GCSE Algebra Worksheets: Practice & Worked Answers

GCSE revision • expressions, equations and graphs

Choose an algebra skill, try a short question, then use the topic library for more practice. This page combines 12 original questions with worked answers, four worked examples and the existing 67 resource links across 31 topics.

The linked resources include 66 PDF entries and one PNG image, with one PDF listed under two topics. They are hosted by M4ths; their availability and answer-key coverage are not guaranteed. Our own questions and answers below are available without opening a download.

Choose the right algebra practice

These routes refer to GCSE (9–1) Mathematics in England. Higher-tier students also need the Foundation skills. The route is a guide to the skill, not a verified tier label for every external worksheet; a file may mix easier and harder questions. Check your exam board's current specification and your teacher's plan.

Build the essentials

Start with like terms, substitution, single brackets, common factors, linear equations and linear sequences. These underpin both tiers.

Connect the skills

Practise expanding two brackets, factorising simple quadratics, two linear simultaneous equations and straight-line graphs. These are not automatically Higher-only topics.

Extend at Higher tier

Move on to completing the square, algebraic fractions, the quadratic formula, linear–quadratic simultaneous equations, quadratic nth terms, inverse/composite functions and graph transformations.

Need broader practice? Explore our algebra worksheet library. If signs are causing mistakes, review negative numbers before tackling substitution and rearrangement.

A useful 20-minute revision routine

Choose one target. For example, “solve equations with x on both sides”, rather than “revise algebra”.
Try three questions unaided. Keep every algebraic line so you can locate the first error.
Check and repair. Substitute solutions into the original equation; expand a factorised answer to check it.
Retry a similar question later. Explain why each step is valid. Correcting a method matters more than copying the answer.

Four worked GCSE algebra examples

1. Expand two brackets and collect like terms

Expand (x + 2)(x + 3). Every term in the first bracket multiplies every term in the second:

(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Area model for (x plus 2)(x plus 3): four regions x squared, 3x, 2x and 6, giving x squared plus 5x plus 6. — Schematic, not to scale. The two middle regions combine: 3x + 2x = 5x. Positive lengths illustrate the model; the algebraic identity holds for every real x.

Common mistake: x² and 5x are unlike terms, so they cannot be combined into 6x². As a quick check, x = 1 gives (1 + 2)(1 + 3) = 12 and 1 + 5 + 6 = 12. One numerical check can catch a mistake, but it does not prove an identity for every x.

2. Solve an equation with x on both sides

Solve 5x − 7 = 2x + 8. Do the same operation to both sides:

Subtract 2x: 3x − 7 = 8
Add 7: 3x = 15
Divide by 3: x = 5

Check: left side 5 × 5 − 7 = 18; right side 2 × 5 + 8 = 18. A solution makes the original equation true. “Move it and change the sign” is a shortcut for applying an inverse operation to both sides, not a separate rule.

3. Find a line's gradient and intercept

The line passes through (1, 1) and (3, 5). The change in y is 5 − 1 = 4, and the change in x is 3 − 1 = 2. Its gradient is m = 4/2 = 2.

Substitute (1, 1) into y = mx + c: 1 = 2 × 1 + c, so c = −1. The equation is y = 2x − 1.

Graph of y equals 2x minus 1 through (1,1) and (3,5), showing a rise of 4, run of 2, gradient 2 and y-intercept minus 1. — Use the same point order in both differences. The y-intercept is the value of y when x = 0.

4. Simplify an algebraic fraction without losing a restriction

For x ≠ 3, factorise the numerator:

(x² − 9)/(x − 3) = [(x − 3)(x + 3)]/(x − 3) = x + 3, with x ≠ 3

We cancel a common factor, not separate terms. The restriction remains because the original denominator is zero at x = 3. In particular, solving (x² − 9)/(x − 3) = 6 appears to give x = 3, but that value is excluded: the original equation has no solution.

12 original practice questions

Work on paper before opening the answers. Questions 1–10 build and connect skills found across the two tiers; questions 11–12 use Higher-tier methods. This is a short diagnostic set, not a complete exam or a grade predictor.

Simplify 7a + 3b − 2a + 5b.
Expand −3(2x − 5).
Factorise fully: 12x² − 18x.
Factorise x² − x − 12.
Solve 4(2x − 1) = 20.
Evaluate 2x² − 3x when x = −2.
Solve 3 − 2x ≤ 11.
Find the nth term of 6, 10, 14, 18, …, with the first term at n = 1. Then find the 20th term.
Solve simultaneously: 2x + y = 11 and x − y = 1.
Find the gradient and equation of the straight line through (−1, −3) and (2, 3).
Write x² + 6x + 5 in completed-square form. State its minimum value and the x-value at which it occurs.
Simplify (2x² − 8)/(x − 2), and state the excluded value.

Worked answers and checks

Show all 12 worked answers

5a + 8b. Combine a-terms and b-terms separately: (7 − 2)a + (3 + 5)b.
−6x + 15. Multiply both terms by −3. The product (−3)(−5) is positive.
6x(2x − 3). The greatest common factor of both terms is 6x. Expanding gives 12x² − 18x again.
(x − 4)(x + 3). The two numbers multiply to −12 and add to −1. Check: x² + 3x − 4x − 12 = x² − x − 12.
x = 3. Divide by 4: 2x − 1 = 5. Add 1, then divide by 2. Check: 4(6 − 1) = 20.
14. Write brackets: 2(−2)² − 3(−2) = 2 × 4 + 6 = 14. Squaring −2 gives 4.
x ≥ −4. Subtract 3: −2x ≤ 8. Dividing by −2 reverses the inequality. Check the boundary x = −4: both sides are 11. Values larger than −4 also work.
4n + 2; the 20th term is 82. The constant difference is 4. Since 4 × 1 + 2 = 6, the adjustment is +2. Then 4 × 20 + 2 = 82.
x = 4, y = 3. Add the equations to eliminate y: 3x = 12. Substitute into x − y = 1. Check 2 × 4 + 3 = 11 and 4 − 3 = 1.
Gradient 2; y = 2x − 1. m = [3 − (−3)]/[2 − (−1)] = 6/3 = 2. Using (2, 3), 3 = 4 + c gives c = −1.
(x + 3)² − 4; minimum −4 at x = −3. Expanding the square gives x² + 6x + 9; subtracting 4 leaves +5. A real square is at least zero.
2(x + 2), with x ≠ 2. Factorise 2x² − 8 = 2(x − 2)(x + 2), then cancel the common factor x − 2. The original expression remains undefined at x = 2.

If you missed several questions, return to one matching topic below. For notation such as ≠ and ≤, see our maths symbols reference.

Questions about the worksheet library

Do the downloaded worksheets include answers?

Answer-key coverage has not been verified. All 66 unique external file URLs returned an access-control response during our 3 October 2026 automated check, so we could not inspect their contents. This does not establish that they are unavailable to every visitor. The 12 original questions on this page have worked answers immediately above.

Are these official GCSE exam papers?

No. This is a topic-practice directory linking to third-party teaching resources. A worksheet title is not evidence that its questions are official exam-board questions, current-specification material or suitable for a particular tier.

How should I print or use a worksheet?

Open the file on its publisher's site, check the content and any usage notice, then use your PDF viewer's print controls if printing is permitted. Check the first page before printing a long file. These resources remain on their original host; HeLovesMath does not claim ownership of them.

Why do some topics have no download?

Four headings in the existing directory have no separate file: factorising expressions, number patterns, polynomial/exponential graphs and trigonometric graphs. They are retained so existing topic links still work. Use the original examples above or a related topic where appropriate; an empty heading is not a download.

Sources & References

Topic worksheet downloads

Source credit: the following 67 entries link to files hosted by M4ths (m4ths.com). Individual authors retain their rights. Titles are the existing directory labels; content, answer keys and exact tier alignment have not been independently verified.

Access note, checked 3 October 2026: our requests to all 66 unique file URLs received HTTP 406 “Not Acceptable” HTML responses rather than the files. Links are preserved for readers who can access them. If a file does not open, use the on-page practice above or ask your teacher for an accessible alternative. No worksheet or answer-key file is rehosted here.

Search the topic library

Showing 67 resource links across 31 topics.

Collecting Like Terms (Simplifying Expressions)

Combine terms with the same variable part. Products and quotients use multiplication, division and index rules rather than the rule for adding like terms.

Collecting Like Terms Worksheet 1

PDF Worksheet • hosted by M4ths

Choose the right algebra practice

Build the essentials

Connect the skills

Extend at Higher tier

A useful 20-minute revision routine

Four worked GCSE algebra examples

1. Expand two brackets and collect like terms

2. Solve an equation with x on both sides

3. Find a line's gradient and intercept

4. Simplify an algebraic fraction without losing a restriction

12 original practice questions

Worked answers and checks

Questions about the worksheet library

Sources & References

Topic worksheet downloads

Collecting Like Terms (Simplifying Expressions)

Collecting Like Terms Worksheet 1

Collecting Like Terms Worksheet 2

The Rules of Indices

Rules of Indices Worksheet 1

Rules of Indices Worksheet 2

Rules of Indices Worksheet 3

Rules of Indices Worksheet 4

Indices Worksheet 5

Identities and Indices Worksheet

Indices Worksheet 6

Rules of Indices Harder

Expanding Brackets (Multiplying Out Brackets)

Expanding Brackets Worksheet 1

Expanding Brackets Worksheet 2

Factoring (Factorising Algebraic Expressions)

Algebraic Fractions

Algebraic Fractions Worksheet 1

Algebraic Fractions Worksheet 2

Forming Algebraic Expressions

Forming Algebraic Expressions Worksheet

Formulae (Substitution)

Substitution Worksheet

Identities

Identities Worksheet 1

Identities Worksheet 2

Algebraic Identities Questions

Identities Worksheet 3

Identities Worksheet 4

Rearranging Equations (Changing the Subject of a Formula)

Rearranging Equations Worksheet 1

Making x the Subject Worksheet 2

Making x the Subject Worksheet 3

Solving Equations (Linear Equations with One Unknown)

Solving Linear Equations Worksheet 1

Solving Linear Equations Worksheet 2

Solving Linear Equations Worksheet 3

Setting Up and Solving Linear Equations

Setting Up and Solving Linear Equations Worksheet

Quadratic Equations and Expressions

Quadratic Expressions and Equations Worksheet 1

Quadratic Worksheet 2

Quadratic Worksheet 3

Quadratic Worksheet 4

Quadratic Worksheet 5

Quadratics Worksheet 6

Quadratic Expressions (Completing the Square)

Completing the Square Worksheet 1

Completing the Square Worksheet 2

Simultaneous Equations (Linear)

Simultaneous Equations Linear Worksheet 1

Basic Simultaneous Equations

Simultaneous Equations Worksheet 2

Simultaneous Equations Image

Simultaneous Equations (Linear and Non-Linear)

Linear and Non-Linear Simultaneous Equations Worksheet

Solving Equations Graphically

Solving Equations Graphically Worksheet

Linear Inequalities

Linear Inequalities Worksheet

Quadratic Inequalities

Quadratic Inequalities Worksheet

Shading Inequalities (Two Variables)

Shading Inequalities Worksheet

Functions