01-01 - Algebraic Expressions

Phase: 1 | Subject: 01-01 Prerequisites: None for this subject Next subject: 01-02-linear-equations.md

Learning Objectives

By the end of this subject, you will be able to:

1. Use variables and constants to represent real-world relationships algebraically
2. Substitute values into expressions and evaluate them correctly
3. Collect like terms to simplify expressions
4. Expand brackets using the distributive law
5. Factorise expressions by taking out common factors

Core Content

Variables and Constants

A variable is a symbol (usually a letter like x, y, or a) that stands for a number we don't know yet or that can change. A constant is a fixed number.

Example: The cost of hiring a bike is $10 plus $3 per hour. If h = hours, the cost C is:

C = 10 + 3h

Here, 10 and 3 are constants. h is a variable. C is also a variable (it depends on h).

Writing Expressions

Turn word problems into algebra by identifying quantities and operations.

Substitution

Replace each variable with its given value and calculate using order of operations.

Example: If a = 4 and b = -2, find 3a^2 - 2b.

3a^2 - 2b = 3(4)^2 - 2(-2) = 3(16) + 4 = 48 + 4 = 52

Common mistake: Forgetting that 3a^2 means 3 * (a^2), not (3a)^2. If a = 4, 3a^2 = 3 * 16 = 48, but (3a)^2 = 12^2 = 144.

Like Terms

Like terms have exactly the same variable part (same letters raised to the same powers).

• Like terms: 5x and 3x, -2y^2 and 7y^2, 4ab and -ab
• Not like terms: 3x and 3y, 2x^2 and 5x, 7xy and 7x

Collecting like terms: 7x + 3y - 2x + 5y = (7x - 2x) + (3y + 5y) = 5x + 8y

Expanding Brackets

The distributive law says: a(b + c) = ab + ac

⚠️ THIS IS CRITICAL — you will use the distributive law constantly throughout all of algebra and beyond. Every expansion, every factorisation, every time you simplify an expression, you're applying this law. Master it now.

Single bracket: 3(x + 4) = 3x + 12

Example with a negative: -2(5 - x) = -10 + 2x = 2x - 10

Be careful with negatives: -(x + 3) = -x - 3, not -x + 3.

Factorising (Common Factors)

Factorising is the reverse of expanding. Find the largest common factor (LCF) and divide each term by it.

Example: Factorise 6x^2 + 9x

1. LCF of 6x^2 and 9x is 3x
2. Divide each term by 3x: 6x^2 / 3x = 2x, 9x / 3x = 3
3. Result: 3x(2x + 3)

Check: 3x(2x + 3) = 6x^2 + 9x [OK]

Example with negatives: Factorise -4xy + 8x

LCF is -4x: -4x(y - 2)

Always factor out the negative if the leading term is negative.

Key Terms

• Answers
• Like terms

Worked Examples

Example 1: Simplify 4x + 7 - 2x + 3

• Group like terms: (4x - 2x) + (7 + 3)
• Combine: 2x + 10
• This cannot be factorised further.

Example 2: Expand and simplify 2(x - 3) + 4(2x + 1)

• Expand first bracket: 2x - 6
• Expand second bracket: 8x + 4
• Add: 2x - 6 + 8x + 4 = 10x - 2

Example 3: If p = 5 and q = -3, find 2pq - p^2 + 3q

• Substitute: 2(5)(-3) - (5)^2 + 3(-3)
• Calculate: -30 - 25 - 9
• Sum: -64

Quiz

Q1: What does the concept of Like terms primarily refer to in this subject?

A) The definition and application of Like terms B) A historical anecdote about Like terms C) A computational error related to Like terms D) A visual representation of Like terms

Correct: A)

• If you chose A: Like terms is defined as: the definition and application of like terms. The other options describe different aspects that are not the primary focus. Correct!
• If you chose B: This is incorrect. Like terms is defined as: the definition and application of like terms. The other options describe different aspects that are not the primary focus.
• If you chose C: This is incorrect. Like terms is defined as: the definition and application of like terms. The other options describe different aspects that are not the primary focus.
• If you chose D: This is incorrect. Like terms is defined as: the definition and application of like terms. The other options describe different aspects that are not the primary focus.

Q2: What is the primary purpose of Answers?

A) It is primarily a historical notation system B) It replaces all other methods in this domain C) It is used to answers in mathematical analysis D) It is used only in advanced research contexts

Correct: C)

• If you chose A: This is incorrect. Answers serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose B: This is incorrect. Answers serves the purpose described in the correct answer. The other options misrepresent its role.
• If you chose C: Answers serves the purpose described in the correct answer. The other options misrepresent its role. Correct!
• If you chose D: This is incorrect. Answers serves the purpose described in the correct answer. The other options misrepresent its role.

Q3: Which statement about Variables And Constants is TRUE?

A) Variables And Constants is mentioned only as a historical footnote B) Variables And Constants is a fundamental concept covered in this subject C) Variables And Constants is not related to this subject D) Variables And Constants is an advanced topic beyond this subject's scope

Correct: B)

• If you chose A: This is incorrect. Variables And Constants is a fundamental concept covered in this subject. This subject covers Variables And Constants as part of its core content.
• If you chose B: Variables And Constants is a fundamental concept covered in this subject. This subject covers Variables And Constants as part of its core content. Correct!
• If you chose C: This is incorrect. Variables And Constants is a fundamental concept covered in this subject. This subject covers Variables And Constants as part of its core content.
• If you chose D: This is incorrect. Variables And Constants is a fundamental concept covered in this subject. This subject covers Variables And Constants as part of its core content.

Q4: Based on the worked examples in this subject, what is the correct result?

A) 3x(2x + 3) B) The inverse of the correct answer C) An unrelated numerical value D) A different result from a common mistake

Correct: A)

• If you chose A: The worked examples show that the result is 3x(2x + 3). The other options represent common errors. Correct!
• If you chose B: This is incorrect. The worked examples show that the result is 3x(2x + 3). The other options represent common errors.
• If you chose C: This is incorrect. The worked examples show that the result is 3x(2x + 3). The other options represent common errors.
• If you chose D: This is incorrect. The worked examples show that the result is 3x(2x + 3). The other options represent common errors.

Q5: How are Variables And Constants and Writing Expressions related?

A) Variables And Constants and Writing Expressions are completely unrelated topics B) Variables And Constants is a special case of Writing Expressions C) Variables And Constants is the inverse of Writing Expressions D) Variables And Constants and Writing Expressions are closely related concepts

Correct: D)

• If you chose A: This is incorrect. Both Variables And Constants and Writing Expressions are covered in this subject as interconnected topics.
• If you chose B: This is incorrect. Both Variables And Constants and Writing Expressions are covered in this subject as interconnected topics.
• If you chose C: This is incorrect. Both Variables And Constants and Writing Expressions are covered in this subject as interconnected topics.
• If you chose D: Both Variables And Constants and Writing Expressions are covered in this subject as interconnected topics. Correct!

Q6: What is a common pitfall when working with Substitution?

A) Substitution is always computed the same way in all contexts B) The main error with Substitution is using it when it is not needed C) A common mistake is confusing Substitution with a similar concept D) Substitution has no common misconceptions

Correct: C)

• If you chose A: This is incorrect. Students often confuse Substitution with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose B: This is incorrect. Students often confuse Substitution with similar-sounding or related concepts. Pay attention to the precise definitions.
• If you chose C: Students often confuse Substitution with similar-sounding or related concepts. Pay attention to the precise definitions. Correct!
• If you chose D: This is incorrect. Students often confuse Substitution with similar-sounding or related concepts. Pay attention to the precise definitions.

Q7: When should you apply Expanding Brackets?

A) Avoid Expanding Brackets unless explicitly instructed B) Expanding Brackets is not practically useful C) Apply Expanding Brackets to solve problems in this subject's domain D) Use Expanding Brackets only in pure mathematics contexts

Correct: C)

• If you chose A: This is incorrect. Expanding Brackets is a practical tool used throughout this subject to solve relevant problems.
• If you chose B: This is incorrect. Expanding Brackets is a practical tool used throughout this subject to solve relevant problems.
• If you chose C: Expanding Brackets is a practical tool used throughout this subject to solve relevant problems. Correct!
• If you chose D: This is incorrect. Expanding Brackets is a practical tool used throughout this subject to solve relevant problems.

Practice Problems

1. Simplify: 8y - 3 + 2y + 7
   Click for answer

10y + 4

2. Expand: 5(2a - 1)

Answers are provided above each problem so you can check your work immediately.

Summary

Key takeaways:

• Variables represent unknown or changing quantities; constants are fixed numbers
• Collect like terms by matching variable parts exactly
• Expanding uses the distributive law: a(b + c) = ab + ac
• Factorising reverses expansion by pulling out the largest common factor
• Always substitute carefully and follow order of operations

Pitfalls

• Confusing 3x² with (3x)². The expression 3x² means 3 × (x²) — only x is squared, then the result is multiplied by 3. (3x)² means (3x) × (3x) = 9x². If x = 4, 3x² = 48 but (3x)² = 144.
• Incorrectly distributing negative signs. The distributive law with −1 means −(x + 3) = −x − 3, NOT −x + 3. Every term inside the bracket gets the sign flip. This is especially tricky with expressions like −2(5 − x) = −10 + 2x.
• Combining unlike terms. 3x + 5y cannot be combined into 8xy. Only terms with exactly the same variable part (same letters, same powers) are like terms and can be added or subtracted.
• Incomplete factorisation. Factorising 8a² + 12a as 4(2a² + 3a) is technically correct but incomplete. The largest common factor is 4a, giving 4a(2a + 3). Always pull out the largest common factor.
• Misinterpreting \"less than\" in word problems. \"5 less than twice a number n\" is 2n − 5, not 5 − 2n. The phrase \"less than\" reverses the order — it means start with the larger quantity and subtract.

Next Steps

Next up: 01-02-linear-equations.md