2 + b b = 5 - 2 b = 3 With m = -1 and b = 3,…
   Full step-by-step solution

   Step 1: Find the slope (m) using the formula m = (y2 - y1)/(x2 - x1) Using points (-2, 5) and (4, -1): m = (-1 - 5)/(4 - (-2)) = (-6)/(6) = -1 Step 2: Use the slope-intercept form y = mx + b and substitute one point to find b Using point (-2, 5): 5 = (-1)(-2) + b 5 = 2 + b b = 5 - 2 b = 3 Step 3: Write the final equation With m = -1 and b = 3, the equation is y = -x + 3

9. Liam is designing a rectangular garden. The length of the garden is 5 feet more than twice its width. If the perimeter of the garden is 82 feet, write an equation in the form y = mx + b that represents the relationship between the length (y) and width (x) of the garden. Answer: y = 2x + 5 Solution: - Width = x - Length = y The problem says: "The length of the garden is 5 feet more than twice its width." "Twice its width" means 2x. "5 feet more than twice its width" means 2x + 5.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let: - Width = x - Length = y --- **Step 2: Translate the first sentence into an equation** The problem says: "The length of the garden is 5 feet more than twice its width." "Twice its width" means 2x. "5 feet more than twice its width" means 2x + 5. So: y = 2x + 5 --- **Step 3: Check if we need the perimeter information** The perimeter of a rectangle is: Perimeter = 2 × (length + width) = 2(y + x) We are told the perimeter is 82 feet: 2(y + x) = 82 --- **Step 4: Substitute y from Step 2 into the perimeter equation** From Step 2: y = 2x + 5 Substitute into 2(y + x) = 82: 2( (2x + 5) + x ) = 82 --- **Step 5: Simplify and solve for x (just to verify)** 2(3x + 5) = 82 6x + 10 = 82 6x = 72 x = 12 --- **Step 6: Find y** y = 2(12) + 5 = 24 + 5 = 29 --- **Step 7: Verify with perimeter** Perimeter = 2(29 + 12) = 2(41) = 82 ✔ --- **Step 8: Conclusion** The equation relating length (y) to width (x) is already found in Step 2: y = 2x + 5 This matches the required form y = mx + b, where m = 2 and b = 5. --- **Final answer:** y = 2x + 5