Substitution into Expressions –

Substitution into expressions is a core algebra skill that appears on almost every GCSE Maths paper. You replace letters with given numbers, then use BIDMAS to evaluate the result accurately.

What Is Substitution?

Substitution means replacing the variables (letters) in an algebraic expression or formula with specific numerical values. For example, if a = 3 and b = 5, then the expression 2a + b becomes 2(3) + 5 = 11.

The most important rule during substitution is to follow BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction). Indices must be dealt with before multiplication, and multiplication before addition. Getting this order wrong is the single most common source of errors.

Special care is needed when substituting negative numbers. Writing the negative value inside brackets helps prevent sign mistakes. For instance, if x = -4, then x² = (-4)² = 16, not -16.

Key Formulas

Step-by-Step Method

1. Write out the expression clearly.
2. Replace each letter with its given value, placing negative numbers in brackets.
3. Work out indices (powers) first.
4. Then carry out multiplication and division from left to right.
5. Finally, perform addition and subtraction from left to right to get the answer.

Worked Example 1 — Foundation Level

Question: Find the value of 3x + 2y when x = 4 and y = -1.

Working:

Step 1 — Substitute: 3(4) + 2(-1).

Step 2 — Multiply: 12 + (-2).

Step 3 — Add: 12 - 2 = 10.

Answer: 10

Worked Example 2 — Higher Level

Question: Find the value of 2a² - 3ab when a = -2 and b = 5.

Working:

Step 1 — Substitute: 2(-2)² - 3(-2)(5).

Step 2 — Indices first: (-2)² = 4, so 2 × 4 = 8.

Step 3 — Multiply: 3 × (-2) × 5 = -30.

Step 4 — Combine: 8 - (-30) = 8 + 30 = 38.

Answer: 38

Worked Example 3 — Exam Style

Question: The formula for the surface area of a cylinder is S = 2πr² + 2πrh. Find S when r = 3 and h = 10. Give your answer to 1 decimal place. (3 marks)

Working:

Step 1 — Substitute: S = 2π(3)² + 2π(3)(10).

Step 2 — Indices: (3)² = 9, so 2π × 9 = 18π.

Step 3 — Multiply: 2π × 3 × 10 = 60π.

Step 4 — Add: S = 18π + 60π = 78π.

Step 5 — Evaluate: 78π = 78 × 3.14159... = 245.04...

Answer: S = 245.0 (1 d.p.)

Common Mistakes

• Confusing -x² and (-x)². If x = 3, then -x² = -(3²) = -9, but (-x)² = (-3)² = 9. Always check where the negative sign sits.
• Ignoring BIDMAS after substitution. Students sometimes evaluate left to right instead of respecting the order of operations, especially with mixed addition and multiplication.
• Forgetting that 2x means 2 × x. When x = 5, the expression 2x = 10, not 25. The multiplication sign is implied.

Exam Tips

• Write out every substitution step — do not try to do it in your head for multi-term expressions.
• Use brackets around negative numbers every time to avoid sign slips.
• If the question says "give your answer to" a certain accuracy, you must round at the end, not during the calculation.

Practice Questions

Q1 (Foundation): Find the value of 5a - 2b when a = 3 and b = 4.

Q2 (Foundation): Find the value of x² + 3x when x = -2.

Q3 (Higher): Given that p = 4, q = -3 and r = 0.5, find the value of pq² - 2r.

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Related Topics

• Simplifying Expressions
• Rearranging Formulae
• Order of Operations (BIDMAS)

Summary

• Substitution means replacing letters with numbers and evaluating the result.
• Always follow BIDMAS: deal with indices before multiplication, and multiplication before addition.
• Place negative numbers in brackets when substituting to prevent sign errors.
• Write out each step to avoid careless mistakes and to earn method marks.
• Check your final answer by estimating or re-substituting.