Module 6 of 30Lesson 1 of 14 · A-F1 The Language of Algebra


Algebra strand · Unit A-F1 · Builds towards C2.1 & C2.2

The Language of Algebra

🟢 Foundation level
Letters are just numbers wearing masks

1 Learn it

Algebra is arithmetic with a twist: some numbers are hiding behind letters. If airtime costs P5 per day, after \(d\) days you have spent \(5 \times d\) — which algebra writes as \(5d\). The letter d is just “a number I don’t know yet”.

The grammar rules: \(3a\) means 3 × a (the × sign is invisible) · \(ab\) means a × b · \(\dfrac{a}{4}\) means a ÷ 4 · \(a^{2}\) means a × a · and a letter alone, like \(x\), means \(1x\).

Vocabulary: \(3a\), \(-2b\) and \(7\) are terms. Joined together, \(3a – 2b + 7\) is an expression. Put an equals sign in, \(3a – 2b + 7 = 10\), and it becomes an equation.

🔑 Like terms — terms with exactly the same letter part — can be collected: \(3a + 5a = 8a\) (three a’s plus five a’s). But \(3a + 2b\) stays as it is: apples and bananas do not add into “apple-bananas”.

Substitution = removing the mask. If \(a = 4\), then \(5a – 3 = 5 \times 4 – 3 = 17\). Write the substitution line first, then calculate with BODMAS (unit N-F2!).

2 Watch it work
EXAMPLE 1

Simplify \(3a + 5a\).

\(3a + 5a = 8a\)

Three of something plus five of the same thing = eight of it.

EXAMPLE 2

Simplify \(4x + 3y – x + 2y\).

Collect the x-family: \(4x – x = 3x\). Collect the y-family: \(3y + 2y = 5y\).

\(4x + 3y – x + 2y = 3x + 5y\)

Each term carries the sign in FRONT of it — the minus belongs to the x.

EXAMPLE 3

Find the value of \(5a – 3\) when \(a = 4\).

\(5a – 3 = 5 \times 4 – 3 = 20 – 3 = 17\)

EXAMPLE 4

A window is \(w\) metres wide and \(h\) metres high. Write an expression for the total length of its frame, and find it when \(w = 3\) and \(h = 2\).

Two widths and two heights: \(2w + 2h\).

\(2w + 2h = 2 \times 3 + 2 \times 2 = 6 + 4 = 10\) metres

3 Your turn — try first, then reveal
🚫 No calculator[1 mark]

Q1. Simplify \(7b – 2b\).

👀 Show answer
✍️ How your working should look in the exam
7b − 2b = 5b   ✓ B1
Why this works
Seven b’s take away two b’s leaves five b’s. The letter never changes — only the count.

🚫 No calculator[2 marks]

Q2. Simplify \(5m + 2n + 3m – n\).

👀 Show answer
✍️ How your working should look in the exam
m-terms: 5m + 3m = 8m   ✓ B1
n-terms: 2n − n = n  →  8m + n   ✓ B1
Why this works
Collect each family separately. Note 2n − n = 1n, written simply as n.

🚫 No calculator[1 mark]

Q3. Find the value of \(4x + 7\) when \(x = 5\).

👀 Show answer
✍️ How your working should look in the exam
4 × 5 + 7 = 20 + 7 = 27   ✓ B1
Why this works
Substitute, then BODMAS: multiply before adding. Writing the substitution line protects the mark even if the arithmetic slips.

🚫 No calculator[2 marks]

Q4. Find the value of \(3p – 4q\) when \(p = 6\) and \(q = 2\).

👀 Show answer
✍️ How your working should look in the exam
3 × 6 − 4 × 2   ✓ M1
= 18 − 8 = 10   ✓ A1
Why this works
Both multiplications happen before the subtraction. Two letters, one substitution line each.

🚫 No calculator[1 mark]

Q5. Kabo buys \(n\) fat cakes at P3 each and pays with a P20 note. Write an expression for his change.

👀 Show answer
✍️ How your working should look in the exam
change = 20 − 3n   ✓ B1
Why this works
The fat cakes cost 3 × n = 3n pula, taken away from 20. Real IGCSE questions dress algebra in shopping clothes exactly like this.

🚫 No calculator[2 marks]

Q6. Find the value of \(x^{2} + 3x\) when \(x = 4\).

👀 Show answer
✍️ How your working should look in the exam
42 + 3 × 4   ✓ M1
= 16 + 12 = 28   ✓ A1
Why this works
\(x^2\) means x × x = 16, and 3x means 3 × x = 12. The classic error is 4² = 8 — squaring is multiplying by itself, not by 2.

4 Common mistakes — don’t lose easy marks
  • Adding unlike terms: \(3a + 2b\) is NOT \(5ab\). Different letters never merge.
  • Losing the invisible 1: \(x\) means \(1x\), so \(4x – x = 3x\), not 4.
  • Ignoring BODMAS in substitution: \(2 + 3a\) with a = 5 is 2 + 15 = 17, never 25.
  • Sign left behind: in \(4x + 3y – x\), the minus travels with the x. Underline terms WITH their signs before collecting.
5 Where this leads

Welcome to the biggest strand in the syllabus! Next: A-F2, where brackets enter the story.

✅ Ready? Take the Check Yourself quiz below to complete this unit.
(more Algebra units arriving — the full strand is being built)

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