EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
Formulae
Algebra · Lesson 4 of 7
Last Lesson and Before
Answer each one, then check.
1. If x = 3, what is 2x + 5?
11
2. If y = −2, what is y²?
4
3. What is (−2)³?
−8
4. Last lesson: solve 3x − 4 = 11.
x = 5
5. Last lesson: solve 2(x + 3) = 14.
x = 4
Learning Objectives
1. Tell the difference between an expression, an equation, a formula and an identity.
2. Substitute positive and negative numbers into formulae.
3. Write a formula from a description.
4. Change the subject of a formula.
5. Change the subject when the subject is squared or appears twice. (Higher)
Four Words, Four Meanings
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Expression Terms with no equals sign: 3x + 2. It cannot be solved. |
Equation Two expressions that are equal for particular values: 3x + 2 = 11 is only true when x = 3. |
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Formula A rule linking quantities, each with its own letter: v = u + at, A = πr². |
Identity Two expressions equal for EVERY value of the letter, written with ≡: 2(x + 3) ≡ 2x + 6. |
Substituting into a Formula
Replace each letter with its value, then follow the order of operations.
▸ Use brackets. Put negative numbers in brackets: if a = −2, then a² = (−2)² = 4.
▸ Powers first. In ½at², square t before multiplying.
▸ Show the substitution. Write the formula with the numbers in before working it out: it earns the method mark.
▸ Units. Give the answer in the right units if the question has them.
Substituting Negative Numbers
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s = ut + ½at². Work out s when u = 3, t = 4 and a = −2. |
1. Substitute
s = 3 × 4 + ½ × (−2) × 4²
2. Powers first
4² = 16
3. Multiply
3 × 4 = 12 and ½ × (−2) × 16 = −16
4. Add
12 + (−16) = −4
Answer: s = −4
The Area of a Trapezium
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The area of a trapezium is A = ½(a + b)h, where a and b are the parallel sides and h is the perpendicular height. With a = 5 cm, b = 9 cm and h = 4 cm: A = ½ × (5 + 9) × 4 = ½ × 14 × 4 = 28 cm². |
A = ½(a + b)h: add the parallel sides, halve, multiply by the height. |
PART ONE
Changing the Subject
Rearranging a formula to work out a different letter.
What Is the Subject?
The subject of a formula is the letter on its own on one side: in v = u + at, the subject is v.
▸ The idea. Rearrange the formula so a different letter is on its own.
▸ The method. Exactly like solving an equation: undo what has been done to the new subject, in reverse order, doing the same to both sides.
▸ Why. To find a when you know v, u and t, it is quicker to rearrange once than to solve a new equation every time.
Changing the Subject
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Make t the subject of v = u + at. |
1. Subtract u from both sides
v − u = at
2. Divide both sides by a
(v − u)/a = t
3. Write the subject on the left
t = (v − u)/a
Answer: t = (v − u)/a
Rearranging the Trapezium Formula
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Make h the subject of A = ½(a + b)h. |
1. Multiply both sides by 2
2A = (a + b)h
2. Divide both sides by (a + b)
2A/(a + b) = h
Answer: h = 2A/(a + b)
PART TWO · HIGHER
Harder Rearranging
When the subject is squared, or appears twice.
The Subject Is Squared
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Make r the subject of A = πr². |
1. Divide both sides by π
A/π = r²
2. Square root both sides
√(A/π) = r
Answer: r = √(A/π)
The Subject Appears Twice
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Make x the subject of y = (x + 2)/(x − 3). |
1. Multiply both sides by (x − 3)
y(x − 3) = x + 2
2. Expand
xy − 3y = x + 2
3. Collect the x terms on one side
xy − x = 3y + 2
4. Factorise out x
x(y − 1) = 3y + 2
5. Divide by (y − 1)
x = (3y + 2)/(y − 1)
Answer: x = (3y + 2)/(y − 1)
Case Study
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CASE STUDY Celsius and Fahrenheit The Fahrenheit temperature scale, still used in the USA, is linked to Celsius by the formula F = 1.8C + 32. Rearranging it gives C = (F − 32)/1.8, so a weather forecast of 77 °F becomes (77 − 32)/1.8 = 25 °C. There is one temperature that reads the same on both scales: solve C = 1.8C + 32 and you get C = −40. So −40 °C and −40 °F are exactly the same temperature. |
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−40° The one temperature that is the same on both scales |
100 °C = 212 °F The boiling point of water |
Key Terms
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Formula A rule linking two or more quantities, written with letters. |
Substitute Replace the letters in a formula with numbers. |
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Subject The letter on its own on one side of a formula. |
Change the subject Rearrange a formula so a different letter is the subject. |
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Identity Two expressions that are equal for every value of the letters, written with ≡. |
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Your Task: Rearrange It
12 minutes
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Make the letter in brackets the subject. (a) P = 4s (s) (b) y = mx + c (x) (c) C = 2πr (r) (d) V = IR (I) (e) A = ½bh (b). Higher: (f) E = ½mv² (v) (g) a(x − 2) = x + 5 (x). 1. Undo operations in reverse order. 2. Do the same to both sides. 3. Higher: collect the subject, then factorise. |
A good answer shows: (a) s = P/4 (b) x = (y − c)/m (c) r = C/2π (d) I = V/R (e) b = 2A/h (f) v = √(2E/m) (g) ax − 2a = x + 5, so ax − x = 5 + 2a, x(a − 1) = 5 + 2a and x = (5 + 2a)/(a − 1)
Can I...?
☐ Tell a formula from an equation, expression and identity.
☐ Substitute positive numbers into a formula.
☐ Substitute negative numbers into a formula.
☐ Write a formula from words.
☐ Change the subject in one step.
☐ Change the subject in two or more steps.
☐ Change the subject when it is squared. (Higher)
☐ Change the subject when it appears twice. (Higher)
Summary
✓ Expression: no equals sign. Equation: true for some values. Formula: links quantities. Identity: true for all values.
✓ Substitute: numbers in, brackets round negatives, powers first.
✓ Change the subject: undo operations in reverse order, doing the same to both sides.
✓ (Higher) Subject twice: collect those terms, factorise, then divide.
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EXAM FOCUS Make t the subject of the formula v = u + at (2 marks) Write the substitution before calculating, and put negative numbers in brackets. Most lost marks on substitution come from squaring a negative number without brackets. |