If v = u + at, find v when u=5, a=3, t=4 v = 5 + 3×4 = 5 + 12 = 17
Negative Substitution
If A = b² − 2c, find A when b=−3, c=4 A = (−3)² − 2×4 = 9 − 8 = 1
Changing the Subject
Make t the subject of v = u + at v − u = at → t = (v − u) ÷ a
Formula Machine Preview 🧪
Area of a circle: A = πr²
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What you'll learn:
Substituting positive values into any formula
Substituting negative values safely (watch out for squares!)
Real formulae: Area, Perimeter, Speed = d/t, v = u + at
Writing formulae from word descriptions
Changing the subject of a formula (basic rearrangement)
Formulae with powers: A = πr², V = lwh, Ek = ½mv²
📖 Learn: Formulae & Substitution
Part 1: What is a Formula?
A formula is a mathematical rule written using letters (variables) and numbers. You substitute (replace) the letters with given values to find the answer.
📌 Formula: P = 2(l + w) (perimeter of a rectangle)
To use it: replace each letter with its value, then calculate using BIDMAS.
Find P when l = 8, w = 3: P = 2(8 + 3) = 2 × 11 = 22
💡 Always write the formula first, then substitute, then calculate — never skip steps!
Part 2: Substituting Positive Values
Replace each variable with its value. Use BIDMAS: Brackets → Indices → Divide/Multiply → Add/Subtract.
Formula: v = u + at (velocity after acceleration)
Find v when u = 3, a = 5, t = 2: v = 3 + 5 × 2 = 3 + 10 = 13
Formula: A = πr² (area of circle, use π ≈ 3.14)
Find A when r = 4: A = 3.14 × 4² = 3.14 × 16 = 50.24
💡 Multiply before adding: 5 × 2 comes before + 3 because × is higher priority than +
Part 3: Substituting Negative Values
Negative values need brackets when you substitute them. This prevents sign errors — especially with squares!
⚠️ Key rule: (−3)² = (−3) × (−3) = +9 (a negative squared is positive)
⚠️ But: −3² = −(3 × 3) = −9 (the square only applies to 3, not to the minus)
Formula: y = x² + 2x
Find y when x = −4: y = (−4)² + 2×(−4) = 16 + (−8) = 16 − 8 = 8
Formula: E = ½mv², find E when m = 2, v = −6: E = ½ × 2 × (−6)² = 1 × 36 = 36
✅ Always put brackets around a negative value when substituting: write (−4) not −4
Part 4: Writing Formulae from Words
You can write your own formula from a word description. Identify what changes (variables) and what stays fixed (constants).
📌 "The cost C of hiring a bike is £5 per hour h, plus a £10 deposit."
Formula: C = 5h + 10
📌 "The area A of a triangle with base b and height h."
Formula: A = ½bh (or A = bh ÷ 2)
📌 "Speed s equals distance d divided by time t."
Formula: s = d ÷ t (usually written s = d/t)
💡 Key words → maths: "per" means ÷ or ×; "plus a fixed amount" means + constant; "product of" means ×
Part 5: Changing the Subject
The subject of a formula is the letter on its own. To change the subject, use inverse operations to isolate the new variable — just like solving equations.
Formula: v = u + at Subject is v.
Make u the subject: Subtract at from both sides → u = v − at
Make a the subject: First v − u = at, then divide by t → a = (v − u) / t
Formula: P = 2l + 2w. Make w the subject:
P − 2l = 2w → w = (P − 2l) / 2 → w = P/2 − l
🔑 Do the same to both sides. Work backwards through BIDMAS (undo + and − first, then × and ÷).
💡 Worked Examples
Example 1: Substituting Positive Values
The formula for kinetic energy is E = ½mv². Find E when m = 4 and v = 5.
Step 1: Write the formula: E = ½mv²
Step 2: Substitute: E = ½ × 4 × 5²
Step 3: Powers first: 5² = 25 → E = ½ × 4 × 25
Step 4: Multiply: ½ × 4 = 2, then 2 × 25 = 50
Answer: E = 50 J
💡 Powers come before multiplication in BIDMAS — always square v before multiplying.
Example 2: Substituting Negative Values
Using y = 3x² − 2x + 1, find y when x = −2.
Step 1: Write formula: y = 3x² − 2x + 1
Step 2: Substitute with brackets: y = 3(−2)² − 2(−2) + 1
Step 3: Powers: (−2)² = 4 → y = 3×4 − 2×(−2) + 1
Step 4: Multiply: y = 12 − (−4) + 1 = 12 + 4 + 1
Answer: y = 17
⚠️ (−2)² = +4 because two negatives multiply to a positive. Never forget the brackets!
Example 3: Using a Real-World Formula
A car travels at 60 km/h for 2.5 hours. Use d = st to find the distance.
Step 1: Identify: d = distance, s = 60, t = 2.5
Step 2: Substitute: d = 60 × 2.5
Answer: d = 150 km
Now rearrange to find time: t = d/s. How long does a 210 km trip at 70 km/h take?
Step 1: t = d/s = 210 / 70
Answer: t = 3 hours
💡 The speed-distance-time triangle: cover the letter you want, the remaining two show what to do (side-by-side = multiply; one on top of other = divide).
Example 4: Changing the Subject
Make r the subject of A = πr².
Step 1: Divide both sides by π: A/π = r²
Step 2: Square-root both sides: √(A/π) = r
Answer: r = √(A/π)
Now use it: find r when A = 78.5 (use π ≈ 3.14)
r = √(78.5 / 3.14) = √25 = 5
🔑 Undo operations in reverse order: the last thing applied to r was "square", so undo it last (take the square root).
🔬 Science Lab — Formula Machine
Pick a formula, fill the beakers with values using the sliders, and watch the output calculate live with step-by-step working!
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BIDMAS Substitution Checker
Watch how BIDMAS controls the order in substitution. Select an example:
Subject Change Step-Builder
Fill in each rearrangement step. The arrow shows your target variable.
✏️ Exercise 1: Substituting Positive Values
Substitute the given values and calculate. Type your answer (you can use decimals).
✏️ Exercise 2: Substituting Negative Values
Careful with squares of negative numbers! Use brackets when substituting.
✏️ Exercise 3: Real-World Formulae
Apply standard formulae to real-world problems. Read carefully!
📝 Exercise 4: Writing Formulae from Words
Write a formula using the variables given, then use it to calculate a value.
🔄 Exercise 5: Changing the Subject
Rearrange each formula to make the indicated variable the subject.
📝 Practice (20 Questions)
Mixed questions covering all topics. Self-marking — type your answer and check!
🏆 Challenge (8 Questions)
Harder problems — multi-step substitution, changing subject with powers, word problems.