Revise Introduction to Algebra for International Mathematics 0607 (O Level) — revision notes and instant AI marking. Free to start.
📖 Revision notes · preview
Cambridge IGCSE · International Maths (Extended)
Introduction to Algebra
The Big Idea: Algebra is just arithmetic with a placeholder — instead of always working with known numbers, we use letters to stand in for numbers we don't know yet (or that can be anything), and all the same rules of add, subtract, multiply, divide, powers and roots still apply.
Summary — What This Chapter Covers
Algebraic notation — the shorthand rules for writing letters, multiplication, division, powers and brackets.
Algebraic vocabulary — the language you need: term, factor, coefficient, expression, equation, formula.
Substitution — replacing letters in a formula with actual numbers to calculate a value.
Collecting like terms — simplifying expressions by combining terms that "match".
Everything in this chapter is — it's the grammar of algebra. Get this rock-solid now, because every later topic (equations, expanding brackets, factorising, graphs) assumes you can do this fluently and without thinking twice.
1. Algebraic Notation
What is algebra, really?
Algebra uses letters (called variables) to represent numbers that are either unknown (we don't yet know their value, e.g. "what number plus 5 equals 12?") or general (they can represent any number, e.g. "for any two numbers x and y..."). Think of a letter like an empty box waiting to be filled with a number — we just don't know the number yet, or we're keeping it flexible on purpose.
Analogy
Imagine a vending machine slot. It's built to accept coin of the right size — you don't know in advance which exact coin will go in. The letter x is that slot: a fixed "shape" in the calculation that can hold different numbers depending on the situation.
The shorthand rules of algebraic notation
Algebra has its own writing conventions — they exist purely to save space and avoid clutter. You must know these cold, because exam questions assume you can read and write them instantly.
Operation
How it's written
What it means
Add / Subtract
a + b, c + d − e
Same as normal — + and − are written out
Multiply
ab, 3ab
No × symbol! ab means a × b; 3ab means 3 × a × b
Divide
a/b (as a fraction)
Means a ÷ b — division is always shown as a fraction, never with a ÷ sign
Powers
a², 4a²
a² means a × a; in 4a², only a is squared, multiplied by 4
Roots
√a
The square root of a — same idea as with plain numbers
Brackets
3(a + b)
Means 3 × (a + b) — work out what's inside the brackets conceptually first
Key Rule
Order of operations still applies in algebra
In ab + c/3, you're meant to work out a × b and c ÷ 3 as separate "chunks" before adding them — exactly like BIDMAS/PEMDAS with ordinary numbers. Algebra doesn't invent new rules; it just hides the symbols.
Watch Out
4a² is not the same as (4a)². In 4a², the power only applies to the a — square a first, then multiply the result by 4. If you wanted to square the whole thing including the 4, it would need to be written (4a)². This tiny detail trips up a huge number of students.
Worked Example — Building Expressions From a Story
(a) Write down the total number of points Raheem has.
He started with q and gained 6 more, so we just add: Raheem has q + 6 points.
(b) Write down his new number of points.
"Doubles his total" means multiply the previous amount by 2. Since the total was (q + 6), we must bracket it before multiplying: Raheem has 2(q + 6) points, which is also equal to 2q + 12.
Notice why the brackets matter here — if you wrote 2q + 6 instead, you'd only be doubling the q part and forgetting to double the 6 points he already earned. Always bracket a "total" before applying an operation to the whole thing.
Practice Question 1
Write an expression for: "Jamal starts with p pencils, buys 4 more, and then gives away half of his total."
Practice Question 2
Is 5x² the same as (5x)²? If x = 2, work out both values to prove your answer.
2. Algebraic Vocabulary
This is the "grammar dictionary" of algebra. Examiners use these words precisely in questions, so if you're fuzzy on what "term" or "factor" or "expression" actually means, you'll misread what's being asked even if you know the maths.
Term
A term is one single "piece" of an algebraic statement — the pieces that get separated by + or − signs. A term can be:
A letter on its own or raised to a power, e.g. x or x²
A number on its own — called a constant because it never changes value, e.g. 20
A number multiplied by a letter, e.g. 5x
The number sitting in front of the letter is called the coefficient. In 6x, the coefficient is 6. In −5y, the coefficient is −5 (don't forget the sign belongs to the coefficient!). Terms can also mix multiple letters and powers together, like 6xy, 4x², or ab³c.
Factor
A factor is any number or letter that divides a term exactly, with no remainder left over. A term can be broken down (or "factorised") into a multiplication of its factors.
Example — Listing Factors
The factors of 3x are: 1, 3, x, 3x
The factors of 4xy are: 1, 2, 4, x, 2x, 4x, y, 2y, 4y, xy, 2xy, 4xy
Comparing 6xy and 4x, their common factors (factors shared by both) are 1, 2, x and 2x. The highest common factor (HCF) is 2x — this becomes crucial later when factorising expressions.
Expression vs. Equation vs. Formula — the trio students always mix up
This is genuinely one of the most commonly confused ideas at this level, so let's nail it with a clear comparison.
Word
Has an "=" sign?
Can it be "solved"?
Example
Expression
No
No — nothing to solve, only simplify
2x + 5y
Equation
Yes
Yes — find the value(s) of the letter
2x + 5 = 4
Formula
Yes
No (on its own) — it's a relationship rule; substitute values to turn it into a solvable equation
w = mg
How To Remember The Difference
Expression = an unfinished sentence (no verdict, nothing to "resolve"). Equation = a specific claim about specific numbers, which you can prove true by solving for the unknown. Formula = a general recipe connecting several different quantities (like a science law) — you plug numbers it, you don't "solve" the formula itself, you use it to build an equation you can then solve.
Worked Example — Formula → Equation
Step 1: Identify the formula — it links four quantities: x, v, t, w.
Step 2: Substitute in the three known values: 10 = 2 × t − 3
Answer:10 = 2t − 3 — this is now a genuine because it has specific numbers and only one unknown (t), which could be solved.
Practice Question 3
From this list, identify which is an expression, which is an equation, and which is a formula: 3x − 7A = πr²4x + 1 = 9
3. Substitution
What is substitution?
Substitution means replacing a letter (variable) in a formula or expression with a given number, so you can calculate an actual numerical value. It's the algebra equivalent of "filling in the blank."
Method — 4 Steps
1. Write down the formula or expression. 2. Replace each letter with its given number — put brackets around negative numbers when you substitute them in. 3. Simplify any numerical calculations, following the order of operations. 4. Calculate the final value (sometimes you'll end up with an equation, which you then solve).
The #1 Substitution Mistake
Forgetting brackets around negative numbers causes chaos. If x = −4, then x² means (−4)² = 16 (positive!) — not−4² = −16. Without the brackets, calculators (and people) sometimes apply the negative sign squaring, giving the wrong sign. Always write (−4) when substituting a negative value, especially before a power.
Worked Example — Substituting Into a Formula, Then Solving
Step 1 — Substitute known values:20 = 2 × l + 2 × 4
Step 2 — Simplify:20 = 2l + 8
Step 3 — Solve the equation (subtract 8 from both sides): 12 = 2l
Step 4 — Divide both sides by 2:l = 6 cm
This shows exactly why formulas matter: substitution turned a general rule into a specific, solvable equation.
Practice Question 4
Given a = −3 and b = 5, find the value of a² + 2b.
Practice Question 5
The formula for weight is w = mg. If w = 50 and m = 5, substitute to form an equation, then solve it to find g.
4. Collecting Like Terms
What is a "like term"?
Like terms are terms that have exactly the same letters raised to exactly the same powers. The coefficients (the numbers in front) are allowed to be different — that's fine, that's actually the whole point of collecting them.
These ARE like terms
These are NOT like terms
2x and 3x
2x and 3y — different letters
4x² and 6x²
4x² and 6x⁴ — different powers
5xy and -7xy
5xy and 7xyz — different letters
xy and yx (multiplication order doesn't matter!)
—
The sign always travels with its term
Every term has a positive or negative sign attached to its front — think of subtraction as "adding a negative." This matters because it means you're allowed to reorder terms, .
Reordering Rule
2x − 3y is exactly the same as −3y + 2x — the − stayed glued to the 3y when it moved. It's just like how 20 − 30 is the same as −30 + 20 (both equal −10). Never let a term "leave its sign behind" when you rearrange!
Key Rule
Collecting like terms = adding/subtracting their coefficients only
If there are several of like terms in one expression (e.g. some x terms and some y terms), collect each group separately — never mix an x term with a y term.
Worked Example — Two Different Letters
Step 1 — Collect the a terms first:8a − 6a = 2a
Step 2 — Collect the b terms (don't lose that minus sign in front of 5b!): −5b + 4b = −b
Step 3 — Combine the two results:2a + (−b)
Step 4 — Simplify the signs:2a − b
Answer: 2a − b
Practice Question 6
Simplify: 2x + 4y + 5x − 3y
Practice Question 7
Simplify: 9m − 2n − 12m − 6n + 3
What to Memorise
Term
Meaning
Variable
A letter representing an unknown or general number
Term
A single piece of an expression: a letter, a number, or a letter × number, separated from others by + or −
Constant
A term that is just a plain number — it never changes value
Coefficient
The number in front of a letter in a term (e.g. 6 in 6x)
Factor
A number or letter that divides a term exactly with no remainder
Factorise
To write something as a multiplication of its factors
Common factor
A factor shared by two or more terms
Expression
An algebraic statement with no equals sign — cannot be "solved," only simplified
Equation
An algebraic statement with an equals sign, connecting a left- and right-hand side — can be solved
Solution
The value of the letter that makes an equation true
Formula
A general rule linking several quantities using letters — substitute values into it to get an equation
Substitution
Replacing a letter with a given number to calculate a value
Like terms
Terms with exactly the same letters and exactly the same powers (coefficients may differ)
Golden Rules Recap
• No symbol for multiplication: ab means a × b
• Division is always written as a fraction
• 4a² means "square a, then × 4" — not the same as (4a)²
• Always bracket negative numbers when substituting
• Only combine terms with identical letters and identical powers
• A sign belongs to the term directly after it and moves with that term
Concepts Checklist
Tick off each concept once you're confident you could explain it to someone else without looking back at your notes.
Exam Tips & Common Mistakes
Squaring negatives without brackets
The single most common error in this topic. If x = −4, always write (−4)² when substituting — not −4². On a calculator, missing brackets around a negative number before a power gives the wrong sign every time.
Misreading "expression" vs "equation" in the question
If a question asks you to "solve," it must be an equation. If it asks you to "simplify," it's an expression — there's nothing to solve, only tidy up. Reading the command word carefully saves you from doing the wrong type of working entirely.
Combining terms that aren't actually "like"
Students often merge 3x and 5x² into 8x² or 8x — this is wrong. Different powers of the same letter are never like terms. They must stay separate in your final simplified answer.
Dropping signs when reordering terms
When you rearrange 2x − 3y to put the y term first, the minus sign MUST travel with the 3y, giving −3y + 2x. Leaving it as 3y + 2x silently flips the sign and changes the value — mark schemes penalise this heavily.
Leaving "1x" or "0" mistakes in a final answer
Always simplify a coefficient of 1 down to just the letter (write x, never 1x). Also double check: if two like terms cancel exactly to zero, don't forget they've genuinely vanished from the expression — don't leave a stray "0x" sitting in your answer.