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From plugging in values to making any letter the subject. Complete IGCSE guide with reverse BIDMAS, inverse operations and worked examples for grades 5-9.
Substitution means replacing letters in a formula with given numerical values and evaluating the result. Rearranging (or changing the subject) means rewriting a formula so that a different letter is isolated on one side. Both skills are tested on Paper 1 and Paper 2, with rearranging questions ranging from Standard tier basics to Higher tier problems where the subject appears twice.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Substitution means replacing every occurrence of a letter in a formula with its given numerical value, then evaluating the result using correct order of operations (BIDMAS).
Always write negative values inside brackets when substituting. This prevents the most common substitution error — confusing the negative sign with a subtraction. For example, if v = u + at with u = 15, a = −9.8, and t = 4:
Without brackets, students often write 15 + −9.8 × 4 and lose track of signs, leading to incorrect answers.
When a formula contains squared or cubed terms, substitute the entire value (including its sign) in brackets before applying the power. For example, if E = mc² with m = 5 and c = −3, then E = 5 × (−3)² = 5 × 9 = 45, not 5 × −9.
Rearranging a formula means isolating a chosen letter (the new subject) on one side of the equation. The method is called the balance method — whatever you do to one side, you must do to the other.
To isolate the subject, undo operations in the reverse of the BIDMAS order:
| Operation | Inverse |
|---|---|
| + (add) | − (subtract) |
| × (multiply) | ÷ (divide) |
| Square (x²) | Square root (√x) |
| nth power (xⁿ) | nth root (ⁿ√x) |
| Reciprocal (1/x) | Reciprocal (1/x) |
At Higher tier, rearranging questions fall into three increasingly difficult categories:
Given v = u + at, find v when u = 15, a = −9.8, t = 4
Step 1 (Substitute with brackets): v = 15 + (−9.8)(4)
Step 2 (Multiply): v = 15 + (−39.2)
Step 3 (Add): v = −24.2
Answer: v = −24.2
Make r the subject of A = 4πr²
Step 1 (Divide both sides by 4π): A/(4π) = r²
Step 2 (Square root both sides): √(A/(4π)) = r
Answer: r = √(A/(4π))
Make x the subject of y = (3x + 5)/(x − 2)
Step 1 (Multiply both sides by (x − 2)): y(x − 2) = 3x + 5
Step 2 (Expand): xy − 2y = 3x + 5
Step 3 (Collect x terms on one side): xy − 3x = 2y + 5
Step 4 (Factorise x out): x(y − 3) = 2y + 5
Step 5 (Divide both sides by (y − 3)): x = (2y + 5)/(y − 3)
Answer: x = (2y + 5)/(y − 3)
Square root trap: For A = 4πr², students square root too early. You must isolate r² first by dividing by 4π, then take the square root. Applying the root before isolating the squared term gives a wrong answer.
Incomplete rearrangement when x appears twice: Students move one x term but forget the other. You must collect ALL terms containing x on the same side, then factorise. If any x terms remain on the wrong side, the rearrangement is incomplete.
Sign errors during substitution: Failing to use brackets for negative values leads to (−3)² being evaluated as −9 instead of +9. Always write negative substitution values in brackets.
Forgetting ± with square roots: When an equation involves squaring and the context allows both positive and negative values, remember that √(x²) = ±x. However, in formula questions the context usually determines which root to use.
Show every step: Method marks are available at each stage of rearranging. Even if you make an arithmetic slip, you can still earn most of the marks by showing clear, logical working.
Use brackets for negatives: This is the simplest habit that prevents the most errors. Write (−9.8) not −9.8 when substituting, especially on Paper 1 where there is no calculator to check.
Check by back-substitution: After rearranging, pick simple values for the variables, calculate the subject from your rearranged formula, then substitute back into the original. If both sides match, your rearrangement is correct.
Identify the rearrangement type first: Before starting, check whether the subject appears once or twice, and whether the formula involves fractions or powers. This tells you which technique to apply and saves time.
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