Understand Slope

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Understand Slope · Grade 8 · Worksheet 1

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

   Step 1: Identify the coordinates: (x₁, y₁) = (3, 7) and (x₂, y₂) = (8, 22) Step 2: Use the slope formula: slope = (y₂ - y₁) ÷ (x₂ - x₁) Step 3: Substitute the values: slope = (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 The slope of the line is 3.

2. Liam is designing a wheelchair ramp for his school's new accessibility project. The building code requires that for every 3 inches of vertical rise, the ramp must extend 36 inches horizontally. If the ramp needs to reach a doorway that is 15 inches above ground level, how many feet long must the ramp be? Answer: 15 Solution: The code says: for every 3 inches of vertical rise, the ramp must extend 36 inches horizontally. 36 / 3 = 12 So for every 1 inch of vertical rise, the ramp must extend 12 inches horizontally.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the ratio from the building code** The code says: for every 3 inches of vertical rise, the ramp must extend 36 inches horizontally. So the ratio of horizontal length to vertical rise is: 36 inches horizontal / 3 inches vertical. --- **Step 2: Simplify the ratio** 36 / 3 = 12 So for every 1 inch of vertical rise, the ramp must extend 12 inches horizontally. --- **Step 3: Apply the ratio to the given rise** The vertical rise needed is 15 inches. Horizontal length required = 15 inches (rise) × 12 (inches horizontal per inch vertical) = 15 × 12 = 180 inches. --- **Step 4: Convert inches to feet** Since 12 inches = 1 foot, 180 inches ÷ 12 = 15 feet. --- **Step 5: Conclusion** The ramp must be **15 feet** long horizontally. --- **Final answer:** 15

3. During a road trip, Olivia notices that the elevation of the highway increases by 46 meters over a horizontal distance of 46 meters. What is the slope of the highway as a simplified fraction? Answer: 1 Solution: The slope is the rise (vertical change) divided by the run (horizontal change): slope = rise/run. Plug in the numbers: slope = 46/46.
   Full step-by-step solution

   Step 1: The slope is the rise (vertical change) divided by the run (horizontal change): slope = rise/run. Step 2: Plug in the numbers: slope = 46/46. Step 3: Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor: slope = 1. The slope of the highway is 1.

4. Liam is designing a wheelchair ramp for a community center. The building code requires that the ramp must have a slope no steeper than 1:12. If the vertical height from the ground to the entrance is 2.5 feet, what is the minimum horizontal length, in feet, that the ramp must have to meet the code? Answer: 30 Solution: The building code says the slope must be no steeper than 1:12. This means: for every 1 foot of vertical rise, the ramp must have at least 12 feet of horizontal run. So slope = rise / run ≤ 1/12.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the slope requirement** The building code says the slope must be no steeper than 1:12. This means: for every 1 foot of vertical rise, the ramp must have at least 12 feet of horizontal run. So slope = rise / run ≤ 1/12. --- **Step 2: Identify given values** Vertical rise (height) = 2.5 feet. Let the horizontal run = \( x \) feet. --- **Step 3: Set up the slope inequality** From slope ≤ 1/12: rise / run ≤ 1/12 2.5 / x ≤ 1/12 --- **Step 4: Solve for \( x \)** Multiply both sides by \( x \) (positive, so inequality direction stays the same): 2.5 ≤ (1/12) * x Multiply both sides by 12: 2.5 * 12 ≤ x 30 ≤ x So \( x \) ≥ 30. --- **Step 5: Interpret the result** The minimum horizontal length is 30 feet. --- **Final Answer:** 30

5. A line passes through the points (3, 7) and (5, 13). What is the slope of this line? Answer: 3 Solution: To find the slope of a line that passes through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Identify the coordinates of the two points.
   Full step-by-step solution

   To find the slope of a line that passes through two points, we use the slope formula: slope = (change in y) / (change in x) = (y2 - y1) / (x2 - x1) Step 1: Identify the coordinates of the two points. Point 1: (x1, y1) = (3, 7) Point 2: (x2, y2) = (5, 13) Step 2: Substitute the coordinates into the slope formula. slope = (13 - 7) / (5 - 3) Step 3: Perform the subtractions inside the parentheses. slope = (6) / (2) Step 4: Divide the numerator by the denominator. slope = 6 / 2 = 3 Therefore, the slope of the line is 3.

6. Liam is designing a wheelchair ramp for his school's new accessibility project. The building code requires that for every 3 inches of vertical rise, the ramp must extend at least 36 inches horizontally. If the ramp needs to reach a doorway that is 15 inches above ground level, what is the minimum horizontal length the ramp must extend? Answer: 180 inches Solution: 1. This means the ratio of vertical rise to horizontal run is 3 : 36. 2.
   Full step-by-step solution

   Let's go step by step. 1. Understand the requirement: The code says: for every 3 inches of vertical rise, the ramp must extend at least 36 inches horizontally. This means the ratio of vertical rise to horizontal run is 3 : 36. 2. Write the ratio as a fraction: Vertical rise / Horizontal run = 3 / 36. This simplifies to 1 / 12 (since 3 ÷ 3 = 1 and 36 ÷ 3 = 12). So for every 1 inch of rise, you need 12 inches of horizontal run. 3. Apply this to the given rise: The doorway is 15 inches above ground level, so the vertical rise = 15 inches. Let the horizontal length be L inches. 4. Set up the proportion: Using the ratio: rise / run = 1 / 12 15 / L = 1 / 12 5. Solve for L: Cross-multiply: 15 × 12 = 1 × L L = 180 6. Conclusion: The minimum horizontal length the ramp must extend is 180 inches. Answer: 180 inches

7. A line is drawn on a coordinate plane passing through points A(-2, 4) and B(4, -2). A second line is drawn perpendicular to this line and passes through point C(1, 5). What is the slope of the second line? Answer: 1 Solution: Find the slope of the first line using points A(-2, 4) and B(4, -2) Slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(4 - (-2)) = (-6)/(6) = -1 Find the slope of a line perpendicular to the first line Perpendicular slope = negative reciprocal of -1 = 1 The second line has slope 1 Point C(1, 5) is not…
   Full step-by-step solution

   Step 1: Find the slope of the first line using points A(-2, 4) and B(4, -2) Slope = (y2 - y1)/(x2 - x1) = (-2 - 4)/(4 - (-2)) = (-6)/(6) = -1 Step 2: Find the slope of a line perpendicular to the first line Perpendicular slope = negative reciprocal of -1 = 1 Step 3: The second line has slope 1 Point C(1, 5) is not needed to find the slope of the perpendicular line The answer is 1.

8. (2x + 8) ÷ 2 = 10 Answer: 6 Solution: The equation is (2x + 8) ÷ 2 = 10 Multiply both sides by 2 to eliminate the division: 2x + 8 = 20 Subtract 8 from both sides to isolate the term with x: 2x = 12 Divide both sides by 2 to solve for x: x = 6 The answer is 6.
   Full step-by-step solution

   Step 1: The equation is (2x + 8) ÷ 2 = 10 Step 2: Multiply both sides by 2 to eliminate the division: 2x + 8 = 20 Step 3: Subtract 8 from both sides to isolate the term with x: 2x = 12 Step 4: Divide both sides by 2 to solve for x: x = 6 The answer is 6.