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Chapter 28: Linear Equations and y = mx + b

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.

28.1 Slope-intercept form

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

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 Slope-intercept form. Show the important steps, include units when needed, and explain how you checked the answer.

28.2 Meaning of y = mx + b

The mean is the sum of all values divided by the number of values. In this section, the focus is Meaning of y = mx + b.

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 y = mx + b. Show the important steps, include units when needed, and explain how you checked the answer.

28.3 Slope m

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

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

28.4 y-intercept b

The y-intercept is the y-value where a graph crosses the y-axis. In this section, the focus is y-intercept b.

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 y-intercept b. Show the important steps, include units when needed, and explain how you checked the answer.

28.5 Graphing from an equation

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Graphing from an equation.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

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

28.6 Writing an equation from a graph

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing an equation from a graph.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

Create one new Grade 8 problem involving Writing an equation from a graph. Show the important steps, include units when needed, and explain how you checked the answer.

28.7 Writing an equation from a table

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing an equation from a table.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

Create one new Grade 8 problem involving Writing an equation from a table. Show the important steps, include units when needed, and explain how you checked the answer.

28.8 Comparing slopes

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

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

28.9 Comparing intercepts

Comparing intercepts is an important Grade 8 concept in Linear Equations and y = mx + b. 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 Comparing intercepts. Show the important steps, include units when needed, and explain how you checked the answer.

28.10 Direct variation

Direct variation is an important Grade 8 concept in Linear Equations and y = mx + b. 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 Direct variation. Show the important steps, include units when needed, and explain how you checked the answer.

28.11 Initial value

Initial value is an important Grade 8 concept in Linear Equations and y = mx + b. 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 Initial value. Show the important steps, include units when needed, and explain how you checked the answer.

28.12 Real-life linear equations

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Real-life linear equations.

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: Solve 2x + 5 = 17.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 2

Problem: Solve 3x - 4 = 11.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 3

Problem: Solve 5x + 7 = 2x + 22.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 4

Problem: Solve 4(x + 2) = 24.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 5

Problem: Solve 7x - 3 = 4x + 12.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 5

Worked Example 6

Problem: Solve 0.5x + 2 = 7.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 10

Worked Example 7

Problem: Solve 2(x-3)+4=10.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 6

Worked Example 8

Problem: Solve 9 - 2x = 1.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: x = 4

Worked Example 9

Problem: Solve 3x + 8 = 3x + 8.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: all real numbers

Worked Example 10

Problem: Solve 4x + 1 = 4x + 9.

  1. Simplify each side if needed.
  2. Use inverse operations to move variable terms and constants.
  3. Keep both sides balanced.
  4. Check by substitution when a single solution exists.

Very beginner explanation: An equation is like a balance: whatever operation is done to one side must preserve equality.

Answer: no solution

Practice exercise

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

28.13 Word problems

Word problems is an important Grade 8 concept in Linear Equations and y = mx + b. 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: Word problems: Explain Word problems in one simple sentence.

  1. Look at the words in “Word problems”.
  2. State what the idea is used for.
  3. Give one small mathematical example.

Very beginner explanation: A beginner should first understand the meaning before memorizing a rule.

Answer: Word problems is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Word problems: A student says, “I can use Word problems without checking units or labels.” Is that a good method?

  1. Read the statement.
  2. Ask whether units, signs, labels, or conditions matter.
  3. Decide whether the method is safe.

Very beginner explanation: Many math mistakes come from using a correct calculation on the wrong quantity.

Answer: No. Important units, labels, signs, and conditions must be checked.

Worked Example 3

Problem: Word problems: What is the first step when solving a problem about Word problems?

  1. Read the whole question.
  2. Underline the known information.
  3. Identify what must be found.

Very beginner explanation: This prevents you from calculating before you understand the problem.

Answer: Identify the given information and the unknown before choosing a rule.

Worked Example 4

Problem: Word problems: After solving a Word problems problem, what should you do before accepting the answer?

  1. Estimate the expected size or direction.
  2. Check units and signs.
  3. Use an inverse method if possible.

Very beginner explanation: Checking is part of solving, not an optional extra.

Answer: Check whether the answer is reasonable and consistent with the problem.

Worked Example 5

Problem: Word problems: Give one way Word problems could appear outside a textbook.

  1. Think about money, measurements, data, maps, geometry, or technology.
  2. Connect the topic to one of those situations.

Very beginner explanation: Connecting math to real situations makes the rule easier to remember.

Answer: Word problems can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Word problems: Which representation could help explain Word problems: a table, graph, diagram, equation, or number line?

  1. Choose the representation that makes the relationship easiest to see.
  2. Label it clearly.

Very beginner explanation: Different representations show different features of the same mathematics.

Answer: Use the representation that best matches the problem; more than one may be valid.

Worked Example 7

Problem: Word problems: A student gets an answer for Word problems but cannot explain the steps. What should be improved?

  1. Rewrite the solution one step at a time.
  2. Name the rule used at each important step.

Very beginner explanation: A correct final number without reasoning may hide an error.

Answer: The reasoning should be shown so the solution can be checked.

Worked Example 8

Problem: Word problems: Why can estimation help before a detailed Word problems calculation?

  1. Round or use benchmark values.
  2. Predict the approximate answer.
  3. Compare the exact result with the estimate.

Very beginner explanation: If the exact answer is far from the estimate, recheck the work.

Answer: Estimation gives a target range for the final answer.

Worked Example 9

Problem: Word problems: How can you test whether your rule for Word problems works?

  1. Choose a very small easy example.
  2. Apply the rule.
  3. Check the result another way.

Very beginner explanation: Small examples expose mistakes quickly.

Answer: Test the rule on a simple case whose answer can be verified.

Worked Example 10

Problem: Word problems: How would you teach Word problems to someone seeing it for the first time?

  1. Define the idea.
  2. Show one easy example.
  3. Explain every step.
  4. Then let the learner try a similar problem.

Very beginner explanation: This sequence reduces memorization without understanding.

Answer: Teach meaning first, then a small worked example, then guided practice.

Practice exercise

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

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Chapter 28 Review Questions and Answers

Q1. What is important to remember about Slope-intercept form?

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

Q2. What is important to remember about Meaning of y = mx + b?

Answer: The mean is the sum of all values divided by the number of values. In this section, the focus is Meaning of y = mx + b.

Q3. What is important to remember about Slope m?

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

Q4. What is important to remember about y-intercept b?

Answer: The y-intercept is the y-value where a graph crosses the y-axis. In this section, the focus is y-intercept b.

Q5. What is important to remember about Graphing from an equation?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Graphing from an equation.

Q6. What is important to remember about Writing an equation from a graph?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing an equation from a graph.

Q7. What is important to remember about Writing an equation from a table?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Writing an equation from a table.

Q8. What is important to remember about Comparing slopes?

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

Q9. What is important to remember about Comparing intercepts?

Answer: Comparing intercepts is an important Grade 8 concept in Linear Equations and y = mx + b. 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.

Q10. What is important to remember about Direct variation?

Answer: Direct variation is an important Grade 8 concept in Linear Equations and y = mx + b. 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.

Q11. What is important to remember about Initial value?

Answer: Initial value is an important Grade 8 concept in Linear Equations and y = mx + b. 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.

Q12. What is important to remember about Real-life linear equations?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Real-life linear equations.

Q13. What is important to remember about Word problems?

Answer: Word problems is an important Grade 8 concept in Linear Equations and y = mx + b. 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.