Mathematics (Specification A) | Expressions and formulae

Welcome to Expressions and Formulae!

Hello future Mathematicians! This chapter is absolutely fundamental to all of algebra. Think of expressions and formulae as the vocabulary and grammar of the mathematical world. Mastering this section will make solving equations and tackling harder problems much, much easier.

Don't worry if letters in maths feel strange right now. We'll break down these concepts step-by-step, using simple rules and lots of examples. Let's get started on turning those variables into valuable knowledge!

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1. Understanding the Language: Terms, Expressions, and Formulae

What are the Building Blocks?

In algebra, we use letters (called variables) to represent unknown numbers.

Key Definitions

• Term: A single number, a single variable, or variables multiplied together.
  Examples: \(5\), \(x\), \(4y^2\), \(-3ab\).
• Expression: A combination of terms joined by addition or subtraction. It does not have an equals sign.
  Example: \(4x + 7\) or \(a^2 - 3b + 1\).
• Formula (plural: Formulae): A mathematical rule that shows the relationship between different quantities (variables). It always has an equals sign.
  Analogy: A recipe! If you know the ingredients (variables), the formula tells you the result.
  Example: The area of a rectangle is length times width: \(A = l \times w\).
• Equation: A statement that two expressions are equal. We solve equations to find the value of the unknown variable(s).
  Example: \(4x + 7 = 15\).
• Identity: An equation that is true for every possible value of the variable. We use the symbol \(\equiv\) (triple equals) instead of \(=\).
  Example: \(3(x+2) \equiv 3x + 6\).

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2. Substitution: Putting Numbers into Letters

Substitution is the process of replacing variables (letters) in an expression or formula with their given numerical values. This allows us to evaluate (find the value of) the expression.

Step-by-Step Guide to Substitution

Rule 1: Always use BIDMAS/BODMAS (Brackets, Indices, Division/Multiplication, Addition/Subtraction) when calculating the final answer.

Rule 2: When substituting, particularly when squaring a negative number, use brackets to avoid confusion.

Example: Evaluate the expression \(3x + y^2\) if \(x = 5\) and \(y = -2\).

1. Write out the expression: \(3x + y^2\)
2. Substitute the values: Remember that \(3x\) means \(3 \times x\).
   \(3x + y^2 = 3(5) + (-2)^2\)
3. Calculate Indices (Powers) first: \( (-2)^2 = (-2) \times (-2) = 4\)
   (Common mistake: \(-2^2\) is NOT the same as \((-2)^2\)! Always use brackets when squaring negatives.)
4. Calculate Multiplication: \(3 \times 5 = 15\)
5. Calculate Addition: \(15 + 4 = 19\)

The value of the expression is 19.

Did you know? In computer programming and science, substitution into complex formulae is one of the most common uses of algebra!

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3. Simplifying Expressions: Collecting Like Terms

Simplifying an expression means making it as short as possible while keeping the same value. The primary way we simplify is by collecting like terms.

What are Like Terms?

Like terms are terms that contain the exact same variables raised to the exact same powers. Only like terms can be added or subtracted together.

Analogy: Imagine sorting fruit. You can add 3 apples and 2 apples to get 5 apples. You cannot add 3 apples and 2 bananas to get 5 apple-bananas!

• Like Terms: \(4x\) and \(-2x\); \(5y^2\) and \(y^2\); \(ab\) and \(7ab\).
• Unlike Terms: \(4x\) and \(4y\); \(x^2\) and \(x\); \(ab\) and \(a^2b\).

The Simplification Process

When collecting, the sign in front of the term belongs to that term.

Example 1: Simplify \(3x + 5y - x + 2y\)

1. Identify the like terms:
   Term 1 (x-terms): \(\color{red}{3x}\) and \(\color{red}{-x}\)
   Term 2 (y-terms): \(\color{blue}{+5y}\) and \(\color{blue}{+2y}\)
2. Combine the x-terms: \(3x - x = 2x\)
3. Combine the y-terms: \(5y + 2y = 7y\)
4. Final simplified expression: \(2x + 7y\)

Example 2: Simplify \(2a^2 + 5a - 3a^2 - a\)

\(2a^2 - 3a^2 = -a^2\)
\(5a - a = 4a\)
Answer: \(-a^2 + 4a\)

Multiplying and Dividing Terms (Basic Index Laws)

When multiplying or dividing terms, you don't need them to be 'like terms'.

1. Multiply the numbers (coefficients).
2. Multiply the letters. If the bases are the same (e.g., \(x\) and \(x\)), add the powers.

The Multiplication Rule: \(x^a \times x^b = x^{a+b}\)

Example: \((4x^2) \times (5x^3)\)
Multiply numbers: \(4 \times 5 = 20\)
Multiply letters (add powers): \(x^{2+3} = x^5\)
Answer: \(20x^5\)

The Division Rule: \(x^a \div x^b = x^{a-b}\)

Example: \(12y^5 \div 4y^2\)
Divide numbers: \(12 \div 4 = 3\)
Divide letters (subtract powers): \(y^{5-2} = y^3\)
Answer: \(3y^3\)

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4. The Power of Brackets: Expanding and Factorising

Brackets are used to group terms together. We often need to remove them (expand) or introduce them (factorise).

4.1 Expanding Single Brackets

Expanding means multiplying the term outside the bracket by every single term inside the bracket. This is often called the distributive law.

Analogy: If you have a tray of sweets (the term outside) and you are distributing them to everyone inside the room (the terms inside the bracket), everyone must get one!

Example 1: Expand \(3(x + 5)\)

Multiply \(3\) by \(x\): \(3 \times x = 3x\)
Multiply \(3\) by \(5\): \(3 \times 5 = 15\)
Answer: \(3x + 15\)

Example 2 (Watch the signs!): Expand \(-4(2y - 3)\)

1. \(-4 \times 2y = -8y\)
2. \(-4 \times -3 = +12\) (A negative times a negative is a positive!)

Answer: \(-8y + 12\)

Common Mistake: Forgetting to multiply the outside term by the second term inside the bracket, or getting the signs wrong when multiplying negatives.

4.2 Factorising into a Single Bracket

Factorising is the reverse of expanding. We are looking for the Highest Common Factor (HCF) shared by all terms and putting it outside the bracket.

How to Factorise:

1. Find the HCF of the numbers (coefficients).
2. Find the common variables. (Choose the letter with the lowest power).
3. Write the HCF outside the bracket.
4. Divide each original term by the HCF to find the terms that go inside the bracket.

Example 1: Factorise \(6x + 9\)

• HCF of 6 and 9 is 3. (No common variables)
• Divide \(6x\) by \(3\): \(2x\)
• Divide \(9\) by \(3\): \(3\)
• Answer: \(3(2x + 3)\)

Example 2: Factorise \(10ab - 15ac\)

• HCF of 10 and 15 is 5.
• Common variable is a. (b and c are not common).
• HCF is 5a.
• Divide \(10ab\) by \(5a\): \(2b\)
• Divide \(-15ac\) by \(5a\): \(-3c\)
• Answer: \(5a(2b - 3c)\)

If you are unsure if you factorised correctly, simply EXPAND your answer. If you get back to the original expression, you are correct!

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Chapter Summary and Review

You have now built a strong foundation in handling algebraic expressions! Remember these key concepts:

1. Substitution requires careful application of BIDMAS/BODMAS, especially when dealing with negative numbers and squares.
2. To simplify by adding/subtracting, you must have like terms.
3. When multiplying terms, add the powers of the same variables.
4. Expanding means multiplying the term outside the bracket by everything inside.
5. Factorising means finding the Highest Common Factor (HCF) and taking it outside the bracket.

Keep practising these skills—they are the backbone of your entire IGCSE Mathematics course! Great work!