Substitution means replace letters with values
What you will learn
Know the rule, then use it
These are the short notes. Read each one, then check you can use it in the worked example below.
Method
Always substitute using brackets around negative values
Write the expression with brackets around each substituted value
2(−3)² − 3(4) + (−2)
Evaluate the power first (BIDMAS — indices before multiplication)
(−3)² = 9
Evaluate the remaining terms
3 × 4 = 12
Watch out
The most common error is writing (−3)² = −9 instead of +9
Replace the letter with the given value before simplifying.
Use brackets: x2 when x = -3 means (-3)2.
Find the value of 2a² − 3b + c when a = −3, b = 4 and c = −2
Write the expression with brackets around each substituted value: 2(−3)² − 3(4) + (−2).Brackets are essential here because (−3)² = 9 (positive), but without brackets you might write −3² = −9.
Evaluate the power first (BIDMAS — indices before multiplication): (−3)² = 9. Then 2 × 9 = 18.
Evaluate the remaining terms: 3 × 4 = 12. The expression becomes 18 − 12 + (−2).
Combine with correct signs: 18 − 12 − 2 = 4.
4
Build up to the hardest questions
Do them in order. If you miss a step, read the solution, then redo the question without looking.
WorkedreasoningFind the value of 2a² − 3b + c when a = −3, b = 4 and c = −2
4 marks4 minsalgebraic-notation-and-substitution-workedShow solution
Find the value of 2a² − 3b + c when a = −3, b = 4 and c = −2
- 1.Write the expression with brackets around each substituted value: 2(−3)² − 3(4) + (−2).Brackets are essential here because (−3)² = 9 (positive), but without brackets you might write −3² = −9.
- 2.Evaluate the power first (BIDMAS — indices before multiplication): (−3)² = 9. Then 2 × 9 = 18.
- 3.Evaluate the remaining terms: 3 × 4 = 12. The expression becomes 18 − 12 + (−2).
- 4.Combine with correct signs: 18 − 12 − 2 = 4.
4
- M1: write the expression with brackets around each substituted value
- M1: evaluate the power first (bidmas — indices before multiplication)
- M1: evaluate the remaining terms
- M1: combine with correct signs
- A1: 4
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
DiagnosticrecallFind 5x − 3 when x = 7
1 mark2 minsalgebraic-notation-and-substitution-q1Show solution
Find 5x − 3 when x = 7
- 1.Spot the skill: Always substitute using brackets around negative values.
- 2.Use the write the expression with brackets around each substituted value stage first, then evaluate the power first (bidmas — indices before multiplication).
- 3.Keep the final answer visible: 32.
32
- M1: use the correct always substitute using brackets around negative values. this prevents sign errors.work through bidmas after substituting: powers before multiplication before addition/subtraction.
- A1: 32
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
EasyprocedureFind t² + 4t when t = −2
2 marks3 minsalgebraic-notation-and-substitution-q2Show solution
Find t² + 4t when t = −2
- 1.Spot the skill: Always substitute using brackets around negative values.
- 2.Use the evaluate the power first (bidmas — indices before multiplication) stage first, then evaluate the remaining terms.
- 3.Keep the final answer visible: −4.
−4
- M1: use the correct always substitute using brackets around negative values. this prevents sign errors.work through bidmas after substituting: powers before multiplication before addition/subtraction.
- A1: −4
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
MediumreasoningFind (p + q)² when p = 3 and q = −5
3 marks4 minsalgebraic-notation-and-substitution-q3Show solution
Find (p + q)² when p = 3 and q = −5
- 1.Spot the skill: Always substitute using brackets around negative values.
- 2.Use the evaluate the remaining terms stage first, then combine with correct signs.
- 3.Keep the final answer visible: 4.
4
- M1: use the correct always substitute using brackets around negative values. this prevents sign errors.work through bidmas after substituting: powers before multiplication before addition/subtraction.
- A1: 4
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
Hardproblem solvingFind 3m − n² + 2mn when m = 2, n = −1
3 marks5 minsalgebraic-notation-and-substitution-q4Show solution
Find 3m − n² + 2mn when m = 2, n = −1
- 1.Spot the skill: Always substitute using brackets around negative values.
- 2.Use the combine with correct signs stage first, then write the expression with brackets around each substituted value.
- 3.Keep the final answer visible: −3.
−3
- M1: use the correct always substitute using brackets around negative values. this prevents sign errors.work through bidmas after substituting: powers before multiplication before addition/subtraction.
- A1: −3
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
Exam-stylemulti-stepFind (a − b)/(a + b) when a = 7 and b = −3, giving your answer as a fraction
4 marks6 minsalgebraic-notation-and-substitution-q5Show solution
Find (a − b)/(a + b) when a = 7 and b = −3, giving your answer as a fraction
- 1.Spot the skill: Always substitute using brackets around negative values.
- 2.Use the write the expression with brackets around each substituted value stage first, then evaluate the power first (bidmas — indices before multiplication).
- 3.Keep the final answer visible: .
- M1: use the correct always substitute using brackets around negative values. this prevents sign errors.work through bidmas after substituting: powers before multiplication before addition/subtraction.
- A1:
The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
Grade 9 stretchproblem solvingFind 2a2 - 3b when a = -3 and b = 4.
4 marks7 minssubstitution-g9Show solution
Find 2a2 - 3b when a = -3 and b = 4.
- 1.Substitute using brackets around the negative value.
- 2.Square before subtracting.
6
- M1: use 2(-3)2 - 3(4)
- A1: 6
Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.
Switch between skills
Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.
1Algebraic notation and substitution - 2 marksFind 5x − 3 when x = 7Mark answer
32
2Simplifying expressions - 2 marksSimplify 6x² − x + 3 − 2x² + 5x − 8Mark answer
4x² + 4x − 5
3Expanding and factorising - 2 marksFactorise x² − 7x + 12Mark answer
(x − 3)(x − 4)
4Linear equations - 3 marksSolve 3(2x − 1) = 2(x + 5)Mark answer
x =
- I can explain the method for algebraic notation and substitution.
- I can show clear working without skipping key steps.
- I can avoid this mistake: The most common error is writing (−3)² = −9 instead of +9.A negative number squared is always positive because (negative) × (negative) = positive.Writing brackets around the substituted value before squaring prevents this.Also, students sometimes forget that +c with c = −2 becomes +(−2) = −2, not +2.
This guide follows the OCR GCSE Mathematics J560 specification. Practice questions are original Learnova questions shaped around official content and exam skills.