IGCSE Maths · Algebra and Graphs · Core & Extended

Introduction to Algebra: Substitution, Simplifying and Expanding

This opens the Algebra and Graphs unit — everything from here on, including equations, graphs, and functions, rests on being fluent with the four skills in this note: using letters to stand for numbers, substituting values in, simplifying expressions, and expanding brackets. All of it is shared between Core and Extended; the harder factorising work (quadratics, difference of two squares, completing the square) is Extended-only and gets its own note.

Why this matters Every formula you’ve used so far — density = mass ÷ volume, speed = distance ÷ time, simple interest = PRT/100 — is algebra. A letter standing in for a number isn’t a new kind of maths; it’s the same arithmetic you already know, just written so it works for any number, not just one specific case.

1. Letters as Generalised Numbers

In algebra, a letter (usually x, y, a, b, n…) stands in for a number we don’t know yet, or a number that can change. An expression is a collection of numbers, letters, and operations, like 3x + 5. A formula is a rule connecting two or more quantities, like A = l × w.

Worked example Write an expression for “a number that is 7 less than double n.”
Double n is 2n. Seven less than that is 2n − 7.
Try it — Writing Expressions
  1. Write an expression for “5 more than three times x.”
  2. Write an expression for “half of y, minus 4.”
  3. Write an expression for “the product of two consecutive whole numbers, starting from n.”
Show answers
  1. 3x + 5
  2. y/2 − 4
  3. n(n + 1)

2. Substituting Into Expressions and Formulas

To substitute means to replace each letter with its given value, then work out the result — following the usual order of operations.

Worked example — expression Find the value of 3x² − 2x + 5 when x = 4.
3(4)² − 2(4) + 5 = 3(16) − 8 + 5 = 48 − 8 + 5 = 45
Worked example — negative values Find the value of 2a − b² when a = −3 and b = 4.
2(−3) − (4)² = −6 − 16 = −22
Worked example — a real formula The formula for converting Celsius to Fahrenheit is F = (9/5)C + 32. Find F when C = 20.
F = (9/5)(20) + 32 = 36 + 32 = 68
Try it — Substitution
  1. Find the value of 4x + 7 when x = 6.
  2. Find the value of x² − 3x + 2 when x = 5.
  3. Find the value of 2p − 3q when p = −2 and q = 5.
  4. The formula for the perimeter of a rectangle is P = 2(l + w). Find P when l = 9 and w = 4.
  5. Find the value of a² + b² when a = −3 and b = −4.
Show answers
  1. 31
  2. 25 − 15 + 2 = 12
  3. −4 − 15 = −19
  4. P = 2(13) = 26
  5. 9 + 16 = 25

3. Simplifying by Collecting Like Terms

Like terms have exactly the same letters raised to exactly the same powers — 3x and 5x are like terms; 3x and 3x² are not. Simplify means combine everything as far as possible.

Worked example Simplify 5x + 3y − 2x + 7y.
Group like terms: (5x − 2x) + (3y + 7y) = 3x + 10y
Worked example — more terms, including squared terms Simplify 4p² + 2pq − 3 + 6p² − 5pq + 7.
Group like terms: (4p² + 6p²) + (2pq − 5pq) + (−3 + 7) = 10p² − 3pq + 4
Try it — Collecting Like Terms
  1. Simplify 6a + 4b − 2a + b.
  2. Simplify 8x − 3y − 5x + 9y.
  3. Simplify 3m² + 2m − 1 + m² − 5m + 6.
  4. Simplify 7xy − 2x + 3xy + 5x.
Show answers
  1. 4a + 5b
  2. 3x + 6y
  3. 4m² − 3m + 5
  4. 10xy + 3x

4. Expanding a Single Bracket

Expanding means multiplying out a bracket — every term inside gets multiplied by whatever is directly outside.

a(b + c) = ab + ac
Worked example Expand 3x(2x − 4y).
3x × 2x = 6x²,   3x × (−4y) = −12xy
Answer: 6x² − 12xy
Worked example Expand 5a(3a + 2b − 1).
5a × 3a = 15a²,   5a × 2b = 10ab,   5a × (−1) = −5a
Answer: 15a² + 10ab − 5a
Try it — Expanding a Single Bracket
  1. Expand 4x(3x + 5).
  2. Expand 2a(5a − 3b).
  3. Expand −3y(2y − 4).
  4. Expand 6p(p² + 2p − 3).
Show answers
  1. 12x² + 20x
  2. 10a² − 6ab
  3. −6y² + 12y
  4. 6p³ + 12p² − 18p

5. Expanding Two Brackets

When two brackets are multiplied together, every term in the first bracket multiplies every term in the second — four multiplications for two two-term brackets.

Worked example Expand (2x + 1)(x − 4).
2x × x = 2x²,   2x × (−4) = −8x,   1 × x = x,   1 × (−4) = −4
Combine: 2x² − 8x + x − 4 = 2x² − 7x − 4
Worked example Expand (x + 3)(x − 5).
x² − 5x + 3x − 15 = x² − 2x − 15
Try it — Expanding Two Brackets
  1. Expand (x + 2)(x + 6).
  2. Expand (x − 3)(x + 7).
  3. Expand (2x − 1)(x + 5).
  4. Expand (3x + 2)(2x − 3).
Show answers
  1. x² + 8x + 12
  2. x² + 4x − 21
  3. 2x² + 9x − 5
  4. 6x² − 5x − 6

6. Factorising by Common Factors

Factorising is the reverse of expanding — writing an expression as a product of brackets instead of a sum of terms. At this stage, that means spotting the highest common factor of every term and pulling it outside a bracket.

Worked example Factorise 9x² + 15xy fully.
HCF of 9 and 15 is 3. Both terms also share a factor of x.
Take out 3x: 9x² + 15xy = 3x(3x + 5y)
Check by expanding back: 3x × 3x = 9x², 3x × 5y = 15xy ✓
Worked example Factorise 12a²b − 8ab² fully.
HCF of 12 and 8 is 4. Both terms share a factor of ab.
Take out 4ab: 4ab(3a − 2b)
Try it — Factorising
  1. Factorise 6x + 9 fully.
  2. Factorise 10a² − 15a fully.
  3. Factorise 8x²y + 12xy² fully.
  4. Factorise 5p³ − 20p² fully.
Show answers
  1. 3(2x + 3)
  2. 5a(2a − 3)
  3. 4xy(2x + 3y)
  4. 5p²(p − 4)
Common mistakes to watch for
  • Only multiplying the first term inside a bracket by the term outside, and forgetting the rest — every term inside must be multiplied.
  • Combining terms that aren’t actually “like” — 3x and 3x² can never be added together into one term.
  • Losing a negative sign when expanding, especially with a negative number outside the bracket, e.g. −3y(2y − 4) = −6y² + 12y, not −6y² − 12y.
  • Factorising “partially” — always check whether the bracket itself could be factorised further, or whether a bigger common factor was missed.
  • In double-bracket expansion, forgetting one of the four multiplications (most often the two “outer” or two “inner” terms).

7. Quick-Fire Challenge Round

True or False?
  1. 3x and 5x are like terms.
  2. 3x and 3x² are like terms.
  3. Expanding and factorising are opposite processes.
  4. −2(x − 5) expands to −2x − 10.
Show answers
  1. True — same letter, same power.
  2. False — different powers of x, so they cannot be combined.
  3. True — factorising “undoes” expanding, and vice versa.
  4. False — −2 × (−5) = +10, so it expands to −2x + 10.
Spot the Error
  1. “5x + 3x² = 8x²”
  2. “4(2x + 3) = 8x + 3”
  3. “Factorise 6x + 8: answer is 2(3x + 8)”
Show answers
  1. 5x and 3x² are not like terms and cannot be combined at all — the expression is already fully simplified as 3x² + 5x.
  2. The 4 must multiply both terms inside the bracket: 4 × 2x = 8x, and 4 × 3 = 12, giving 8x + 12, not 8x + 3.
  3. The 8 wasn’t divided by the common factor of 2 — it should be 2(3x + 4), not 2(3x + 8).

8. Mixed Practice — Bringing It All Together

  1. Find the value of 2x² + 5 when x = −3.
  2. Simplify 7a − 4b + 2a + 9b.
  3. Expand 6x(2x − 5).
  4. Expand (x + 4)(x − 2).
  5. Factorise 14x²y − 21xy² fully.
Show answers
  1. 2(9) + 5 = 23
  2. 9a + 5b
  3. 12x² − 30x
  4. x² + 2x − 8
  5. 7xy(2x − 3y)

Practice at Home

A large set to open the Algebra unit — work through it in stages.

Part A — Writing expressions and substitution
  1. Write an expression for “4 less than twice y.”
  2. Find the value of 5x − 3 when x = 8.
  3. Find the value of 2x² + 3x − 1 when x = −2.
  4. The formula for the area of a triangle is A = ½bh. Find A when b = 14 and h = 9.
Part B — Collecting like terms
  1. Simplify 9x + 2y − 4x + 6y.
  2. Simplify 5p² − 3p + 8 + 2p² + 7p − 10.
  3. Simplify 4ab + 3a − 2ab + 5a.
Part C — Expanding brackets
  1. Expand 5x(3x − 2).
  2. Expand −4a(2a + 3).
  3. Expand (x + 5)(x + 3).
  4. Expand (x − 6)(x − 2).
  5. Expand (3x + 1)(2x − 5).
Part D — Factorising
  1. Factorise 15x + 20 fully.
  2. Factorise 6a² − 9a fully.
  3. Factorise 10x²y + 25xy² fully.
  4. Factorise 8p²q − 12pq² fully.
Part E — Applied / challenge
  1. A rectangle has length (x + 4) and width x. Write an expression for its area, expanded fully.
  2. Show that (x + 2)(x + 3) and x² + 5x + 6 are the same expression, by expanding the brackets.
  3. Factorise 12x²y − 18xy² + 6xy fully (three terms this time — find what all three share).

Key Vocabulary Recap

Expression — a combination of numbers, letters and operations, e.g. 3x + 5
Formula — a rule connecting two or more quantities, e.g. A = l × w
Substitute — replace a letter with a given value
Like terms — terms with the same letters raised to the same powers
Expand — multiply out a bracket
Factorise — write an expression as a product of brackets

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) sections C2.1 and C2.2 (Core subject content) and E2.1 and E2.2 (Extended subject content) — Introduction to algebra, and Algebraic manipulation (points 1–3 only: collecting like terms, expanding, and factorising by common factor). Extended’s additional factorising work — difference of two squares, trinomials of the form ax² + bx + c, cubics, and completing the square — is covered in a separate note, since it’s substantial enough to stand on its own.

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