EASYTUTORGUIDE

Practical tutorials, tools, courses, digital skills, and business promotion.

Free Learning
Google Translate — English / فارسی / العربية
EasyTutorGuide LogoLearn, Build, & Grow

Chapter 27: Slope and Rate of Change

Learn Grade 8 Math from beginner foundations through advanced Grade 8 problem solving with detailed explanations, at least five examples per topic, practice exercises, and review questions.

Grade 8Beginner Friendly5+ Examples Per TopicPractice30 Q&A
Estimated reading time0% read
Advertisement area — AdSense / Auto Ads

What this chapter covers

This chapter contains 13 topics. Technical terms are followed by plain-language meanings where they first appear. Each topic includes at least five worked examples. Coding is included only where it naturally supports the Grade 8 coding expectations.

27.1 Meaning of slope

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Meaning of slope.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Meaning of slope. Show the important steps, include units when needed, and explain how you checked the answer.

27.2 Rise

Rise is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Rise. Show the important steps, include units when needed, and explain how you checked the answer.

27.3 Run

Run is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Run. Show the important steps, include units when needed, and explain how you checked the answer.

27.4 Rise over run

Rise over run is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Rise over run. Show the important steps, include units when needed, and explain how you checked the answer.

27.5 Positive slope

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Positive slope.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Positive slope. Show the important steps, include units when needed, and explain how you checked the answer.

27.6 Negative slope

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Negative slope.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Negative slope. Show the important steps, include units when needed, and explain how you checked the answer.

27.7 Zero slope

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Zero slope.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Zero slope. Show the important steps, include units when needed, and explain how you checked the answer.

27.8 Undefined slope

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Undefined slope.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Undefined slope. Show the important steps, include units when needed, and explain how you checked the answer.

27.9 Finding slope from graphs

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from graphs.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Finding slope from graphs. Show the important steps, include units when needed, and explain how you checked the answer.

27.10 Finding slope from two points

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from two points.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Finding slope from two points. Show the important steps, include units when needed, and explain how you checked the answer.

27.11 Finding slope from tables

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from tables.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Finding slope from tables. Show the important steps, include units when needed, and explain how you checked the answer.

27.12 Rate of change

A rate compares quantities measured in different units. In this section, the focus is Rate of change.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: A quantity of 80 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 80 ÷ 2 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 2

Problem: A quantity of 120 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 120 ÷ 3 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 3

Problem: A quantity of 190 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 190 ÷ 4 = 47.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 47.5 per unit

Worked Example 4

Problem: A quantity of 250 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 250 ÷ 5 = 50.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 50 per unit

Worked Example 5

Problem: A quantity of 220 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 220 ÷ 2 = 110.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 110 per unit

Worked Example 6

Problem: A quantity of 160 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 160 ÷ 3 = 53.3333.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 53.3333 per unit

Worked Example 7

Problem: A quantity of 130 units is used over 4 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 130 ÷ 4 = 32.5.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 32.5 per unit

Worked Example 8

Problem: A quantity of 200 units is used over 5 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 200 ÷ 5 = 40.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 40 per unit

Worked Example 9

Problem: A quantity of 290 units is used over 2 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 290 ÷ 2 = 145.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 145 per unit

Worked Example 10

Problem: A quantity of 180 units is used over 3 units of time. Find the unit rate.

  1. Divide the first quantity by the second.
  2. 180 ÷ 3 = 60.

Very beginner explanation: A unit rate answers “how much for one?”

Answer: 60 per unit

Practice exercise

Create one new Grade 8 problem involving Rate of change. Show the important steps, include units when needed, and explain how you checked the answer.

27.13 Real-life slope applications

Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Real-life slope applications.

For a beginner, focus on the meaning before memorizing a procedure. Start with a small example, name the quantities and units, apply one rule at a time, and connect the result back to the question.

10 Worked Examples with Answers and Very-Beginner Explanations

Worked Example 1

Problem: For y = 2x + (1), find y when x = 1.

  1. Substitute the x-value.
  2. y = 2(1) + (1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 3

Worked Example 2

Problem: For y = -1x + (4), find y when x = 2.

  1. Substitute the x-value.
  2. y = -1(2) + (4).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 2

Worked Example 3

Problem: For y = 0.5x + (-2), find y when x = 3.

  1. Substitute the x-value.
  2. y = 0.5(3) + (-2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -0.5

Worked Example 4

Problem: For y = 3x + (0), find y when x = 4.

  1. Substitute the x-value.
  2. y = 3(4) + (0).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 12

Worked Example 5

Problem: For y = -2x + (5), find y when x = 5.

  1. Substitute the x-value.
  2. y = -2(5) + (5).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: -5

Worked Example 6

Problem: For y = 4x + (-3), find y when x = 1.

  1. Substitute the x-value.
  2. y = 4(1) + (-3).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 1

Worked Example 7

Problem: For y = 1.5x + (2), find y when x = 2.

  1. Substitute the x-value.
  2. y = 1.5(2) + (2).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 5

Worked Example 8

Problem: For y = -0.5x + (6), find y when x = 3.

  1. Substitute the x-value.
  2. y = -0.5(3) + (6).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 4.5

Worked Example 9

Problem: For y = 5x + (-1), find y when x = 4.

  1. Substitute the x-value.
  2. y = 5(4) + (-1).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 19

Worked Example 10

Problem: For y = 0x + (7), find y when x = 5.

  1. Substitute the x-value.
  2. y = 0(5) + (7).
  3. Multiply first, then add.

Very beginner explanation: In y = mx + b, m is the rate of change and b is the starting value or y-intercept.

Answer: 7

Practice exercise

Create one new Grade 8 problem involving Real-life slope applications. Show the important steps, include units when needed, and explain how you checked the answer.

Advertisement area — AdSense / Auto Ads

Chapter 27 Review Questions and Answers

Q1. What is important to remember about Meaning of slope?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Meaning of slope.

Q2. What is important to remember about Rise?

Answer: Rise is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Run?

Answer: Run is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q4. What is important to remember about Rise over run?

Answer: Rise over run is an important Grade 8 concept in Slope and Rate of Change. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q5. What is important to remember about Positive slope?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Positive slope.

Q6. What is important to remember about Negative slope?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Negative slope.

Q7. What is important to remember about Zero slope?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Zero slope.

Q8. What is important to remember about Undefined slope?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Undefined slope.

Q9. What is important to remember about Finding slope from graphs?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from graphs.

Q10. What is important to remember about Finding slope from two points?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from two points.

Q11. What is important to remember about Finding slope from tables?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Finding slope from tables.

Q12. What is important to remember about Rate of change?

Answer: A rate compares quantities measured in different units. In this section, the focus is Rate of change.

Q13. What is important to remember about Real-life slope applications?

Answer: Slope measures a line’s rate of change and is commonly calculated as rise divided by run. In this section, the focus is Real-life slope applications.

Q14. Why should you show your steps?

Answer: Showing steps makes reasoning easier to verify and helps locate mistakes.

Q15. Why is estimation useful?

Answer: Estimation helps you judge whether a calculated answer is reasonable.

Q16. Why do units matter?

Answer: Units tell what a value measures and prevent incompatible quantities from being mixed.

Q17. When should you round?

Answer: Usually round near the end unless the question specifically asks for earlier rounding.

Q18. How can you check an answer?

Answer: Use an inverse operation, substitution, estimation, a graph, or a second method.

Q19. What should you identify first in a word problem?

Answer: Identify what is known, what is unknown, and the mathematical relationship connecting them.

Q20. What makes a final answer complete?

Answer: Give the value, correct units when needed, and a brief interpretation in context.

Q21. What should you do if an answer seems unreasonable?

Answer: Re-read the question and check copied values, signs, units, formulas, and calculations.

Q22. Why are diagrams useful?

Answer: A labelled diagram can reveal relationships among lengths, angles, areas, and coordinates.

Q23. Why are tables useful?

Answer: Tables organize values and help reveal patterns, rates, relationships, and missing information.

Q24. Why should you explain your reasoning?

Answer: Reasoning shows why a method works, not just what answer was obtained.

Q25. How do mistakes help learning?

Answer: Analyzing a mistake identifies the misunderstood rule or step and helps prevent repetition.

Q26. When is a calculator useful?

Answer: A calculator helps with lengthy arithmetic after the mathematical setup is understood and estimated.

Q27. Why compare more than one strategy?

Answer: Different strategies may make a problem easier and provide a way to verify the result.

Q28. What is mathematical communication?

Answer: It is presenting ideas clearly with words, symbols, diagrams, tables, graphs, and justified steps.

Q29. What is mathematical modelling?

Answer: It is representing a real situation with mathematics, testing the model, and revising it if needed.

Q30. Why should assumptions be stated?

Answer: Assumptions show the conditions the model depends on and help readers judge whether it is reasonable.