IGCSE Mathematics: Algebraic Manipulation Practice Questions
Algebraic manipulation means simplifying expressions, substituting values and rearranging formulae. To change the subject of a formula, apply inverse operations to both sides until the required letter stands alone.
Topic 2.2 of Cambridge IGCSE Mathematics 0580 underpins every other algebra question on the paper. Rearranging formulae is the part that separates grades, particularly when the required letter appears twice. The questions below build up to that case.
What you need to know for Algebraic Manipulation
- SubstitutionReplace each letter with its value, keeping brackets around negative numbers, then follow the order of operations.
- Like termsTerms with exactly the same letters raised to the same powers. 3x and 5x are like terms, but 3x and 3x2 are not and cannot be combined.
- Changing the subjectUse inverse operations on both sides, in the reverse order to how the expression was built, until the required letter is alone on one side.
- Subject appearing twiceCollect every term containing that letter on one side, factorise it out, then divide by the bracket.
- Order of inverse operationsUndo addition and subtraction first, then multiplication and division, then powers and roots last.
- Checking a rearrangementSubstitute simple numbers into both the original and the rearranged formula. If they agree, the rearrangement is almost certainly right.
IGCSE Mathematics Algebraic Manipulation questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Find the value of 2a2 minus 3b when a = 3 and b = minus 2.
Show the worked answer
- Substitute the values, keeping the negative number in brackets: 2 x (3)2 minus 3 x (minus 2).
- Work out the power first: 32 = 9, so the first term is 2 x 9 = 18.
- The second term is 3 x (minus 2) = minus 6, and the formula subtracts it, so minus (minus 6) = plus 6.
- 18 + 6 = 24.
Simplify 5x + 3y minus 2x + 7y.
Show the worked answer
- Group the like terms, keeping the sign in front of each: 5x minus 2x, and 3y plus 7y.
- 5x minus 2x = 3x.
- 3y plus 7y = 10y.
- The x and y terms cannot be combined with each other, so the simplified expression is 3x + 10y.
Make a the subject of the formula v = u + at.
Show the worked answer
- The term containing a is at, which is added to u. Undo the addition first.
- Subtract u from both sides: v minus u = at.
- a is now multiplied by t, so divide both sides by t.
- a = (v minus u) divided by t. Check with u = 2, a = 3, t = 4: v = 14, and (14 minus 2) divided by 4 = 3.
Make x the subject of y = (3x + 2) divided by (x minus 1).
Show the worked answer
- Multiply both sides by (x minus 1) to clear the fraction: y(x minus 1) = 3x + 2.
- Expand the left hand side: xy minus y = 3x + 2.
- Collect every term containing x on one side and everything else on the other: xy minus 3x = y + 2.
- Factorise x out of the left hand side: x(y minus 3) = y + 2. Divide by the bracket: x = (y + 2) divided by (y minus 3).
Common mistakes in this topic
- Losing brackets when substituting negative values.
- Applying an index to a coefficient as well as the letter.
- Combining unlike terms.
- Omitting brackets in a rearranged formula.
- Dividing only one term on a side instead of the whole side.
Exam tips
- Write out the substitution line in full before evaluating. It is a method mark on its own.
- When changing the subject, undo operations in the reverse of the order they were applied.
- If the required letter appears twice, expect to factorise. That is always the route.
- Check a rearrangement by substituting easy numbers into both versions.
- Keep brackets around any expression that ends up as a numerator or denominator.
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Algebraic Manipulation FAQs
How do I change the subject of a formula?
Apply inverse operations to both sides until the required letter stands alone, working in the reverse order to how the expression was built. Undo addition and subtraction first, then multiplication and division, then powers and roots. Whatever you do to one side you must do to the whole of the other side.
What do I do when the new subject appears twice?
Clear any fractions first, then expand the brackets. Collect every term containing that letter on one side and everything else on the other. Factorise the letter out, which leaves it multiplied by a bracket, then divide both sides by that bracket.
What are like terms?
Like terms have exactly the same letters raised to exactly the same powers. 4x and 7x are like terms and combine to 11x. 4x and 4x squared are not like terms, and neither are 4x and 4y, so neither pair can be combined.
Why do brackets matter when substituting negatives?
Without brackets the sign can attach to the wrong part of the expression. Substituting b equals minus 2 into 3b squared gives 3 multiplied by (minus 2) squared, which is 12. Writing it without brackets invites the error of squaring only the 2 and losing the sign.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.