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Chapter 20: Multi-Term Equations with Integers and Decimals

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.

20.1 Multi-term equations

A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Multi-term 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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

20.2 Simplifying both sides

Simplifying both sides is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. 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: Simplifying both sides: Explain Simplifying both sides in one simple sentence.

  1. Look at the words in “Simplifying both sides”.
  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: Simplifying both sides is a Grade 8 math idea used to describe or solve a specific mathematical relationship.

Worked Example 2

Problem: Simplifying both sides: A student says, “I can use Simplifying both sides 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: Simplifying both sides: What is the first step when solving a problem about Simplifying both sides?

  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: Simplifying both sides: After solving a Simplifying both sides 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: Simplifying both sides: Give one way Simplifying both sides 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: Simplifying both sides can be used in a real-life situation where quantities must be compared, measured, predicted, or calculated.

Worked Example 6

Problem: Simplifying both sides: Which representation could help explain Simplifying both sides: 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: Simplifying both sides: A student gets an answer for Simplifying both sides 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: Simplifying both sides: Why can estimation help before a detailed Simplifying both sides 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: Simplifying both sides: How can you test whether your rule for Simplifying both sides 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: Simplifying both sides: How would you teach Simplifying both sides 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 Simplifying both sides. Show the important steps, include units when needed, and explain how you checked the answer.

20.3 Combining like terms before solving

A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Combining like terms before solving.

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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

Create one new Grade 8 problem involving Combining like terms before solving. Show the important steps, include units when needed, and explain how you checked the answer.

20.4 Distribution before solving

Distribution before solving is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. 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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

20.5 Variables on one side

A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Variables on one side.

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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

20.6 Variables on both sides

A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Variables on both sides.

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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

20.7 Integer coefficients

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer coefficients.

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: Calculate -7 + (5).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

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

20.8 Decimal coefficients

A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Decimal coefficients.

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: Which is greater: 3.6 or 0.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 3.6

Worked Example 2

Problem: Which is greater: 8.25 or 1.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 8.25

Worked Example 3

Problem: Which is greater: 12.75 or 2.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 12.75

Worked Example 4

Problem: Which is greater: 0.96 or 0.3?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 0.96

Worked Example 5

Problem: Which is greater: 5.04 or 1.2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 5.04

Worked Example 6

Problem: Which is greater: 14.8 or 2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 14.8

Worked Example 7

Problem: Which is greater: 7.125 or 0.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 7.125

Worked Example 8

Problem: Which is greater: 20.05 or 3.75?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 20.05

Worked Example 9

Problem: Which is greater: 2.4 or 0.06?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 2.4

Worked Example 10

Problem: Which is greater: 100.5 or 4.02?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 100.5

Practice exercise

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

20.9 Integer constants

Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer constants.

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: Calculate -7 + (5).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -2

Worked Example 2

Problem: Calculate 4 + (-9).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -5

Worked Example 3

Problem: Calculate -6 + (-3).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -9

Worked Example 4

Problem: Calculate 12 + (-8).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 4

Worked Example 5

Problem: Calculate -15 + (20).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 6

Problem: Calculate -11 + (-4).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -15

Worked Example 7

Problem: Calculate 9 + (13).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 22

Worked Example 8

Problem: Calculate -2 + (7).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: 5

Worked Example 9

Problem: Calculate 18 + (-25).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -7

Worked Example 10

Problem: Calculate -30 + (12).

  1. If signs match, add absolute values and keep the sign.
  2. If signs differ, subtract the smaller absolute value from the larger and keep the sign of the larger absolute value.

Very beginner explanation: Adding integers can be understood as movement left or right on a number line.

Answer: -18

Practice exercise

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

20.10 Decimal constants

A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Decimal constants.

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: Which is greater: 3.6 or 0.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 3.6

Worked Example 2

Problem: Which is greater: 8.25 or 1.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 8.25

Worked Example 3

Problem: Which is greater: 12.75 or 2.4?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 12.75

Worked Example 4

Problem: Which is greater: 0.96 or 0.3?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 0.96

Worked Example 5

Problem: Which is greater: 5.04 or 1.2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 5.04

Worked Example 6

Problem: Which is greater: 14.8 or 2?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 14.8

Worked Example 7

Problem: Which is greater: 7.125 or 0.5?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 7.125

Worked Example 8

Problem: Which is greater: 20.05 or 3.75?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 20.05

Worked Example 9

Problem: Which is greater: 2.4 or 0.06?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 2.4

Worked Example 10

Problem: Which is greater: 100.5 or 4.02?

  1. Compare whole-number parts first.
  2. Then compare tenths, hundredths, and later decimal places.

Very beginner explanation: Decimal comparison is place-value comparison.

Answer: 100.5

Practice exercise

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

20.11 Identity equations introduction

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Identity equations introduction.

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

20.12 No-solution equations introduction

An equation is a mathematical statement that two expressions have the same value. In this section, the focus is No-solution equations introduction.

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 No-solution equations introduction. Show the important steps, include units when needed, and explain how you checked the answer.

20.13 Checking by substitution

Checking by substitution is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. 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: Simplify 3x + 2x.

  1. 3x and 2x are like terms.
  2. Add the coefficients: 3 + 2 = 5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5x

Worked Example 2

Problem: Simplify 7y - 4y + 3.

  1. 7y and -4y are like terms.
  2. Combine them; keep the constant 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3y + 3

Worked Example 3

Problem: Simplify 4(a + 3).

  1. Multiply 4 by a.
  2. Multiply 4 by 3.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 4a + 12

Worked Example 4

Problem: Simplify 2(3x - 5) + x.

  1. Distribute 2.
  2. 2(3x - 5)=6x-10.
  3. Combine 6x+x.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 10

Worked Example 5

Problem: Simplify 5m + 8 - 2m - 3.

  1. Combine variable terms.
  2. Combine constant terms.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 3m + 5

Worked Example 6

Problem: Simplify -3(2p + 4).

  1. Multiply -3 by both terms.
  2. Keep signs carefully.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: -6p - 12

Worked Example 7

Problem: Simplify 6x + 4 + x - 9.

  1. Combine x-terms.
  2. Combine constants.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 7x - 5

Worked Example 8

Problem: Simplify 0.5x + 1.5x.

  1. Both terms have x.
  2. Add decimal coefficients 0.5 + 1.5.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 2x

Worked Example 9

Problem: Simplify 3(2a + 1) - a.

  1. Distribute 3.
  2. Combine 6a-a.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 5a + 3

Worked Example 10

Problem: Simplify 8q - 2(q + 3).

  1. Distribute -2.
  2. Combine 8q-2q.

Very beginner explanation: Like terms have exactly the same variable part. Distribution multiplies every term inside parentheses.

Answer: 6q - 6

Practice exercise

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

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Chapter 20 Review Questions and Answers

Q1. What is important to remember about Multi-term equations?

Answer: A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Multi-term equations.

Q2. What is important to remember about Simplifying both sides?

Answer: Simplifying both sides is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. Start by identifying the quantities, relationships, units, or rules involved. Work with a small example first, show the important steps, and check whether the result is reasonable.

Q3. What is important to remember about Combining like terms before solving?

Answer: A term is one part of an algebraic expression separated by addition or subtraction signs. In this section, the focus is Combining like terms before solving.

Q4. What is important to remember about Distribution before solving?

Answer: Distribution before solving is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. 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 Variables on one side?

Answer: A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Variables on one side.

Q6. What is important to remember about Variables on both sides?

Answer: A variable is a letter or symbol that represents a value that may be unknown or may change. In this section, the focus is Variables on both sides.

Q7. What is important to remember about Integer coefficients?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer coefficients.

Q8. What is important to remember about Decimal coefficients?

Answer: A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Decimal coefficients.

Q9. What is important to remember about Integer constants?

Answer: Integers (negative whole numbers, zero, and positive whole numbers) are useful for temperature, elevation, scores, gains, and losses. In this section, the focus is Integer constants.

Q10. What is important to remember about Decimal constants?

Answer: A decimal uses place value to represent parts of one, including tenths, hundredths, thousandths, and smaller values. In this section, the focus is Decimal constants.

Q11. What is important to remember about Identity equations introduction?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is Identity equations introduction.

Q12. What is important to remember about No-solution equations introduction?

Answer: An equation is a mathematical statement that two expressions have the same value. In this section, the focus is No-solution equations introduction.

Q13. What is important to remember about Checking by substitution?

Answer: Checking by substitution is an important Grade 8 concept in Multi-Term Equations with Integers and Decimals. 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.