Understand Slope

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Answer Key & Explanations

Understand Slope · Grade 8 · Worksheet 2

1. Mason is analyzing the steepness of a ramp on a coordinate grid. The ramp passes through points (2, 7) and (12, 17). What is the slope of the ramp? Answer: 1 Solution: Identify the coordinates. Point 1: (2, 7) and Point 2: (12, 17). Use the slope formula: slope = (y2 - y1) / (x2 - x1).
   Full step-by-step solution

   Step 1: Identify the coordinates. Point 1: (2, 7) and Point 2: (12, 17). Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1). Step 3: Substitute the values: slope = (17 - 7) / (12 - 2). Step 4: Simplify: slope = 10 / 10. Step 5: slope = 1. The answer is 1.

2. Tane plots two points on a coordinate grid: A(10, 14) and B(22, 38). He draws a straight line through these points. What is the slope of this line? Answer: 2 Solution: Identify the coordinates. Point A is (10, 14) and point B is (22, 38). Use the slope formula: slope = (y2 - y1) / (x2 - x1).
   Full step-by-step solution

   Step 1: Identify the coordinates. Point A is (10, 14) and point B is (22, 38). Step 2: Use the slope formula: slope = (y2 - y1) / (x2 - x1). Step 3: Substitute the values: slope = (38 - 14) / (22 - 10). Step 4: Simplify: slope = 24 / 12. Step 5: Divide: 24 / 12 = 2. The slope of the line is 2.

3. Nikau is hiking up a mountain trail. The trail rises 20 meters vertically for every 5 meters horizontally. What is the slope of the trail? Answer: 4 Solution: Identify the rise and run. The vertical rise is 20 meters, and the horizontal run is 5 meters. Use the slope formula: slope = rise / run.
   Full step-by-step solution

   Step 1: Identify the rise and run. The vertical rise is 20 meters, and the horizontal run is 5 meters. Step 2: Use the slope formula: slope = rise / run. Step 3: Substitute the values: slope = 20 / 5. Step 4: Simplify: 20 / 5 = 4. The slope of the trail is 4.

4. Moana is watching a hot air balloon rise. The balloon rises 30 meters while moving horizontally 5 meters. What is the slope of the balloon's path? Answer: 6 Solution: Identify the rise and run. The vertical rise is 30 meters, and the horizontal run is 5 meters. Use the slope formula: slope = rise / run.
   Full step-by-step solution

   Step 1: Identify the rise and run. The vertical rise is 30 meters, and the horizontal run is 5 meters. Step 2: Use the slope formula: slope = rise / run. Step 3: Substitute the values: slope = 30 / 5. Step 4: Simplify: 30 / 5 = 6. The slope of the balloon's path is 6.

5. Find the slope of the line passing through points (2, 8) and (10, 20). Answer: 1.5 Solution: Identify the coordinates: (x₁, y₁) = (2, 8) and (x₂, y₂) = (10, 20) Apply the slope formula: slope = (y₂ - y₁) / (x₂ - x₁) Substitute the values: slope = (20 - 8) / (10 - 2) Simplify the numerator: 20 - 8 = 12 Simplify the denominator: 10 - 2 = 8 Divide: 12 ÷ 8 = 1.5 The slope of the line is 1.5.
   Full step-by-step solution

   Step 1: Identify the coordinates: (x₁, y₁) = (2, 8) and (x₂, y₂) = (10, 20) Step 2: Apply the slope formula: slope = (y₂ - y₁) / (x₂ - x₁) Step 3: Substitute the values: slope = (20 - 8) / (10 - 2) Step 4: Simplify the numerator: 20 - 8 = 12 Step 5: Simplify the denominator: 10 - 2 = 8 Step 6: Divide: 12 ÷ 8 = 1.5 The slope of the line is 1.5.

6. (2/3) + (5/6) = ? Answer: 3/2 Solution: Identify the problem. We need to add the fractions 2/3 and 5/6. Check if the denominators are the same.
   Full step-by-step solution

   Step 1: Identify the problem. We need to add the fractions 2/3 and 5/6. Step 2: Check if the denominators are the same. The denominators are 3 and 6, which are not the same. We cannot add them directly. Step 3: Find a common denominator. The least common multiple of 3 and 6 is 6. So, the common denominator will be 6. Step 4: Rewrite each fraction with denominator 6. The fraction 5/6 already has denominator 6, so it stays as 5/6. For 2/3, we multiply numerator and denominator by 2: (2 × 2)/(3 × 2) = 4/6. Step 5: Now add the fractions with the same denominator. 4/6 + 5/6 = (4 + 5)/6 = 9/6. Step 6: Simplify the result. Both 9 and 6 can be divided by 3: 9 ÷ 3 = 3, 6 ÷ 3 = 2. So, 9/6 simplifies to 3/2. Final answer: 3/2

7. Mere is designing a wheelchair ramp for a building. The ramp needs to go up 21 inches over a horizontal distance of 7 inches. What is the slope of the ramp? Answer: 3 Solution: Identify the rise and run. The rise is 21 inches, and the run is 7 inches. Use the slope formula: slope = rise / run.
   Full step-by-step solution

   Step 1: Identify the rise and run. The rise is 21 inches, and the run is 7 inches. Step 2: Use the slope formula: slope = rise / run. Step 3: Substitute the values: slope = 21 / 7. Step 4: Simplify: 21 / 7 = 3. The slope of the ramp is 3.

8. 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

9. (-3x + 15) ÷ 3 = ? Answer: -x + 5 Solution: The expression is (-3x + 15) ÷ 3. Divide each term inside the parentheses by 3. -3x ÷ 3 = -x 15 ÷ 3 = 5 Combine the results: -x + 5 The simplified expression is -x + 5.
   Full step-by-step solution

   Step 1: The expression is (-3x + 15) ÷ 3. Step 2: Divide each term inside the parentheses by 3. Step 3: -3x ÷ 3 = -x Step 4: 15 ÷ 3 = 5 Step 5: Combine the results: -x + 5 The simplified expression is -x + 5.