7.1 Substitution & Formulas
Key Takeaways
- Substitution replaces each letter with its value; the normal order of operations then applies, so 2p² means 2 × p × p, never (2p)²
- Write substituted negatives in brackets: for x = -3, x² = (-3)² = 9 while -x² = -9
- Core formulas include P = 2(l + w), A = ½bh, v = d/t and F = 9C/5 + 32 — ICAS expects accurate substitution without a calculator
- To change the subject of a formula, undo the operations in reverse order, doing the same thing to both sides, exactly like solving an equation
- When dividing by a decimal such as 150 ÷ 2.5, scale both numbers (300 ÷ 5) to clear the decimal before dividing
From Paper D onwards, ICAS Mathematics regularly asks you to find the value of an expression when the letters are replaced by numbers, and Papers E-J build whole questions around formulas such as speed = distance ÷ time. Because personal calculators are not allowed, you need fluent written and mental methods, especially when negative numbers or fractions are involved. A single sign slip is the most common way students lose these otherwise easy marks.
What Substitution Means
An algebraic expression like 3x + 2 is a recipe: multiply the value of x by 3, then add 2. Substitution means replacing each letter with its given value and then carrying out the arithmetic.
Example 1. If x = 5, find 3x + 2. Replace x with 5: 3 × 5 + 2 = 15 + 2 = 17.
Example 2. If a = 4 and b = -2, find 5a - 3b. Substitute both values: 5 × 4 - 3 × (-2) = 20 - (-6) = 20 + 6 = 26. Notice the double negative: subtracting 3b when b is negative turns into addition.
Order of Operations Still Applies
After substituting, evaluate using the normal order of operations (brackets, indices, multiplication and division, then addition and subtraction).
Example 3. If p = -3, find 2p² - 4p + 1.
- p² means (-3)² = (-3) × (-3) = 9, so 2p² = 18.
- -4p = -4 × (-3) = +12.
- Total: 18 + 12 + 1 = 31.
A classic trap is writing 2p² as (2p)². These are different: 2p² = 2 × p × p = 18, while (2p)² = (-6)² = 36. Only the p is squared unless brackets say otherwise.
Substituting Fractions
Fractions test whether you can keep your arithmetic tidy without a calculator.
Example 4. If y = 3/4, find 8y + 5. 8 × 3/4 = 24/4 = 6, so 8y + 5 = 11. Dividing first (8 ÷ 4 = 2, then 2 × 3 = 6) keeps the numbers small.
Quick Reference: Signs When Substituting
| Expression | Value of letter | Result |
|---|---|---|
| -a | a = -5 | -(-5) = 5 |
| a² | a = -4 | (-4)² = 16 |
| -a² | a = -4 | -(16) = -16 |
| 1/a | a = 1/2 | 1 ÷ 1/2 = 2 |
| ab | a = -3, b = -2 | (-3)(-2) = 6 |
The row for -a² catches many students: the square applies only to a, and the negative sign is applied afterwards.
Formulas You Must Know
A formula is a rule written in letters that connects quantities. ICAS expects you to substitute into these accurately, and sometimes to work backwards.
| Formula | What it gives | Letters mean |
|---|---|---|
| P = 2(l + w) | Perimeter of a rectangle | l = length, w = width |
| A = lw | Area of a rectangle | l = length, w = width |
| A = ½bh | Area of a triangle | b = base, h = perpendicular height |
| v = d/t | Average speed | d = distance, t = time |
| F = 9C/5 + 32 | Fahrenheit from Celsius | C = temperature in °C |
| C = 5(F - 32)/9 | Celsius from Fahrenheit | F = temperature in °F |
Worked example (speed). A car travels 150 km in 2.5 hours. Find its average speed. v = d/t = 150 ÷ 2.5. Without a calculator, double both numbers to remove the decimal: 300 ÷ 5 = 60 km/h. Estimation check: 150 ÷ 3 = 50, so an answer a little above 50 is sensible.
Worked example (temperature). Convert 20°C to Fahrenheit. F = 9 × 20 ÷ 5 + 32 = 36 + 32 = 68°F. Dividing before multiplying (20 ÷ 5 = 4, then 9 × 4 = 36) is the friendlier order.
Changing the Subject
The subject of a formula is the letter standing alone on one side. To make a different letter the subject, undo the operations in reverse order, doing the same thing to both sides — exactly like solving an equation.
Example 5. Make d the subject of v = d/t. Multiply both sides by t: vt = d, so d = vt.
Example 6. Make C the subject of F = 9C/5 + 32.
- Subtract 32 from both sides: F - 32 = 9C/5.
- Multiply both sides by 5: 5(F - 32) = 9C.
- Divide both sides by 9: C = 5(F - 32)/9.
Undo the last operation first — deal with addition and subtraction before multiplication and division.
Common Traps in ICAS Substitution Questions
- Sign errors with negatives. Write the substituted value in brackets first: for x = -3, write 2(-3)² rather than 2 × -3². Brackets make the intention visible.
- Forgetting that 1/a flips the fraction. If a = 2/3, then 1/a = 3/2.
- Treating 2x² as (2x)², as shown in Example 3.
- Skipping the magnitude check. After computing a speed or an area, ask whether the size is reasonable; a car averaging 6 km/h or 600 km/h should trigger a re-check.
How ICAS Phrases These Questions
Typical stems include 'If x = -2, the value of 3x² - 5x is ...' and 'The formula h = 5t² gives the height after t seconds; find h when t = 3'. Multi-step items in Papers E-F may ask you to substitute and then feed the result into a second formula, so keep your working exact — fractions rather than rounded decimals — until the final line.
If a = -3 and b = 4, what is the value of 2a² - b?
The formula F = 9C/5 + 32 converts a Celsius temperature to Fahrenheit. What is 20°C in Fahrenheit?