Equations, formulae and identities: Use of symbols to Linear equations for Edexcel IGCSE Mathematics Specification A: Methods and Worked Examples

Algebra is the language maths uses to describe problems you have not met yet. Once you can read it fluently, questions that look intimidating become straightforward.

The Equations, formulae and identities section of the Pearson Edexcel IGCSE Mathematics Specification A syllabus is where many students feel the subject shift from "doing sums" to "doing maths." Algebra introduces letters where numbers used to be, and that change unsettles people. But think of it this way: when you calculate 3 x 7 + 2, you are following a recipe. When you write 3x + 2, you are writing the recipe itself, so it works for any ingredient. That is all algebra is: a recipe written in general terms.

These edexcel igcse mathematics specification a revision notes cover the first four topics in this section: Use of symbols, Algebraic manipulation, Expressions and formulae, and Linear equations. These are tested in every exam session across both papers of the 4MA1 qualification. If algebra feels like your weakest area, this is the place to start building confidence.

Use of symbols

The igcse specification requires you to understand that letters can represent unknown numbers (in equations) or variables (in expressions and formulae). The conventions are simple but important:

• Multiplication is implied. 3a means 3 x a. ab means a x b. You do not write the multiplication sign.
• Division uses fraction notation. a divided by b is written as a/b, not a divided by b.
• Powers use index notation. a x a = a². a x a x a = a³. The index laws from the number section apply to algebraic expressions too.

The index laws for algebra work the same way as for numbers:

At Higher tier, you also need fractional and negative powers in algebraic contexts. x^-1 = 1/x, and x^1/2 = the square root of x. These come up regularly in questions combining algebra and number work on the edexcel exam.

Algebraic manipulation

This is the heart of the algebra section and one of the most commonly tested areas in the edexcel igcse. You need to be able to:

Collect like terms

Like terms have the same letter(s) raised to the same power. 3x and 5x are like terms; 3x and 5x² are not.

Simplify: 4a + 3b - 2a + 5b = (4a - 2a) + (3b + 5b) = 2a + 8b

Expand brackets

Multiply each term inside the bracket by the term outside:

3(2x - 5) = 6x - 15

For two brackets, use FOIL or the grid method:

Factorise

Factorising is the reverse of expanding. Take out the highest common factor from each term:

6x² + 9x = 3x(2x + 3)

For quadratic expressions of the form x² + bx + c, find two numbers that multiply to c and add to b:

At Higher tier, you will also need to factorise expressions of the form ax² + bx + c where a is not 1. The method involves finding two numbers that multiply to ac and add to b, then splitting the middle term. You will also be expected to complete the square and manipulate algebraic fractions.

Expressions and formulae

An expression is a mathematical phrase (like 3x + 2). A formula is an equation that links variables (like A = pi x r²). An equation states that two things are equal and usually has one or more solutions. An identity is true for all values of the variable (like 2(x + 1) = 2x + 2). The igcse specification expects you to know the difference.

Substitution

Replace the letters with given values and calculate. This sounds simple, but sign errors are common:

The trap is the double negative: subtracting a negative number means adding. If you substitute b = -2 and forget the negative, the whole answer goes wrong.

Changing the subject of a formula

Rearranging a formula means isolating a particular variable. The key principle: whatever you do to one side, you must do to the other.

At Higher tier, the subject may appear twice in the formula, which requires collecting the target variable on one side and factorising it out.

Linear equations

A linear equation is one where the unknown appears to the power of 1 only. Solving it means finding the value of the unknown that makes the equation true. The edexcel igcse mathematics specification a equations, formulae and identities: use of symbols to linear equations section is fundamental to everything else in algebra.

Equations with the unknown on one side

Equations with the unknown on both sides

Equations with fractions

Setting up equations from context

Exam questions often describe a situation in words and expect you to translate it into an equation. Think of this as "turning English into algebra." The trick is identifying what the unknown represents and what the equation states.

Common mistakes across these algebra topics

• Sign errors when expanding: -2(x - 3) = -2x + 6, not -2x - 6. The negative outside the bracket changes the sign of every term inside.
• Incorrect factorisation check: Always expand your answer to verify. If you factorise x² - x - 12 as (x - 3)(x + 4), expand it: x² + 4x - 3x - 12 = x² + x - 12. That does not match. The correct factorisation is (x - 4)(x + 3).
• Substitution with negatives: (-2)² = 4 (positive), but -2² = -4 (negative, because only the 2 is squared). The brackets make all the difference.
• Forgetting to check the solution: Substituting your answer back into the original equation takes 10 seconds and catches arithmetic errors that would cost marks.

Self-check questions

1. Simplify: 5x² + 3x - 2x² + x. (Answer: 3x² + 4x)
2. Expand and simplify: (x - 5)(x + 8). (Answer: x² + 3x - 40)
3. Factorise: x² - 9x + 20. (Answer: (x - 4)(x - 5))
4. If p = -3, find 2p² - 4p + 1. (Answer: 2(9) - 4(-3) + 1 = 18 + 12 + 1 = 31)
5. Solve: 8x - 5 = 3x + 20. (Answer: x = 5)
6. A number is tripled and then 7 is added. The result is 28. Find the number. (Answer: 3n + 7 = 28, n = 7)

These equations, formulae and identities: use of symbols to linear equations edexcel igcse concepts appear across every section of the edexcel igcse mathematics specification a explained specification. Being fluent in expanding, factorising, substituting, and solving linear equations is not optional: it is the prerequisite for simultaneous equations, quadratics, trigonometry, and most multi-step problems. For more igcse 4ma1 equations, formulae and identities: use of symbols to linear equations practice, work through edexcel igcse mathematics specification a practice questions on the Green Bridge CBT platform, where you can filter by topic and track which edexcel igcse mathematics specification a notes areas need more attention.