Understand Slope

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Answer Key & Explanations

Understand Slope · Grade 8 · Worksheet 2

1. Find the slope of the line passing through points (3, 7) and (5, 13) = ? Answer: 3 Solution: Identify the coordinates: (3, 7) and (5, 13) Use the slope formula: slope = (y₂ - y₁) ÷ (x₂ - x₁) Substitute the values: slope = (13 - 7) ÷ (5 - 3) Calculate the differences: slope = 6 ÷ 2 Simplify: slope = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates: (3, 7) and (5, 13) Step 2: Use the slope formula: slope = (y₂ - y₁) ÷ (x₂ - x₁) Step 3: Substitute the values: slope = (13 - 7) ÷ (5 - 3) Step 4: Calculate the differences: slope = 6 ÷ 2 Step 5: Simplify: slope = 3 The answer is 3.

2. A line passes through the points (2, 8) and (6, 20). What is the slope of this line? Answer: 3 Solution: Identify the coordinates of the two points: (2, 8) and (6, 20) Use the slope formula: slope = (y2 - y1) / (x2 - x1) Substitute the values: slope = (20 - 8) / (6 - 2) Calculate the numerator: 20 - 8 = 12 Calculate the denominator: 6 - 2 = 4 Divide: 12 ÷ 4 = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Identify the coordinates of the two points: (2, 8) and (6, 20) Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1) Step 3: Substitute the values: slope = (20 - 8) / (6 - 2) Step 4: Calculate the numerator: 20 - 8 = 12 Step 5: Calculate the denominator: 6 - 2 = 4 Step 6: Divide: 12 ÷ 4 = 3 The answer is 3.

3. Liam is designing a wheelchair ramp for his school's new accessibility project. The building code requires that for every 12 inches of horizontal distance, the ramp can only rise 1 inch. If the ramp needs to reach a platform that is 18 inches high, what is the minimum horizontal distance in feet that the ramp must cover? Answer: 18 Solution: The building code says: for every 12 inches of horizontal distance, the ramp can rise 1 inch. This means the ratio of rise to run is 1 inch rise / 12 inches horizontal. Identify the total rise needed.
   Full step-by-step solution

   Let's solve this step-by-step. Step 1: Understand the slope requirement. The building code says: for every 12 inches of horizontal distance, the ramp can rise 1 inch. This means the ratio of rise to run is 1 inch rise / 12 inches horizontal. Step 2: Identify the total rise needed. The platform height is 18 inches, so the ramp must rise 18 inches. Step 3: Set up a proportion using the slope rule. Let the horizontal distance in inches be \( x \). From the slope: 1 inch rise / 12 inches horizontal = 18 inches rise / \( x \) inches horizontal. So: 1 / 12 = 18 / \( x \) Step 4: Solve for \( x \). Cross-multiply: 1 * \( x \) = 18 * 12 \( x \) = 216 inches. Step 5: Convert inches to feet. Since 1 foot = 12 inches, \( x \) in feet = 216 / 12 = 18 feet. Step 6: Conclusion. The minimum horizontal distance the ramp must cover is 18 feet. ANSWER: 18

4. Aisha is hiking up a mountain trail. For every 150 meters she hikes horizontally, she gains 45 meters in elevation. If she starts at an elevation of 800 meters and wants to reach a viewpoint at 1,070 meters, how many meters horizontally must she hike along this trail? Answer: 900 Solution: Calculate the total elevation gain needed: 1,070 m - 800 m = 270 m The slope relationship is 45 m elevation gain per 150 m horizontal distance Set up a proportion: 45/150 = 270/x Cross-multiply: 45x = 150 × 270 Calculate: 45x = 40,500 Divide both sides by 45: x = 40,500 ÷ 45 x = 900 The answer…
   Full step-by-step solution

   Step 1: Calculate the total elevation gain needed: 1,070 m - 800 m = 270 m Step 2: The slope relationship is 45 m elevation gain per 150 m horizontal distance Step 3: Set up a proportion: 45/150 = 270/x Step 4: Cross-multiply: 45x = 150 × 270 Step 5: Calculate: 45x = 40,500 Step 6: Divide both sides by 45: x = 40,500 ÷ 45 Step 7: x = 900 The answer is 900 meters.

5. A line passes through the points (3, 5) and (7, 13). What is the slope of this line? Answer: 2 Solution: To find the slope of a line passing 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 passing 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: (3, 5) so x1 = 3 and y1 = 5 Point 2: (7, 13) so x2 = 7 and y2 = 13 Step 2: Substitute these values into the slope formula. slope = (y2 - y1) / (x2 - x1) slope = (13 - 5) / (7 - 3) Step 3: Calculate the difference in the y-values (the rise). 13 - 5 = 8 Step 4: Calculate the difference in the x-values (the run). 7 - 3 = 4 Step 5: Divide the rise by the run to get the slope. slope = 8 / 4 Step 6: Simplify the fraction. 8 / 4 = 2 Therefore, the slope of the line is 2.

6. Find the slope of the line passing through points (11, 28) and (17, 52). Answer: 4 Solution: Identify the coordinates: (x₁, y₁) = (11, 28) and (x₂, y₂) = (17, 52) Use the slope formula: slope = (y₂ - y₁) / (x₂ - x₁) Substitute the values: slope = (52 - 28) / (17 - 11) Simplify the numerator: 52 - 28 = 24 Simplify the denominator: 17 - 11 = 6 Divide: 24 ÷ 6 = 4 The slope of the line is 4.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (11, 28) and (x₂, y₂) = (17, 52) Step 2: Use the slope formula: slope = (y₂ - y₁) / (x₂ - x₁) Step 3: Substitute the values: slope = (52 - 28) / (17 - 11) Step 4: Simplify the numerator: 52 - 28 = 24 Step 5: Simplify the denominator: 17 - 11 = 6 Step 6: Divide: 24 ÷ 6 = 4 The slope of the line is 4.

7. A line is drawn on a coordinate plane that passes through points (2, 7) and (5, 1). Another line is drawn perpendicular to this line and passes through the point (3, 4). What is the slope of this perpendicular line? Answer: 0.5 Solution: Find the slope of the first line using the points (2, 7) and (5, 1) Slope = (y2 - y1)/(x2 - x1) = (1 - 7)/(5 - 2) = (-6)/3 = -2 For perpendicular lines, the slopes are negative reciprocals If slope1 = -2, then slope2 = -1/(-2) = 1/2 = 0.5 The product of slopes of perpendicular lines is -1: (-2)…
   Full step-by-step solution

   Step 1: Find the slope of the first line using the points (2, 7) and (5, 1) Slope = (y2 - y1)/(x2 - x1) = (1 - 7)/(5 - 2) = (-6)/3 = -2 Step 2: Find the slope of the perpendicular line For perpendicular lines, the slopes are negative reciprocals If slope1 = -2, then slope2 = -1/(-2) = 1/2 = 0.5 Step 3: Verify this relationship The product of slopes of perpendicular lines is -1: (-2) × (0.5) = -1 ✓ The slope of the perpendicular line is 0.5.

8. A line is drawn on a coordinate plane that passes through points (3, 7) and (9, 19). Another line is drawn perpendicular to this line and passes through point (6, 13). What is the slope of the perpendicular line? Answer: -0.5 Solution: Find the slope of the original line using points (3, 7) and (9, 19) Slope = (y2 - y1)/(x2 - x1) = (19 - 7)/(9 - 3) = 12/6 = 2 Find the slope of a line perpendicular to this line Perpendicular slope = negative reciprocal of 2 = -1/2 = -0.5 The point (6, 13) is given but not needed to find the…
   Full step-by-step solution

   Step 1: Find the slope of the original line using points (3, 7) and (9, 19) Slope = (y2 - y1)/(x2 - x1) = (19 - 7)/(9 - 3) = 12/6 = 2 Step 2: Find the slope of a line perpendicular to this line Perpendicular slope = negative reciprocal of 2 = -1/2 = -0.5 Step 3: The point (6, 13) is given but not needed to find the slope of the perpendicular line The answer is -0.5.

9. Find the slope of the line passing through points (3, 5) and (7, 13). Answer: 2 Solution: Identify the coordinates of the two points: (3, 5) and (7, 13) Apply the slope formula: slope = (y₂ - y₁) / (x₂ - x₁) Substitute the values: slope = (13 - 5) / (7 - 3) Calculate the numerator: 13 - 5 = 8 Calculate the denominator: 7 - 3 = 4 Divide the results: 8 ÷ 4 = 2 The slope of the line is 2.
   Full step-by-step solution

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