All of GCSE Algebra in One Guide: From Number Lines to Simultaneous Equations

Algebra is the spine of GCSE Maths. It runs through Number (percentage multipliers), Geometry (angle equations, Pythagoras) and Statistics (interpreting the gradient of a scatter graph). Get algebra solid and everything else becomes easier. Leave it shaky and you will keep losing marks in topics that do not even look like algebra.

Here is the whole of GCSE Algebra, in the order it actually builds.

Start with the number line

Before any letters appear, you need the number line — a road stretching infinitely in both directions, positive one way, negative the other. Every algebraic answer you ever produce sits somewhere on it.

This matters more than it sounds. Students who are shaky with negatives lose marks all over the paper: substituting a negative into a formula, solving −3x = 12, reading an inequality range. Fix negatives first.

Tip: If you are unsure about negatives, write the number line on your rough paper at the start of the exam. Counting along a drawn line beats guessing every time.

Variables, coefficients and terms

A variable is a letter that stores a value. In the expression 2x:

• 2 is the coefficient — how many lots of x you have
• x is the variable
• Together, 2x is one term

The single most common beginner error is confusing 2x with x². They are not the same:

Expression	Means	Example when x = 5
2x	x + x	10
x²	x × x	25
2x²	2 × (x × x)	50
(2x)²	4 × x × x	100

Why “squared”? Picture a square with side length x. All sides are equal, and the area of a rectangle is length × width — so the area is x × x = x². The word is geometric, not arbitrary. That link between algebra and shape shows up again in trigonometry and area formulas.

Order of operations: BIDMAS

Brackets, Indices, Division, Multiplication, Addition, Subtraction. BODMAS, BIDMAS, PEMDAS — different acronyms, identical rule.

Brackets come first, but what “first” means depends on what is inside them:

• x(5 + 6) — the inside is all numbers, so add first: x × 11 = 11x
• x(5 + y) — you cannot add 5 to y, so expand instead: 5x + xy

That second case is the one students freeze on. You are not stuck. When terms inside a bracket cannot be combined, expansion is the next step.

Expanding brackets with FOIL

For double brackets, multiply everything outside by everything inside. FOIL keeps you honest:

(x + 1)(x + 2)

1. First: x × x = x²
2. Outer: x × 2 = 2x
3. Inner: 1 × x = x
4. Last: 1 × 2 = 2

That gives x² + 2x + x + 2, which simplifies to x² + 3x + 2.

Remember: write all four terms before collecting like terms. Collecting in your head is where the sign errors live — and where method marks quietly disappear.

Simplifying means grouping terms with the same variable and the same power. 2x and x are like terms. 2x and x² are not.

Once you can expand double brackets confidently, running the process backwards gives you factorising — which is the gateway to solving quadratic equations, a guaranteed Higher tier topic.

The index laws

Powers follow three rules you must know cold:

Rule	In symbols	Example
Multiplying	aᵐ × aⁿ = aᵐ⁺ⁿ	x³ × x⁴ = x⁷
Dividing	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	x⁷ ÷ x² = x⁵
Power of a power	(aᵐ)ⁿ = aᵐⁿ	(x²)³ = x⁶
Zero power	a⁰ = 1	7⁰ = 1

Deal with coefficients separately from the powers. For 3x² × 4x:

• Coefficients: 3 × 4 = 12
• Powers: x² × x¹ = x³
• Answer: 12x³

And a useful shortcut: if two sides of an equation have the same base and the same coefficient, the powers must be equal. So 2ˣ = 2⁵ tells you immediately that x = 5. That trick turns several Higher tier index questions into one-liners.

Inequalities

Inequalities describe a range rather than a single answer:

Symbol	Meaning
<	less than
>	greater than
≤	less than or equal to
≥	greater than or equal to

So −6 < x ≤ 2 means x is anything between −6 and 2, not including −6 but including 2. On a number line, use an open circle for strict inequalities and a filled circle for “or equal to”.

Solve them exactly like equations, with one critical exception:

Did you know? If you multiply or divide both sides by a negative number, the inequality sign flips. Divide −2x > 6 by −2 and you get x < −3, not x > −3. The same flip happens when you take reciprocals of both sides.

This single rule accounts for a startling share of lost marks. Our step-by-step guide to solving linear inequalities works through the sign flip with worked examples.

Straight-line graphs: y = mx + c

Every straight line on a graph obeys y = mx + c:

• m is the gradient — the steepness
• c is the y-intercept — where the line crosses the y-axis

The equation is answering a question: for this x value, what is y? The gradient scales x; the intercept shifts the whole line up or down.

Things worth memorising:

• Parallel lines have the same gradient
• Perpendicular gradients multiply to give −1 (Higher tier)
• Gradient = change in y ÷ change in x
• Set y = 0 to find the x-intercept; set x = 0 to find the y-intercept

Simultaneous equations

A pair of equations with two unknowns, solved together. Geometrically, you are finding where two lines cross. Two methods:

Elimination — make the coefficients of one variable match, then add or subtract to remove it. If you have 3x + 2y = 7 and 4x + 5y = 9, multiply the first by 5 and the second by 2 to match the y coefficients, subtract, and you are left with a single equation in x. Solve it, then substitute back into the simpler original equation to find y.

Substitution — rearrange one equation to get a variable on its own, then substitute that expression into the other. This is the method for a linear-and-quadratic pair at Higher tier.

Tip: always substitute your answer back into the simplest original equation, and check it satisfies both. A 30-second check protects four or five marks.

What is not on the GCSE spec

The source video also covers logarithms, natural logs and sigma notation. These are genuinely useful — but they are A-Level topics, not GCSE. Do not spend revision time on them now.

What is worth taking from them: logs are built on the index laws above, and sigma notation is just shorthand for adding up a sequence. GCSE covers sequences (arithmetic, geometric, quadratic and the nth term), so the underlying idea is already on your paper — just written differently.

If you are chasing top grades, the grade 9 techniques guide covers where the hardest algebra marks actually sit.

Practise algebra until it is automatic

Reading about FOIL takes two minutes. Being able to expand double brackets under exam pressure, at speed, without a sign error, takes repetition. That is the entire difference between understanding a topic and scoring on it.

GCSE Algebra covers exactly what is above and more — expressions, equations, inequalities, sequences and graphs, across both Foundation and Higher tier. Every question has a worked solution, so when you get one wrong you see precisely where the method broke down rather than just a red cross.

If algebra is one weak area among several, GCSE Maths is the complete bundle — Number, Algebra, Geometry and Statistics together, 2900+ questions across the four subject apps. That matters more than it sounds: real exam questions mix topics, asking you to find a length with Pythagoras and then solve the quadratic it produces. Practising across areas in one place trains you for that.

Start with 20 questions a day on expanding and solving. Within a fortnight, the algebra that currently makes you pause will be the part of the paper you finish first.