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Chapter 26: Linear Relations

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
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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.

26.1 Meaning of linear relation

A relation pairs input values with output values. In this section, the focus is Meaning of linear relation.

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 linear relation. Show the important steps, include units when needed, and explain how you checked the answer.

26.2 Constant rate of change

A rate compares quantities measured in different units. In this section, the focus is Constant 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 Constant rate of change. Show the important steps, include units when needed, and explain how you checked the answer.

26.3 Tables of values

Tables of values is an important Grade 8 concept in Linear Relations. 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

26.4 Graphing ordered pairs

Graphing ordered pairs is an important Grade 8 concept in Linear Relations. 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: Where is the point (2,3) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant I

Worked Example 2

Problem: Where is the point (-4,5) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant II

Worked Example 3

Problem: Where is the point (-3,-2) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant III

Worked Example 4

Problem: Where is the point (6,-1) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant IV

Worked Example 5

Problem: Where is the point (0,4) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: on an axis

Worked Example 6

Problem: Where is the point (5,0) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: on an axis

Worked Example 7

Problem: Where is the point (-7,2) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant II

Worked Example 8

Problem: Where is the point (1,-6) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant IV

Worked Example 9

Problem: Where is the point (-2,-8) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant III

Worked Example 10

Problem: Where is the point (8,7) located?

  1. Move horizontally using x.
  2. Move vertically using y.
  3. Identify the quadrant or axis.

Very beginner explanation: Ordered pairs are always written (x, y): horizontal first, vertical second.

Answer: Quadrant I

Practice exercise

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

26.5 Straight-line graphs

Straight-line graphs is an important Grade 8 concept in Linear Relations. 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: Find the supplementary angle to 35°.

  1. Supplementary angles total 180°.
  2. 180° - 35° = 145°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 145°

Worked Example 2

Problem: Find the supplementary angle to 48°.

  1. Supplementary angles total 180°.
  2. 180° - 48° = 132°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 132°

Worked Example 3

Problem: Find the supplementary angle to 67°.

  1. Supplementary angles total 180°.
  2. 180° - 67° = 113°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 113°

Worked Example 4

Problem: Find the supplementary angle to 72°.

  1. Supplementary angles total 180°.
  2. 180° - 72° = 108°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 108°

Worked Example 5

Problem: Find the supplementary angle to 110°.

  1. Supplementary angles total 180°.
  2. 180° - 110° = 70°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 70°

Worked Example 6

Problem: Find the supplementary angle to 25°.

  1. Supplementary angles total 180°.
  2. 180° - 25° = 155°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 155°

Worked Example 7

Problem: Find the supplementary angle to 58°.

  1. Supplementary angles total 180°.
  2. 180° - 58° = 122°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 122°

Worked Example 8

Problem: Find the supplementary angle to 83°.

  1. Supplementary angles total 180°.
  2. 180° - 83° = 97°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 97°

Worked Example 9

Problem: Find the supplementary angle to 95°.

  1. Supplementary angles total 180°.
  2. 180° - 95° = 85°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 85°

Worked Example 10

Problem: Find the supplementary angle to 120°.

  1. Supplementary angles total 180°.
  2. 180° - 120° = 60°.

Very beginner explanation: Angles on a straight line are supplementary.

Answer: 60°

Practice exercise

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

26.6 Increasing linear relations

A relation pairs input values with output values. In this section, the focus is Increasing linear relations.

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 Increasing linear relations. Show the important steps, include units when needed, and explain how you checked the answer.

26.7 Decreasing linear relations

A relation pairs input values with output values. In this section, the focus is Decreasing linear relations.

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 Decreasing linear relations. Show the important steps, include units when needed, and explain how you checked the answer.

26.8 Horizontal relations

A relation pairs input values with output values. In this section, the focus is Horizontal relations.

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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

26.9 Comparing linear relations

A relation pairs input values with output values. In this section, the focus is Comparing linear relations.

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 Comparing linear relations. Show the important steps, include units when needed, and explain how you checked the answer.

26.10 Linear versus nonlinear relations

A relation pairs input values with output values. In this section, the focus is Linear versus nonlinear relations.

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 Linear versus nonlinear relations. Show the important steps, include units when needed, and explain how you checked the answer.

26.11 Real-life linear models

The mode is the value that occurs most often. In this section, the focus is Real-life linear models.

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 linear models. Show the important steps, include units when needed, and explain how you checked the answer.

26.12 Interpolation

Interpolation is an important Grade 8 concept in Linear Relations. 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

26.13 Extrapolation introduction

Extrapolation introduction is an important Grade 8 concept in Linear Relations. 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: Continue the pattern 2, 5, 8, ... for two more terms.

  1. Find the common difference: 3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 11, 14

Worked Example 2

Problem: Continue the pattern 5, 9, 13, ... for two more terms.

  1. Find the common difference: 4.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 17, 21

Worked Example 3

Problem: Continue the pattern 10, 8, 6, ... for two more terms.

  1. Find the common difference: -2.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4, 2

Worked Example 4

Problem: Continue the pattern 1.5, 2, 2.5, ... for two more terms.

  1. Find the common difference: 0.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 3.5

Worked Example 5

Problem: Continue the pattern 20, 17.5, 15, ... for two more terms.

  1. Find the common difference: -2.5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12.5, 10

Worked Example 6

Problem: Continue the pattern 7, 13, 19, ... for two more terms.

  1. Find the common difference: 6.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 25, 31

Worked Example 7

Problem: Continue the pattern -3, 2, 7, ... for two more terms.

  1. Find the common difference: 5.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 12, 17

Worked Example 8

Problem: Continue the pattern 0.5, 1.75, 3, ... for two more terms.

  1. Find the common difference: 1.25.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 4.25, 5.5

Worked Example 9

Problem: Continue the pattern 12, 9, 6, ... for two more terms.

  1. Find the common difference: -3.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 3, 0

Worked Example 10

Problem: Continue the pattern 100, 110, 120, ... for two more terms.

  1. Find the common difference: 10.
  2. Add the same difference each time.

Very beginner explanation: An arithmetic pattern changes by the same amount from one term to the next.

Answer: 130, 140

Practice exercise

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

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Chapter 26 Review Questions and Answers

Q1. What is important to remember about Meaning of linear relation?

Answer: A relation pairs input values with output values. In this section, the focus is Meaning of linear relation.

Q2. What is important to remember about Constant rate of change?

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

Q3. What is important to remember about Tables of values?

Answer: Tables of values is an important Grade 8 concept in Linear Relations. 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 Graphing ordered pairs?

Answer: Graphing ordered pairs is an important Grade 8 concept in Linear Relations. 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 Straight-line graphs?

Answer: Straight-line graphs is an important Grade 8 concept in Linear Relations. 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.

Q6. What is important to remember about Increasing linear relations?

Answer: A relation pairs input values with output values. In this section, the focus is Increasing linear relations.

Q7. What is important to remember about Decreasing linear relations?

Answer: A relation pairs input values with output values. In this section, the focus is Decreasing linear relations.

Q8. What is important to remember about Horizontal relations?

Answer: A relation pairs input values with output values. In this section, the focus is Horizontal relations.

Q9. What is important to remember about Comparing linear relations?

Answer: A relation pairs input values with output values. In this section, the focus is Comparing linear relations.

Q10. What is important to remember about Linear versus nonlinear relations?

Answer: A relation pairs input values with output values. In this section, the focus is Linear versus nonlinear relations.

Q11. What is important to remember about Real-life linear models?

Answer: The mode is the value that occurs most often. In this section, the focus is Real-life linear models.

Q12. What is important to remember about Interpolation?

Answer: Interpolation is an important Grade 8 concept in Linear Relations. 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.

Q13. What is important to remember about Extrapolation introduction?

Answer: Extrapolation introduction is an important Grade 8 concept in Linear Relations. 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.

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.