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Algebra and equations

Use algebraic notation, linear equations, formulae and inequalities to model and solve problems.

12 free questions · Notes · Worked solutions

Practise this topic

What you will practise

  • Simplify expressions by collecting like terms and expanding brackets.
  • Solve linear equations and inequalities.
  • Rearrange formulae and substitute values with units.
  • Form equations from short contextual descriptions.

Revision notes

Expressions

Like terms have the same variable part, so 4x+3x=7x but 4x+3y cannot be combined. Expand by multiplying every term inside a bracket. Factorising reverses expansion by taking out a common factor.

Equations

An equation stays balanced when the same operation is applied to both sides. Undo operations in reverse order to isolate the unknown. Check a solution by substitution into the original equation.

Formulae and inequalities

To change the subject, use inverse operations while keeping the formula balanced. When multiplying or dividing an inequality by a negative number, reverse its direction.

Common checks

Distinguish 3x from x+3. Brackets affect every term inside. A solution to an equation is a value; an inequality solution is usually a range.

A worked example

A taxi fare is £4 booking fee plus £1.80 per kilometre. A journey costs £17.50. How far was the journey?

  1. Let the distance be d km.
  2. Write 4+1.8d=17.5.
  3. Subtract 4: 1.8d=13.5.
  4. Divide by 1.8: d=7.5.

7.5 km

Try it yourself

These are original practice questions, not official past-paper questions. Practice points count correct final answers; working is not automatically assessed.

Choose a question to begin.
Question 1 · Standard1 practice point

Simplify 5a−2b+3a+7b.

Need a hint?

Collect the a terms and b terms separately.

Worked solution
  1. 5a+3a=8a.
  2. −2b+7b=5b.
Question 2 · Standard1 practice point

Expand 3(2x−5).

Need a hint?

Multiply both terms by 3.

Worked solution
  1. 3×2x=6x.
  2. 3×(−5)=−15.
Question 3 · Standard1 practice point

Solve 4x+7=31.

Need a hint?

Subtract 7 before dividing by 4.

Worked solution
  1. 4x=24.
  2. x=6.
Question 4 · Standard1 practice point

Which inequality is equivalent to −3x<12?

Need a hint?

Divide by −3 and reverse the inequality sign.

Worked solution
  1. Divide both sides by −3.
  2. Reverse < to >, giving x>−4.
Question 5 · Foundation1 practice point

For the expression 7p−3+2p+11, find the coefficient of p after simplifying.

Need a hint?

Group variable terms and constants.

Worked solution
  1. 7p+2p=9p.
  2. −3+11=8.
  3. The simplified expression is 9p+8, so the coefficient of p is 9.
Question 6 · Foundation2 practice points

Solve 5x−9=26.

Need a hint?

Undo subtraction before division.

Worked solution
  1. Add 9 to both sides: 5x=35.
  2. Divide by 5: x=7.
Question 7 · Standard2 practice points

Solve 3(2y+1)=27.

Need a hint?

You may divide by 3 first.

Worked solution
  1. Divide both sides by 3: 2y+1=9.
  2. Subtract 1: 2y=8.
  3. Divide by 2: y=4.
Question 8 · Standard2 practice points

An object changes velocity from u=3 m/s to v=15 m/s at a constant acceleration of a=2 m/s². Find the time taken in seconds.

Need a hint?

Use v=u+at and rearrange to find t.

Worked solution
  1. Subtract u: v−u=at.
  2. Divide by a: t=(v−u)/a.
  3. t=(15−3)/2=6 s.
Question 9 · Standard2 practice points

Solve 2(x−4)=3x+1.

Need a hint?

Expand before collecting x terms.

Worked solution
  1. Expand: 2x−8=3x+1.
  2. Subtract 2x: −8=x+1.
  3. Subtract 1: x=−9.
Question 10 · Foundation2 practice points

A number n is multiplied by 4 and then 6 is subtracted. The result is 38. Find n.

Need a hint?

Translate the sentence into an equation.

Worked solution
  1. Write 4n−6=38.
  2. Add 6: 4n=44.
  3. Divide by 4: n=11.
Question 11 · Standard2 practice points

What is the greatest integer x satisfying 2x+5≤17?

Need a hint?

Solve the inequality, then identify its greatest integer value.

Worked solution
  1. Subtract 5: 2x≤12.
  2. Divide by 2: x≤6.
  3. The greatest integer solution is 6.
Question 12 · Stretch3 practice points

The perimeter of a rectangle is 46 cm. Its length is 3 cm more than twice its width. Find its width in cm, to 2 decimal places.

Need a hint?

Use P=2L+2W and L=2W+3.

Worked solution
  1. Let the width be w; the length is 2w+3.
  2. 46=2(2w+3)+2w.
  3. 46=6w+6, so 40=6w and w=20/3≈6.67 cm.

Coverage and sources

Selected IGCSE concepts, with a mix of introductory and more demanding questions. Check your own specification. AI drafted; independent teacher review is pending. Editorial details.

Cambridge IGCSE Mathematics 0580 syllabus

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