# Expressions and formulae

> Edexcel International GCSE Mathematics A · 4MA1 2016 Spec
> Source: https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-expressions-and-formulae/

This guide covers all core skills for expressions and formulae in Edexcel IGCSE Maths A (4MA1), including notation rules, substitution, deriving formulae, and rearranging formula subjects for both tiers.

**Prerequisites:** [Basic arithmetic with positive/negative numbers](https://www.owlsprep.com/study/edexcel-igcse-math-a-s1-number-operations/); [Basic algebraic term understanding](https://www.owlsprep.com/study/edexcel-igcse-math-a-s2-algebra-basics/)

## Learning objectives

- Understand variables and correct algebraic notation conventions
- Substitute positive/negative integers, decimals, and fractions into expressions and formulae
- Derive formulae from word or diagrammatic real-life contexts
- Change the subject of a formula (Foundation: subject appears once; Higher: subject appears twice or has powers/roots)

## Algebraic Notation & Variables (Foundation + Higher)

A variable is a letter used to represent an unknown number or a quantity that can change. Following standard notation rules avoids confusion in algebraic work and ensures you do not lose marks for poor presentation.

**Algebraic expression** — A combination of variables, numbers, and operations (e.g. +, -, ×, ÷) with no equals sign.

*Example:* $3x + 2y - 7$ is an expression

> **info**
>
> Correct notational conventions: omit × signs (write 3a not 3×a), write numbers before letters (5x not x5), use fractions instead of ÷ signs ($\frac{a}{2}$ not a÷2), and use brackets to clarify order of operations.

**Worked example:** Rewrite the following using correct algebraic notation: 7 multiplied by b, subtract 2 divided by c.

1. Omit the × sign between 7 and b: 7b
2. Replace 'divided by' with a fraction bar: $\frac{2}{c}$
3. Combine to get the final expression: $7b - \frac{2}{c}$

*Calculator:* allowed

## Substitution into Expressions & Formulae (Foundation + Higher)

Substitution means replacing variables in an expression or formula with given numerical values, then calculating the result. You will substitute positive/negative integers, decimals, and fractions in exam questions.

**Formula** — A rule that relates two or more variables, written with an equals sign.

*Example:* The formula for the area of a circle is $A = \pi r^2$

**Worked example:** Evaluate $2x - 3y$ when $x = 4$ and $y = -5$.

1. Replace x with 4 and y with -5, adding brackets around negative values to avoid sign errors: $2(4) - 3(-5)$
2. Calculate each term separately: $2×4 = 8$, $-3×-5 = +15$
3. Add the terms together: $8 + 15 = 23$

> **Exam tip:** Always put negative numbers in brackets when substituting to avoid mistakes with sign operations, e.g. 3(-2) not 3-2.

*Calculator:* allowed

## Deriving Formulae from Context (Foundation + Higher)

You will often be given a scenario in words or diagram form, and asked to write an expression or formula to represent the relationship between quantities.

**Worked example:** A taxi charges a £3 fixed fee plus £1.20 per kilometre travelled. Derive a formula for the total cost C (in £) for a journey of k kilometres.

1. Identify the fixed cost: £3, which is added regardless of distance.
2. Identify the variable cost: £1.20 per km, so for k km this is $1.20k$
3. Add fixed and variable costs to get the formula: $C = 3 + 1.20k$

> **tip**
>
> Define any variables you use if they are not given in the question, to make your formula clear to examiners.

*Calculator:* allowed

## Rearranging Formulae: Subject Appears Once (Foundation + Higher)

The subject of a formula is the single variable that appears on its own on one side of the equals sign, usually the left. For Foundation tier, you only rearrange formulae where the subject appears once. Use inverse operations applied to both sides of the equals sign to isolate the subject.

**Subject of a formula** — The variable that is expressed in terms of other variables in a formula, written alone on one side of the equals sign.

*Example:* In $v = u + at$, the subject is v.

**Worked example:** Make t the subject of the formula $v = u + at$.

1. Identify the operations applied to t: it is multiplied by a, then u is added to the result. Invert these operations in reverse order.
2. First subtract u from both sides: $v - u = at$
3. Divide both sides by a: $t = \frac{v - u}{a}$

> **Exam tip:** Check your rearranged formula by substituting a simple set of values into the original and your rearranged version to confirm they give the same result.

*Calculator:* allowed

## Rearranging Formulae: Higher Tier Only

For Higher tier, you will rearrange formulae where the subject appears more than once, or where the subject is raised to a power or under a root. When the subject appears twice, collect all terms with the subject on one side, then factorise the subject out before isolating it.

**Worked example:** Make a the subject of $3a + 5 = \frac{4 - a}{r}$.

1. Multiply both sides by r to eliminate the fraction: $r(3a + 5) = 4 - a$
2. Expand the left hand side: $3ar + 5r = 4 - a$
3. Collect all terms with a on the left, others on the right: $3ar + a = 4 - 5r$
4. Factorise a out of the left hand side: $a(3r + 1) = 4 - 5r$
5. Divide both sides by $3r + 1$: $a = \frac{4 - 5r}{3r + 1}$

**Worked example:** Make l the subject of $T = 2\pi\sqrt{\frac{l}{g}}$.

1. Divide both sides by $2\pi$: $\frac{T}{2\pi} = \sqrt{\frac{l}{g}}$
2. Square both sides to eliminate the square root: $\left(\frac{T}{2\pi}\right)^2 = \frac{l}{g}$
3. Multiply both sides by g: $l = \frac{gT^2}{4\pi^2}$

> **warning**
>
> When squaring both sides of a formula, remember to square all terms on both sides, not just the part under the root.

*Calculator:* allowed

## Common pitfalls

- **Wrong:** Writing $x5$ instead of $5x$ or using × signs unnecessarily.
  - Why it fails: Examiners penalise non-standard notation, and it can lead to confusion with multiplication of terms.
  - Correct: Always write numerical coefficients before variables, omit × signs between numbers and letters or between letters.
- **Wrong:** Substituting negative numbers without brackets, e.g. writing $3 - 2$ instead of $3(-2)$ when substituting $x=-2$ into $3x$.
  - Why it fails: This leads to sign errors, e.g. 3-2=1 instead of the correct -6.
  - Correct: Always enclose negative values in brackets when substituting them into expressions.
- **Wrong:** Forgetting to apply inverse operations to both sides of the formula when rearranging, e.g. subtracting u only from the left side of $v = u + at$.
  - Why it fails: This breaks the equality of the formula, leading to an incorrect rearrangement.
  - Correct: Every operation you apply to one side of the equals sign must be applied to the other side to preserve equality.
- **Wrong:** (Higher tier) Trying to isolate a subject that appears twice before collecting and factorising, e.g. dividing by 3 early in $3a + 5 = \frac{4 - a}{r}$.
  - Why it fails: This leaves the subject on both sides, making it impossible to isolate correctly.
  - Correct: Collect all terms containing the subject on one side first, then factorise the subject out of the combined terms.
- **Wrong:** Squaring only the root term when rearranging formulae with roots, e.g. writing $T^2 = 2\pi \frac{l}{g}$ instead of squaring the $2\pi$ term too.
  - Why it fails: This incorrectly modifies the formula, leading to an invalid rearranged version.
  - Correct: When squaring or raising both sides to a power, apply the power to every term on both sides of the equals sign.

## Cheatsheet

| Skill | Foundation Tier Rule | Higher Tier Addition |
| --- | --- | --- |
| Notation | Use $ab$ instead of $a×b$, $\frac{a}{b}$ instead of $a÷b$, coefficients before letters | Same as Foundation, plus correct bracket use for factorisation |
| Substitution | Substitute positive/negative integers, decimals, fractions, use brackets for negatives | Same as Foundation, plus substitute into formulae with roots/powers |
| Derive formulae | Convert word/diagram contexts to algebraic formulae, define variables | Same as Foundation, plus derive formulae with multiple variable terms |
| Rearrange subject | Subject appears once only, use inverse operations on both sides | Subject appears twice: collect terms, factorise, isolate; for powers/roots: use inverse power operations |
| Exactness | Keep $\pi$, fractions, surds unless decimal is requested | Same as Foundation |

## What's next

Mastering expressions and formulae is a foundational algebra skill that underpins almost all other topics in your Edexcel IGCSE Maths A exam, from solving linear equations to calculating geometric measures and interpreting statistical models. Once you are confident with the content in this guide, you can move on to solving linear equations, the next core sub-topic in the algebra unit, where you will apply your rearrangement skills to find unknown values in single-variable equations. For Higher tier students, you will also use these rearrangement skills when working with quadratic formulae and kinematics problems in later units. Be sure to practice past paper questions targeted at your tier to reinforce your understanding, paying close attention to notation and sign errors that are common in this topic.

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