Substitution and formulae

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Notes de leçon

What is a formula?

A formula is a rule that links quantities together using symbols, usually with an equals sign.
Formulae use single letters for quantities, e.g. P = F / A (P = pressure, F = force, A = area).
The plural can be formulas or formulae — both are correct.
A formula lets you work out one quantity when you know the others.

Expressions vs formulae

An expression is a mathematical object made of numbers, letters and operations — it has no equals or inequality sign.
A formula is a statement that two expressions are equal (or unequal), so it contains =, <, >, ≤ or ≥.
Example: 8x − 5 is an expression; 8x − 5 ≥ 3 is a formula.
Expressions can be evaluated by substituting values for the letters.

Substituting numbers into expressions

Substitution means replacing a letter with a given number.
Replace the letter with the number, then follow the normal order of operations (BIDMAS).
Example: if x = 4, then 3x + 2 = 3 × 4 + 2 = 14.
Use brackets when substituting negative numbers: if x = −2, then x² = (−2)² = 4, not −4.

Substituting into formulae

Write the formula, substitute the known values, then calculate the unknown.
Example: v = u + at (v = final velocity, u = initial velocity, a = acceleration, t = time).
If u = 3, a = 2 and t = 5, then v = 3 + 2 × 5 = 13.
Keep units consistent and include the correct unit in your answer.

Standard formulae to know

Area of a rectangle: A = l × w (A = area, l = length, w = width).
Area of a triangle: A = 1/2 × b × h (b = base, h = perpendicular height).
Speed, distance and time: s = d / t (s = speed, d = distance, t = time).
Volume of a sphere: V = \frac{4}{3}\π r³ (V = volume, r = radius).
Force: F = ma (F = force, m = mass, a = acceleration).

Writing formulae from real life

Choose a letter for each quantity and say what it stands for.
Look for the relationship: total cost = number of items × price each, so C = n × p.
Check the formula works with simple numbers before using it.
Use the correct units for each quantity.

Rearranging a formula

Rearranging means changing the subject of the formula — the letter on its own on one side.
Do the same operation to both sides to keep the formula balanced.
Use inverse operations: undo addition with subtraction, multiplication with division.
Example: make r the subject of V = \frac{4}{3}\π r³: multiply by 3, divide by 4π, then take the cube root.
Example: make t the subject of s = d / t: multiply both sides by t, then divide by s, giving t = d / s.

Common mistakes to avoid

Forgetting to square a negative number correctly: (−3)² = 9, not −9.
Ignoring order of operations: 2 + 3 × 4 = 14, not 20.
Mixing up the subject: the subject is the single letter on its own.
Leaving out units or using inconsistent units in a formula.

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1.Which of the following is a formula rather than an expression?
Easy
- A8x − 5
- B8x − 5 ≥ 3
- C3y + 2
- Dx² + 4x
2.In the formula F = ma, what does F represent?
Medium
- Aforce
- Bfrequency
- Cfriction
- Dmass
3.In the formula F = ma, m stands for mass.
Easy
True or false?
4.The formula for speed is speed = distance ÷ time. A car travels 150 km in 3 hours. What is its average speed?
Medium
- A50 km/h
- B450 km/h
- C0.02 km/h
- D150 km/h
5.The area of a triangle is A = 1/2 × b × h. Find A when b = 10 cm and h = 6 cm.
Medium
- A30 cm²
- B60 cm²
- C16 cm²
- D8 cm²
6.Which of the following is an expression, not a formula?
Medium
- A3x + 2
- B3x + 2 = 11
- Cv = u + at
- DA = lw
7.Match each formula with its meaning.
Medium
- F = ma
- v = u + at
- A = 1/2 × b × h
- V = 4/3 π r³
- Newton's second law
- equation of motion
- area of a triangle
- volume of a sphere
8.Rearrange the steps to make x the subject of the formula y = 3x + 2.
Medium
- y = 3x + 2
- y − 2 = 3x
- x = (y − 2) / 3
- x = y/3 − 2/3

Plus de questions

9.Which of the following are formulae? (select all that apply)
Easy
- AA = l × w
- B3x + 5
- Cs = d ÷ t
- D8x − 5 ≥ 3
- E2y − 7
10.A rectangle has length l and width w. Which expressions give its area? (select all that apply)
Medium
- Al × w
- B2l + 2w
- Clw
- Dl + w
- E2(l + w)
11.Which of these are correct rearrangements of the formula y = mx + c to make x the subject? (select all that apply)
Medium
- Ax = (y − c) / m
- Bx = \frac{y - c}{m}
- Cx = y − c / m
- Dx = (c − y) / m
- Ex = y/m − c
12.Match each formula with the quantity it calculates.
Medium
- A = l × w
- A = 1/2 × b × h
- s = d ÷ t
- V = l × w × h
- Area of a rectangle
- Area of a triangle
- Average speed
- Volume of a cuboid
13.Match each expression or formula with its description.
Medium
- 3x + 5
- 3x + 5 = 20
- 3x + 5 > 20
- 3x + 5 ≤ 20
- An expression
- An equation (formula)
- An inequality (formula)
14.Put the steps in the correct order to make x the subject of the formula y = 4x − 3.
Medium
- Add 3 to both sides: y + 3 = 4x
- Divide both sides by 4: x = (y + 3) / 4
- Start with y = 4x − 3
15.Put the steps in the correct order to make r the subject of the formula V = (4/3)πr³.
Medium
- Multiply both sides by 3: 3V = 4πr³
- Divide both sides by 4π: 3V / (4π) = r³
- Take the cube root of both sides: r = cube root of (3V / (4π))
- Start with V = (4/3)πr³
16.An expression contains an equals sign.
Easy
True or false?
17.The formula for the area of a triangle is A = 1/2 × b × h.
Easy
True or false?
18.Substitute x = 4 into the expression 3x + 5.
Easy
- A12
- B17
- C23
- D35
19.Use the formula P = 2(l + w) to find P when l = 7 and w = 3.
Easy
- A10
- B20
- C21
- D42
20.Given s = ut + 1/2 at², find s when u = 5, t = 2 and a = 3.
Medium
- A16
- B22
- C28
- D31
21.Substitute y = −2 into the expression y² + 3y.
Medium
- A−2
- B2
- C10
- D−10
22.In the formula v = u + at, which letter represents acceleration?
Easy
- Av
- Bu
- Ca
- Dt
23.Use the formula D = s × t to find D when s = 12 m/s and t = 5 s.
Easy
- A17 m
- B60 m
- C2.4 m
- D120 m
24.Which of the following are correct substitutions into the formula A = l × w? (select all that apply)
Medium
- Al = 4, w = 6 gives A = 24
- Bl = 3, w = 5 gives A = 15
- Cl = 2, w = 8 gives A = 16
- Dl = 5, w = 5 gives A = 10
- El = 6, w = 2 gives A = 8
25.Substituting x = 5 into 2x + 1 gives 11.
Easy
True or false?
26.Match each formula with the value obtained when the given number is substituted.
Medium
- y = 2x + 3, x = 4
- y = x² − 1, x = 3
- y = 10 − x, x = 2
- y = 3x, x = 5
- 8
- 11
- 15
27.A taxi charges a fixed fee of £3 plus £2 per mile travelled. Which formula gives the total cost C (in pounds) for a journey of m miles?
Medium
- AC = 3m + 2
- BC = 3 + 2m
- CC = 5m
- DC = 3m − 2
28.A cuboid has length l = 8 cm, width w = 5 cm and height h = 4 cm. Its volume is V = l × w × h. What is V?
Hard
- A17 cm³
- B40 cm³
- C160 cm³
- D320 cm³
29.The perimeter of a rectangle is P = 2(l + w). Which of the following pairs give P = 24? (select all that apply)
Hard
- Al = 7, w = 5
- Bl = 10, w = 2
- Cl = 6, w = 6
- Dl = 8, w = 3
- El = 9, w = 4
30.In the formula for the area of a circle, A = πr², the letter r represents the radius.
Medium
True or false?
31.Match each formula with the value obtained when the given numbers are substituted.
Medium
- A = 1/2 × b × h with b = 12, h = 7
- P = 2(l + w) with l = 9, w = 4
- V = l × w × h with l = 2, w = 3, h = 5
- 42
- 26
- 30
32.Put the steps in the correct order to make h the subject of the formula V = l × w × h.
Hard
- Divide both sides by l × w
- Write the formula V = l × w × h
- h = V ÷ (l × w)
33.A cone has volume V = (1/3)πr²h. Find V when r = 3 cm and h = 10 cm, using π ≈ 3.14. Give your answer to 1 decimal place.
Hard
- A94.2 cm³
- B282.6 cm³
- C31.4 cm³
- D18.8 cm³
34.A plumber charges a call-out fee of £40 plus £25 per hour. The total cost is C = 40 + 25h, where h is the number of hours. Which of the following statements are true? (select all that apply)
Hard
- AFor 3 hours the cost is £115
- BFor 5 hours the cost is £125
- CThe call-out fee is £25
- DFor 0 hours the cost is £40
- EFor 2 hours the cost is £90
35.Which of these is the plural of 'formula' used under the influence of scientific Latin?
Easy
- Aformulae
- Bformulaes
- Cformulas
- Dformulai
36.In the formula V = \frac{4}{3}\π r³, the letter V stands for the volume and r stands for the radius.
Easy
True or false?
37.A formula always contains which of the following?
Easy
- Aa relation such as = or <
- Ba bracket
- Ca fraction
- Da power
38.Which of the following are true about expressions and formulas? (select all that apply)
Medium
- AAn expression denotes a mathematical object.
- BA formula denotes a statement about mathematical objects.
- CAn expression usually contains an equality or inequality sign.
- D8x − 5 is an expression.
- E8x − 5 ≥ 3 is an expression.
39.Match each formula with the quantity it gives.
Medium
- V = \frac{4}{3}\π r³
- F = ma
- speed = distance ÷ time
- A = 1/2 × b × h
- volume of a sphere
- force
- average speed
- area of a triangle
40.A formula is used to calculate the volume of a sphere. Which of the following is that formula?
Medium
- AV = \frac{4}{3}\π r³
- BV = \frac{4}{3}\π r²
- CV = \frac{3}{4}\π r³
- DV = \frac{1}{3}\π r³
41.What is the plural of 'formula' in scientific Latin?
Easy
42.Complete the formula for the volume of a sphere.
Easy
V = ____ × π × r³
43.The area of a rectangle is given by A = l × w. Find A when l = 8 cm and w = 5 cm.
Medium
44.Use the formula A = 1/2 × b × h to find A when b = 9 cm and h = 4 cm. Give your answer in cm².
Medium
45.Complete the statements about expressions and formulae.
Easy
An ____ has no equals sign, while a ____ contains an equals sign or inequality.
46.Use the formula A = 1/2 × b × h to find A when b = 12 cm and h = 5 cm. Give your answer in cm².
Easy
47.Complete the substitution into the formula F = ma.
Medium
When m = 4 and a = 6, F = ____.
48.Use the formula A = l × w to find A when l = 11 cm and w = 4 cm. Give your answer in cm².
Easy
49.The formula for speed is speed = distance ÷ time. A cyclist travels 84 km in 4 hours. What is the cyclist's average speed in km/h?
Medium
50.Use the formula A = 1/2 × b × h to find A when b = 15 cm and h = 8 cm. Give your answer in cm².
Medium
51.Given s = ut + 1/2 at², find s when u = 8, t = 3 and a = 4.
Medium
52.Use the formula v = u + at to find v when u = 12, a = -3 and t = 5.
Hard
53.Explain the difference between an expression and a formula, giving an example of each.
Hard
54.Explain why it is useful to use single letters in formulae such as F = ma.
Hard
55.The formula for the volume of a sphere is V = (4/3)πr³. Explain how you would find V when r = 3 cm, and give the value to 1 decimal place.
Medium
56.Explain how you would find the value of v in the formula v = u + at when u = 20, a = −4 and t = 3. Give the final value.
Hard
57.Explain why mathematicians and scientists use single letters such as V and r in formulae, rather than writing the words in full.
Medium
58.The area of a rectangle is given by A = l × w. Find A when l = 8 cm and w = 5 cm.
Medium
- A44 cm²
- B36 cm²
- C48 cm²
- D40 cm²
59.Use the formula A = 1/2 × b × h to find A when b = 9 cm and h = 4 cm. Give your answer in cm².
Medium
- A19 cm²
- B17 cm²
- C20 cm²
- D18 cm²
60.Use the formula A = 1/2 × b × h to find A when b = 12 cm and h = 5 cm. Give your answer in cm².
Easy
- A30 cm²
- B36 cm²
- C27 cm²
- D33 cm²
61.Use the formula A = l × w to find A when l = 11 cm and w = 4 cm. Give your answer in cm².
Easy
- A48 cm²
- B44 cm²
- C40 cm²
- D52 cm²
62.The formula for speed is speed = distance ÷ time. A cyclist travels 84 km in 4 hours. What is the cyclist's average speed in km/h?
Medium
- A19 km/h
- B25 km/h
- C21 km/h
- D23 km/h
63.Use the formula A = 1/2 × b × h to find A when b = 15 cm and h = 8 cm. Give your answer in cm².
Medium
- A72 cm²
- B66 cm²
- C54 cm²
- D60 cm²
64.Given s = ut + 1/2 at², find s when u = 8, t = 3 and a = 4.
Medium
- A42
- B38
- C46
- D50
65.Use the formula v = u + at to find v when u = 12, a = -3 and t = 5.
Hard
- A-2
- B-3
- C-4
- D-1

Notes de révision

## Substitution and Formulae **Expression vs formula** - An **expression** is a mathematical object, e.g. 8x − 5. It has no = or < sign. - A **formula** is a statement relating two expressions, e.g. 8x − 5 ≥ 3, or F = ma. It contains a relation such as = or ≥. **Substitution** Replace each letter with its given value, then work out the result. Use brackets when the value is negative. - If x = 4: 8x − 5 = 8 × 4 − 5 = 27 - If x = −2: 8x − 5 = 8 × (−2) − 5 = −21 - If a = 3, b = −4: a^2 + b = 9 + (−4) = 5 **Writing a formula from a situation** Choose a single letter for each quantity, then link them with an = sign. - Speed: s = d / t (s = speed, d = distance, t = time) - Cost: C = 5n + 2 (C = cost in pounds, n = number of items) **Standard formulae to know** - Area of a rectangle: A = l × w (A = area, l = length, w = width) - Area of a triangle: A = 1/2 × b × h (b = base, h = perpendicular height) - Area of a circle: A = πr^2 (r = radius) - Circumference of a circle: C = 2πr - Volume of a sphere: \(V = \frac{4}{3}\pi r^{3}\) (V = volume, r = radius) - Speed: s = d / t - Density: \(D = \frac{m}{V}\) (D = density, m = mass, V = volume) - Newton's second law: F = ma (F = force, m = mass, a = acceleration) **Rearranging a formula (changing the subject)** The subject is the letter on its own, usually on the left. Do the same operation to both sides. - Make d the subject of s = d / t: multiply both sides by t → d = st - Make t the subject: divide both sides by s → t = d / s - Make a the subject of F = ma: divide both sides by m → a = F / m - Make r the subject of A = πr^2: divide by π, then take the square root → \(r = \sqrt{\frac{A}{\pi}}\) **Exam tips** - Always substitute with brackets for negative numbers. - Follow order of operations: powers, then × ÷, then + −. - Check units match the situation. - After rearranging, test with easy numbers to confirm your new formula works.

Formula: A statement relating two expressions, usually with an = or inequality sign, e.g. F = ma or 8x − 5 ≥ 3. The plural is formulas or formulae.
Expression: A mathematical object made from numbers, letters and operations but with no relation sign, e.g. 8x − 5.
Substitution: Replacing each letter in an expression or formula with a given number and working out the result.
Subject of a formula: The single letter that the formula is written for, usually alone on one side, e.g. s is the subject of s = d / t.
Rearranging: Changing the subject of a formula by doing the same operation to both sides, e.g. turning s = d / t into d = st.
Speed: Distance travelled per unit time: s = d / t, where s = speed, d = distance, t = time.
Area of a triangle: A = 1/2 × b × h, where b = base and h = perpendicular height.
Volume of a sphere: \(V = \frac{4}{3}\pi r^{3}\), where V = volume and r = radius.
Newton's second law: F = ma, where F = force, m = mass and a = acceleration.

Cartes mémo

What is a formula?

A formula is an equation or inequality that relates one expression to another, showing a relationship between quantities.

What is an expression?

An expression is a mathematical object made from numbers, variables and operations, without an equals or inequality sign.

Give an example of an expression and a formula.

8x − 5 is an expression; 8x − 5 ≥ 3 is a formula.

What does 'substitution' mean in algebra?

Substitution means replacing a variable with a given number to work out the value of an expression or formula.

Substitute x = 4 into 3x + 2.

3 × 4 + 2 = 14.

Substitute x = −2 into x² + 5.

(−2)² + 5 = 4 + 5 = 9.

Substitute a = −3 and b = 2 into ab − a.

(−3)(2) − (−3) = −6 + 3 = −3.

What is the formula for the area of a rectangle?

A = l × w, where A = area, l = length, w = width.

What is the formula for the area of a triangle?

A = \frac{1}{2} \times b \times h, where A = area, b = base, h = perpendicular height.

What is the formula for speed?

s = d ÷ t, where s = speed, d = distance, t = time.

What is the formula for density?

D = \frac{m}{V}, where D = density, m = mass, V = volume.

What is the formula for pressure?

P = \frac{F}{A}, where P = pressure, F = force, A = area.

What is the formula for force?

F = ma, where F = force, m = mass, a = acceleration.

What is the formula for the volume of a sphere?

V = \frac{4}{3} \π r³, where V = volume, r = radius.

What does it mean to rearrange a formula?

Rearranging a formula means changing its subject by using inverse operations, while keeping both sides equal.

Make r the subject of the formula C = 2πr.

r = \frac{C}{2\π}.

Make t the subject of the formula s = d ÷ t.

t = \frac{d}{s}.

Make x the subject of the formula y = 3x + 1.

x = \frac{y - 1}{3}.

In a formula, why are variables usually written as single letters?

Single letters make formulas quicker to write and easier to manipulate, especially when they are large or complex.

What is the plural of 'formula'?

Both 'formulas' and 'formulae' are correct.

Write a formula for the total cost C of n books at £4 each plus a £3 delivery charge.

C = 4n + 3.

Write a formula for the perimeter P of a rectangle with length l and width w.

P = 2l + 2w.

Substitute x = 5 into 2x² − 3x.

2(5)² − 3(5) = 50 − 15 = 35.

Substitute a = −1 and b = −4 into a² + b.

(−1)² + (−4) = 1 − 4 = −3.

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