Substituting numerical values into formulae and expressions
Substitution is the bridge between algebra and arithmetic. Replace each letter with the given number, then evaluate using BIDMAS. The hardest marks are lost not on the algebra but on order-of-operations slips and sign errors.
The technique
- Write the expression with brackets around each substituted value. This single habit prevents 80% of substitution errors.
- Apply BIDMAS strictly: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right).
- Show the substitution line in your working — examiners give a method mark for it even if your arithmetic is wrong.
✦Worked example— Worked example 1
Find the value of 3x² - 4x + 7 when x = -2.
Substitute (with brackets!): 3(-2)² - 4(-2) + 7.
Indices first: (-2)² = 4. So: 3(4) - 4(-2) + 7.
Multiplications: 12 - (-8) + 7 = 12 + 8 + 7.
Total: 27.
A common slip is writing -2² = -4. Without brackets your calculator does -(2²) = -4, but the algebra means (-2)² = +4. Always bracket negatives.
✦Worked example— Worked example 2 (formula)
The volume of a cone is V = ⅓πr²h. Find V when r = 6, h = 7, leaving your answer in terms of π.
V = ⅓ × π × (6)² × 7 = ⅓ × π × 36 × 7 = ⅓ × 252π = 84π.
So V = 84π cm³ (about 263.9 cm³).
✦Worked example— Worked example 3 (with fractions)
Evaluate (a + b)/(a - b) when a = 5, b = -3.
Numerator: 5 + (-3) = 2. Denominator: 5 - (-3) = 5 + 3 = 8. Quotient: 2/8 = 1/4.
✦Worked example— Worked example 4 (kinematics)
v² = u² + 2as. Find v when u = 3, a = 4, s = 5.
v² = 3² + 2 × 4 × 5 = 9 + 40 = 49, so v = 7 (taking the positive root in context).
⚠Common mistakes— Common mistakes (examiner traps)
- Forgetting brackets around negatives.
-3²on a calculator returns-9. The algebra wants(-3)² = 9. Always write the bracket. - Misordering BIDMAS. In
3x² - 4xwithx = -2, square first, then multiply, then subtract. - Sign error on
- (-). Subtracting a negative becomes addition:5 - (-3) = 8. - Ignoring the
⅓or½in formulae. Take the fractional coefficient seriously — it's worth at least one mark. - Rounding too early. Keep π or surds exact through the calculation; only round at the end if asked.
➜Try this— Quick check
If P = 2(l + w) and the perimeter is to equal 30 with w = 4, find l.
30 = 2(l + 4) → 15 = l + 4 → l = 11. Substitution and rearrangement combined.
AI-generated · claude-opus-4-7 · v3-deep-algebra