Substitution and formulae

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课程笔记

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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练习题

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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

复习笔记

## 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.

闪卡

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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