⚙️ Formulae & Expressions

Grade 5 · Algebra · Cambridge Primary Stage 5

Algebraic Notation

Letters, coefficients, like terms

Function Machines

Input → operation → output

Substitution

Replace letters with numbers

Simple Equations

Solve x + 7 = 12

Formulae in Context

Area = length × width

1. Algebraic Notation

In algebra, we use letters to represent unknown numbers. This lets us write rules and patterns in a short way.

a + 3 means "a number plus 3"
2b means "2 multiplied by b" (we write 2b, not 2×b)
3c − 5 means "3 multiplied by c, then subtract 5"
The coefficient is the number in front of the letter: in 3c, the coefficient is 3.
Like terms have the same letter and can be collected together:
3a + 2a = 5a  |  4b − b = 3b  |  7c + 2c − c = 8c
You cannot add different letters: 3a + 2b stays as 3a + 2b (unlike terms)
💡 Think of 3a as "3 apples". 3 apples + 2 apples = 5 apples. But you can't add apples and bananas!

2. Function Machines

A function machine takes an input, applies an operation, and gives an output.

Input: 4
→
× 3
→
Output: 12
Two-step machine: Apply two operations in order.
4
→
× 2
→
8
→
+ 1
→
9
Reverse function machine: Work backwards to find the input from the output.
Output is 17. Operations were ×3 then +5.
Reverse: 17 − 5 = 12, then 12 ÷ 3 = 4 (input was 4)

3. Substitution

Substitution means replacing the letter in an expression with a number and calculating the result.

If a = 4, find 3a + 2
3 × 4 + 2 = 12 + 2 = 14
If x = 5, evaluate:
2x = 2 × 5 = 10
x² = 5 × 5 = 25
3x − 4 = 3 × 5 − 4 = 15 − 4 = 11
💡 Always multiply before you add or subtract (BIDMAS/BODMAS). So 3a + 2 means (3×a) + 2, not 3 × (a+2).

4. Simple Equations

An equation has an equals sign. To solve it, find the value of the unknown letter.

One-step: x + 7 = 12
To find x, subtract 7 from both sides: x = 12 − 7 = 5
Two-step: 2x − 3 = 9
Step 1: add 3 to both sides → 2x = 12
Step 2: divide both sides by 2 → x = 6
Check: Always substitute your answer back in to check.
2 × 6 − 3 = 12 − 3 = 9 ✓
💡 Whatever you do to one side of the equation, you must do to the other side too — it must stay balanced!

5. Formulae in Context

Formulae are rules written using letters. We use them in maths and science all the time.

Area of a rectangle: A = l × w
Perimeter of a rectangle: P = 2(l + w)
(l = length, w = width)
A rectangle has length 8 cm and width 5 cm:
Area = 8 × 5 = 40 cm²
Perimeter = 2 × (8 + 5) = 2 × 13 = 26 cm
Temperature formula: F = 9C/5 + 32 (Celsius to Fahrenheit)
If C = 20: F = 9 × 20 ÷ 5 + 32 = 36 + 32 = 68°F

Example 1 — Simplifying Expressions

Simplify 5a + 3a − 2a
Collect like terms: (5 + 3 − 2)a = 6a

Example 2 — Function Machine

Input = 7, machine = ×3 then +4. Find output.
7 × 3 = 21, then 21 + 4 = 25

Example 3 — Substitution

If b = 6, find 4b − 5
4 × 6 − 5 = 24 − 5 = 19

Example 4 — Solving an Equation

Solve 3x + 4 = 19
3x = 19 − 4 = 15,   x = 15 ÷ 3 = 5

Example 5 — Using a Formula

A rectangle is 12 cm long and 7 cm wide. Find the area and perimeter.
Area = 12 × 7 = 84 cm²
Perimeter = 2 × (12 + 7) = 2 × 19 = 38 cm

Two-Step Function Machine

Enter an input and two operations to see the output step by step.

Equation Solver (ax + b = c)

Enter values for a, b, c to solve ax + b = c step by step.

Exercise 1 — Simplifying Expressions

Collect like terms. Enter the coefficient of the simplified expression.

1. 3a + 5a = ?a

2. 7b − 3b = ?b

3. 4c + 6c − 2c = ?c

4. 9x − 4x = ?x

5. 2m + 8m + m = ?m

6. 10n − 3n − 2n = ?n

7. 5p + 4p − 7p = ?p

8. 6q + q − 3q = ?q

9. 12r − 5r + 2r = ?r

10. 3s + 3s + 3s = ?s

Exercise 2 — Function Machines

Find the output of each function machine.

1. Input 5 → × 3 → Output ?

2. Input 8 → + 6 → Output ?

3. Input 4 → × 2 → + 1 → Output ?

4. Input 7 → × 3 → + 4 → Output ?

5. Input 9 → − 3 → × 2 → Output ?

6. Input 12 → ÷ 4 → + 5 → Output ?

7. Input 6 → × 5 → − 8 → Output ?

8. Input 10 → ÷ 2 → × 3 → Output ?

9. Input 3 → × 4 → + 7 → Output ?

10. Input 20 → − 5 → ÷ 3 → Output ?

Exercise 3 — Substitution

Evaluate each expression for the given value.

1. If a = 4, find 3a + 2

2. If b = 6, find 4b − 5

3. If x = 3, find 5x + 1

4. If y = 7, find 2y − 3

5. If n = 5, find n² (n × n)

6. If m = 8, find 3m − 10

7. If t = 2, find 6t + 7

8. If k = 9, find k + 15

9. If p = 4, find 4p − 9

10. If w = 10, find 2w + 3w (= 5w)

Exercise 4 — Solving Simple Equations

Solve each equation. Find the value of the unknown.

1. x + 7 = 12. Find x.

2. y − 4 = 9. Find y.

3. 2m = 14. Find m.

4. n ÷ 3 = 5. Find n.

5. 2x − 3 = 9. Find x.

6. 3a + 1 = 16. Find a.

7. 5b − 2 = 23. Find b.

8. 4k + 7 = 35. Find k.

9. t + 13 = 30. Find t.

10. 6p = 54. Find p.

Exercise 5 — Using Formulae

Substitute into the formula to find the answer.

1. Area = l × w. Length = 8, width = 5. Find area.

2. Perimeter = 2(l + w). l = 9, w = 4. Find perimeter.

3. Area = l × w. l = 12, w = 7. Find area.

4. Perimeter = 2(l + w). l = 15, w = 6. Find perimeter.

5. Speed = distance ÷ time. Distance = 60, time = 4. Find speed.

6. Cost = price × quantity. Price = 3, quantity = 12. Find cost.

7. Area = l × w. l = 6, w = 6. Find area.

8. Perimeter of square = 4 × side. Side = 9. Find perimeter.

9. V = l × w × h (volume). l=4, w=3, h=5. Find V.

10. P = 2l + 2w. l = 11, w = 8. Find P.

🏋️ Practice — Mixed Algebra (20 Questions)

1. 4a + 7a = ?a

2. Input 5 → × 4 → − 3 → Output?

3. If x = 6, find 3x + 1

4. Solve: y + 9 = 20. Find y.

5. Area = l × w. l = 10, w = 4. Area?

6. 8b − 3b = ?b

7. Input 3 → × 5 → + 2 → Output?

8. If n = 7, find 4n − 6

9. Solve: 3m = 24. Find m.

10. Perimeter = 2(l + w). l = 8, w = 5. Perimeter?

11. 6c + c − 2c = ?c

12. Input 9 → ÷ 3 → × 4 → Output?

13. If t = 4, find t² + 1 (4×4+1=?)

14. Solve: 2x + 5 = 17. Find x.

15. V = l × w × h. l=5, w=4, h=3. V?

16. 5p + 2p + p = ?p

17. Input 6 → × 6 → − 10 → Output?

18. If a = 3, find 7a + a (= 8a)

19. Solve: 4k − 1 = 15. Find k.

20. Perimeter of square = 4 × side. Side = 7. Perimeter?

🏆 Challenge — Advanced Algebra (8 Questions)

1. Solve: 3x + 7 = 28. Find x.

2. Reverse function machine: output is 29, operations were ×4 then +1. What was the input?

3. A rectangle has area 60 cm² and width 5 cm. Find the length.

4. If y = 3, find 4y² − 5 (= 4×9 − 5)

5. Form and solve: "I think of a number, multiply by 3, then subtract 8. The answer is 19." Find the number.

6. Simplify 3a + 5b + 4a − 2b. What is the coefficient of a in the simplified expression?

7. A rectangle has perimeter 36 cm. The width is 7 cm. Find the length.

8. Solve: 5x + 3 = 3x + 11. Find x.