Overview
What you’ll learn
- Represent unknown quantities with letters, and identify terms, coefficients and like terms.
- Simplify linear expressions by collecting like terms, including expressions with fractional coefficients.
- Expand expressions of the form a(bx + c) and factorise linear expressions using the HCF.
- Substitute values into expressions and formulae, and interpret formulae in context.
- Find the nth term of a linear number sequence and test whether a number is a term of it.
In 30 seconds
- 1A term is a chunk joined by + or − signs; only like terms (same letter part) can be collected.
- 2The coefficient is the number in front: in −5x the coefficient is −5, sign included.
- 3Expand: multiply every term inside the bracket — 3(2x − 7) = 6x − 21.
- 4Factorise: pull out the HCF of every term — 10x + 15 = 5(2x + 3).
- 5Substitute carefully: replace each letter with brackets, then compute — order of operations still rules.
- 6A sequence rising by d each time has nth term dn + (first term − d).
Notes · 1 of 5
Letters, terms and like terms
Algebra writes general rules with letters — a term is one connected chunk, and only matching chunks combine.
A variable is a letter standing for an unknown or changing number. An expression is built from terms: 4x − 3y + 7 has three terms. A coefficient is the multiplier of a letter: the coefficient of x in 4x − 3y + 7 is 4, and the coefficient of y is −3 (the sign travels with the term).
- Like terms
- Same letter part: 6a and −2a, or 5x²y and x²y. They combine.
- Unlike terms
- Different letter parts: 3x and 3x², or 4a and 4b. They never merge.
- Number terms
- Plain numbers are like terms with each other: 7 and −9 combine to −2.
Multiplication ignores order: 3 × a = 3a, and a × b = ab. Division is written as a fraction: p ÷ 5 = p/5. A term with a fractional coefficient, like x/2 + x, still simplifies — think of x/2 as (1/2)x so x/2 + x = (3/2)x.
Notes · 2 of 5
Simplifying linear expressions
Collect like terms by adding their coefficients — the letter part never changes.
- 1Group the families
In 5a + 2b − 3a + b, circle the a-terms and underline the b-terms.
- 2Combine each family
a: 5 − 3 = 2, so 2a. b: 2 + 1 = 3, so 3b.
- 3Write the tidied form
2a + 3b — both families stay separate.
Watch the signs on the doorstep: in 6k + 4 − 2k − 9, the − belongs to 2k and the second − belongs to 9, giving 4k − 5. A minus sign in front of a bracket flips every sign inside: 7 − (x − 3) = 7 − x + 3 = 10 − x.
Notes · 3 of 5
Expanding and factorising brackets
Expanding multiplies every inside term; factorising undoes it by pulling out the HCF.
| Direction | Rule | Example |
|---|---|---|
| Expand | Multiply each term inside the bracket | 4(3x − 5) = 12x − 20 |
| Factorise | Divide each term by the HCF and write it outside | 12x − 20 = 4(3x − 5) |
| Check | Expand your answer — you must get back to the start | 4 × 3x − 4 × 5 = 12x − 20 ✓ |
The classic expansion slip is multiplying only the first inside term: 5(2x − 4) is 10x − 20, not 10x − 4. The classic factorising slip is leaving a factor behind: 8x + 12 = 4(2x + 3), not 4(2x + 12). Both are caught by the same habit — expand your answer to check.
To find what to pull out, take the HCF of the coefficients (from Chapter 1) and the lowest power of any shared letter: 9x²y and 6xy share HCF 3xy, so 9x²y − 6xy = 3xy(3x − 2).
Notes · 4 of 5
Expressions, substitution and formulae
A formula connects several variables — substitute known values and compute in the right order.
An expression like 5x − 2y has a numerical value once x and y are known. Substitute with brackets to protect the signs: with x = −2 and y = 3, 5x − 2y = 5(−2) − 2(3) = −10 − 6 = −16.
| Quantity | Formula in words | Formula in letters |
|---|---|---|
| Perimeter of a rectangle | twice the sum of length and breadth | P = 2(l + b) |
| Area of a triangle | half the base times the height | A = bh/2 |
| Area of a trapezium | half the sum of the parallel sides times the height | A = h(a + b)/2 |
| Average speed | distance divided by time | s = d/t |
Notes · 5 of 5
Number sequences and the nth term
A linear sequence steps by the same amount each time — the nth term is the rule machine.
- 1Check the step
The sequence 4, 9, 14, 19, … rises by 5 each time, so the rule starts 5n.
- 2Correct the constant
5 × 1 = 5, but the first term is 4 — one less. Rule: 5n − 1.
- 3Verify twice
n = 2: 5 × 2 − 1 = 9 ✓. n = 4: 19 ✓.
For a sequence that falls by the same amount, the rule subtracts: 96, 91, 86, … falls by 5, giving 101 − 5n (check: n = 1 gives 96 ✓). Testing whether a number is a term means solving: is 100 a term of 5n − 1? Then 5n = 101, so n = 20.2 — not a whole number, so no.
Revise
Key terms
The words examiners expect you to use precisely.
- Variable
- A letter used to stand for an unknown or changing number.
- Term
- One part of an expression, separated by + or − signs, e.g. 4x in 4x − 3y + 7.
- Coefficient
- The number multiplying a letter in a term; in −5x it is −5.
- Like terms
- Terms with exactly the same letter part, which can be collected together.
- Linear expression
- An expression where each letter appears to the power 1 at most, e.g. 3x + 2y − 7.
- Expand
- Multiply out a bracket over every term inside it.
Show 6 more key termsShow fewer key terms
- Factorise
- Write an expression as a product by taking out the HCF of all terms.
- HCF (of terms)
- The largest factor common to every term, from the coefficients and shared letters.
- Substitution
- Replacing letters with their values and evaluating the result.
- Formula
- A rule connecting variables, such as A = bh/2 for the area of a triangle.
- nth term
- The rule giving any term of a sequence from its position n, e.g. 5n − 1.
- Linear sequence
- A sequence with a constant difference between consecutive terms.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Combining unlike terms, e.g. writing 3x + 2x² = 5x³.
The letter parts x and x² are different, so the terms cannot be counted together — the exponents differ.
Instead: 3x + 2x² stays as 3x + 2x²; only matching letter parts combine.
Mistake: Expanding 5(2x - 4) as 10x - 4.
Only the first term was multiplied; the −4 was left inside untouched.
Instead: Every inside term is multiplied: 5 × 2x − 5 × 4 = 10x − 20.
Show 8 more mistakesShow fewer mistakes
Mistake: Writing 8x + 12 = 4(2x + 12).
Factorising divides every term by the common factor; the 12 was copied instead of divided.
Instead: 8x + 12 = 4(2x + 3). Check by expanding: 4 × 2x + 4 × 3 = 8x + 12.
Mistake: Dropping the sign of a coefficient, e.g. treating the coefficient of y in 4x − 3y as 3.
The term is −3y, so the multiplier of y includes the minus sign.
Instead: The coefficient is −3. Signs belong to the term that follows them.
Mistake: Substituting x = −3 into 2x² and computing (2 × (−3))² = 36.
Substitution replaces only the letter: 2x² means 2 × (x²), so it is 2 × 9 = 18.
Instead: Powers apply to the letter before any coefficient multiplies it.
Mistake: Claiming a sequence with rule 3n + 2 contains the term 100 because “100 ÷ 3 is nearly 34”.
Nearly is not a term number. The test is exact: 3n + 2 = 100 gives n = 32.67, not a whole number.
Instead: Solve for n exactly; the number is a term only when n is a positive integer.
Mistake: Simplifying 7 − (x − 3) as 7 − x − 3.
The minus sign in front of the bracket flips both signs inside: −(x − 3) = −x + 3.
Instead: 7 − (x − 3) = 7 − x + 3 = 10 − x.
Mistake: Expanding 2(a + b) as 2a + b.
The 2 multiplies every inside term, not just the first one.
Instead: 2(a + b) = 2a + 2b. Expand back as a check.
Mistake: Adding fractional coefficients top-and-bottom, e.g. x/2 + x/3 = 2x/5.
The coefficients 1/2 and 1/3 are fractions, which need a common denominator before adding — denominators never just add.
Instead: x/2 + x/3 = 3x/6 + 2x/6 = 5x/6.
Mistake: Undoing a formula in the wrong order, e.g. solving 2 + 1.5h = 11 as h = (11 − 2) × 1.5 or 11 ÷ 1.5 − 2.
Operations are undone in reverse order: the division by 1.5 was applied last, so it comes off first.
Instead: Subtract the constant, then divide: 1.5h = 9, so h = 6.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Simplify linear expressions by collecting like terms
Simplify 5a + 3a - 2a.
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 6Foundation
- 5Standard
- 4Challenging
15 questions worth 53 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 4 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 4.
Mark scheme
PROChapter 4 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 1Coming soonWeighted Assessment 1 (WA1)
- Term 2Coming soonWeighted Assessment 2 (WA2)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 1 Math Chapter 4 about?
Chapter 4 introduces algebra: letters for numbers, identifying and collecting like terms, expanding and factorising single brackets, substituting into formulae, and finding the nth term of linear number sequences.
How do I know which terms I can combine?
Only like terms combine — terms with exactly the same letter part. 6a and −2a combine to 4a; 6a and 6a² do not. Plain numbers combine with plain numbers.
What is the difference between expanding and factorising?
Expanding multiplies out a bracket over every inside term: 4(3x − 5) = 12x − 20. Factorising reverses it by pulling out the HCF: 12x − 20 = 4(3x − 5). Always check by expanding your factorisation back.
How do I find the nth term of a sequence?
If the terms step by the same amount d, the rule starts with dn. Compare dn at n = 1 with the actual first term to find the constant to add or subtract: 4, 9, 14, … steps by 5 and starts one below 5, so the rule is 5n − 1.
When is this chapter tested?
Algebra foundations like these usually appear in Weighted Assessment 1, Weighted Assessment 2 and the end-of-year exam. Every school sets its own scope, so check your school’s assessment outline.
Up next
Chapter 5: Linear Equations and Simple Inequalities