Algebra Worksheets with Answers (Free PDFs)

Choose your algebra worksheet topic

Each link below opens a topic section on this page. Within that section, choose the worksheet PDF to print, the answer file to mark your work, or the Word version if you need to adapt a task. These are third-party resources; their creator is credited with the library.

Choose a starting level, then build up

A topic name alone does not tell you how demanding a worksheet will be. “Equations” can mean x + 3 = 8 or a problem with brackets, fractions and unknowns on both sides. Look at a few questions before choosing. These four routes are practical study suggestions, not official exam grades or a mapping of every file to a specification.

1. Build the foundations

   Be comfortable with negative numbers, multiplication, division and the order of operations. Then practise algebraic notation, collecting like terms, substitution and one-step equations. Begin with simplifying by adding and subtracting, substitution into expressions or one-step equations. If arithmetic is causing the mistakes, pause for the negative numbers guide or the number worksheet library.

2. Connect the core skills

   Move to single brackets, common factors, two-step equations, simple rearranging, linear sequences and straight-line graphs. Aim to explain why each step is valid. For example, factorising reverses expanding, while solving an equation preserves the equality. Use the expanding and factorising, sequences and linear graphs groups.

3. Combine methods

   Try equations with unknowns on both sides, simultaneous equations, quadratic factorising and more involved inequalities. You will need to choose a method rather than repeat one procedure throughout. Once a focused sheet feels secure, pick a revision grid or a mixed task. Return to a single skill if you can start a question but repeatedly lose signs or omit a solution.

4. Choose more demanding topics for your course

   Algebraic fractions, completing the square, composite and inverse functions, and non-linear simultaneous equations need stronger manipulation skills. Check your own course before selecting these. The calculus group contains differentiation and is an extension beyond standard GCSE Mathematics. It may support a further or advanced mathematics course; use your teacher’s topic list and the Further Mathematics resource page to plan relevant practice.

KS3 learners will often begin with the first two routes, but a year-group label is not a prerequisite check. GCSE students should select across the routes according to their specification; some topics in the fourth route are GCSE Higher content. AQA’s algebra specification and England’s national curriculum are listed in Sources & References; use your own board’s current specification for exact coverage.

Worked examples: understand the method before practising

1. Expand carefully, then collect terms

Simplify 3(2x − 5) + 4x.

1. Multiply every term inside the bracket by 3: 3 × 2x = 6x and 3 × (−5) = −15.
2. The expression becomes 6x − 15 + 4x.
3. Collect the x terms: 10x − 15. The constant −15 is not a like term with 10x.

Quick error check: let x = 2. The original expression gives 3(4 − 5) + 8 = 5, and the simplified expression gives 20 − 15 = 5. A single substitution can catch an error; it does not prove two expressions are identical for every x. The expansion and collection above establish the equivalence.

Practise this with expanding single brackets, then combine it with a simplifying expressions practice grid.

2. Solve an equation with brackets and unknowns on both sides

Solve 3(2x − 5) = 4x + 9. An equation says that both sides have the same value. Applying the same reversible operation to both sides keeps the same solutions.

1. Expand: 6x − 15 = 4x + 9
2. Subtract 4x from both sides: 2x − 15 = 9
3. Add 15 to both sides: 2x = 24
4. Divide both sides by 2: x = 12

Check in the original equation: the left side is 3(24 − 5) = 57; the right side is 48 + 9 = 57. Both match, so x = 12 is a solution. Start a similar task in the equations and inequalities library.

Avoid relying on “move it across and change the sign.” Writing the operation helps explain the sign change and makes it easier to spot a missing term. Also remember that an inequality reverses direction when you multiply or divide both sides by a negative number: −2x > 6 gives x < −3.

3. Simplify a fraction without losing its restriction

Simplify (x² − 9)/(x − 3).

1. The denominator cannot be zero, so record x ≠ 3.
2. Factorise the numerator: x² − 9 = (x − 3)(x + 3).
3. Cancel the common non-zero factor x − 3. The result is x + 3, for x ≠ 3.

You can cancel factors that multiply a whole numerator and denominator. You cannot simply cancel matching terms separated by addition or subtraction. The restriction remains because the original fraction is undefined at x = 3, even though x + 3 can be evaluated there. Use the algebraic fractions group once ordinary fraction arithmetic and factorising are secure.

A short algebra diagnostic, with worked answers

Try these original questions without the answers first. Write a line of working for each one. The questions sample different skills rather than predicting an exam grade; use individual mistakes to choose your next worksheet.

1. Simplify 4a + 3b − a + 2b.
2. Evaluate 2x² − 3y when x = −2 and y = 3.
3. Expand −3(2x − 5).
4. Factorise fully 6x² − 9x.
5. Solve 4x − 7 = 13.
6. Solve 5 − 2x > 11.
7. The first four terms of an arithmetic sequence are 7, 11, 15, 19. Find an expression for the nth term, with n = 1 for the first term.
8. Find the equation of the straight line through (0, −3) and (2, 5).
9. Solve x² + x − 12 = 0.
10. Solve the simultaneous equations 2x + y = 11 and x − y = 1.

Choose by error, not just score: if questions 1–4 were difficult, work on manipulation before combining methods. If questions 5–6 were difficult, focus on equality, inverse operations and inequality signs. Questions 7–8 check connections to patterns and graphs; questions 9–10 introduce different solving strategies. One correct answer is a starting signal, not proof that you have mastered a whole topic.

How to use an algebra worksheet effectively

1. Pick one target. Write a specific goal, such as “expand brackets with a negative multiplier” rather than “get better at algebra.” Use a focused sheet before a mixed revision task.
2. Check the file before printing. Read the title, question range and any instructions about calculators. Make sure the Answers link belongs to the same task. A resource labelled “harder” is a relative description within that topic, not an exam grade.
3. Work a small set independently. Try a manageable group of questions with enough working to see your method. Cover the answers while you do so. Use a fill-in-the-blanks activity when you need structure; try a less scaffolded task when you can explain the steps.
4. Mark the method as well as the result. A matching final answer can hide two cancelling mistakes. Compare each line and locate the first incorrect step. Equivalent forms may both be valid, but follow an instruction such as “factorise fully” or “give an exact answer.”
5. Repair, then retest. Label an error briefly: sign, arithmetic, notation, wrong method or missed restriction. Rework the question without copying. On a later session, attempt a different question with the same structure before returning to mixed practice.

For a parent or tutor, ask “Why is that step allowed?” and “How could you check your result?” For a class, a common wrong answer can be a useful discussion prompt. Keep the creator’s attribution when using their resources and follow their terms for editing or sharing files.

When the core methods are secure, use the mixed-topic revision library to practise selecting a method in context. For additional explanations before a worksheet, visit Algebra Resources.

Algebra worksheet questions

Where are the worksheet answers?

Each listing below retains separate links labelled Editable Word, PDF and Answers. Choose Answers for the matching task. An answer file may give final results rather than a fully explained solution; the worked examples above show how to record and check a method.

Are these suitable for KS3, GCSE or A level?

The library spans a wide range. Start with the skill and prerequisites, then check the actual questions against your course. A whole category should not be treated as a single year group or exam tier. Calculus is separately marked as extension material; it is not a requirement of standard GCSE Mathematics.

Should I use a calculator?

Follow the worksheet or teacher’s instructions. A calculator can check numerical substitution, but it does not replace showing algebraic reasoning. When practising basic manipulation, choose numbers you can handle accurately so that the algebra remains the main focus.

What if a download is unavailable?

The files are hosted by Dr Austin Maths, so their locations can change. Use the creator’s algebra index in Sources & References and search for the worksheet title. The library preserves the existing links; a sample availability check is not a guarantee that every file will remain available.

Sources & References

Worksheet creator and file host: Dr Austin Maths. The download library below links to the creator’s files. HeLovesMath has written the study routes, examples and diagnostic on this page; the linked third-party worksheets remain the work of their creator. No third-party worksheet or answer-file content is reproduced here.

• Dr Austin Maths: current algebra resource index. Use this index to find a title if an older direct file link changes.
• Dr Austin Maths: about the resources and their creator.

Curriculum references

These references support the general curriculum context, not an endorsement or an item-by-item mapping of the download library.

• Department for Education: mathematics programmes of study for England.
• AQA GCSE Mathematics 8300: algebra subject content.

Download library: algebra worksheets and answers

The 300 listings below preserve the existing topic library, including its cross-listed resources. Download links open in a new tab. The links retain their original labels: PDF for the worksheet, Answers for its answer file, and Editable Word for the document version. File availability can change. We have not independently checked every third-party answer file, so preview a task and its answers before assigning it.