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Algebra is the branch of mathematics that uses letters and symbols to represent numbers and quantities. This lesson covers the essential building blocks of algebra for AQA GCSE Mathematics: writing algebraic expressions, collecting like terms, substitution, and using index notation. These skills underpin almost every other algebra topic, so a confident grasp here will pay dividends throughout your course.
Before diving in, make sure you understand the following terms:
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter that represents an unknown value | x, y, n |
| Term | A single number, variable, or number multiplied by a variable | 5, 3x, -2y |
| Coefficient | The number in front of a variable | In 7x, the coefficient is 7 |
| Constant | A term that is just a number (no variable) | 4, -3, 12 |
| Expression | A collection of terms joined by + or - signs (no equals sign) | 3x + 2y - 5 |
| Equation | A statement that two expressions are equal | 3x + 2 = 14 |
| Formula | An equation that shows the relationship between variables | A = l x w |
| Identity | An equation that is true for all values of the variable | 2(x + 1) = 2x + 2 |
Exam Tip: The AQA exam may ask you to state whether a mathematical statement is an expression, equation, formula, or identity. Remember: an expression has no equals sign, an equation is true for particular values, a formula links variables, and an identity is always true.
In algebra, we follow certain conventions:
Write an expression for each of the following:
(a) I think of a number n, multiply it by 4 and add 3.
Answer: 4n + 3
(b) I think of a number x, square it, then subtract 7.
Answer: x^2 - 7
(c) A rectangle has length (2a + 1) and width 3. Write an expression for the perimeter.
Answer: Perimeter = 2 x (2a + 1) + 2 x 3 = 4a + 2 + 6 = 4a + 8
Like terms are terms that have exactly the same variable raised to the same power. You can add or subtract like terms to simplify an expression.
| Like Terms | Not Like Terms |
|---|---|
| 3x and 7x | 3x and 3y |
| 5y^2 and -2y^2 | 5y^2 and 5y |
| 4ab and -ab | 4ab and 4a |
Simplify: 5x + 3y - 2x + 7y - 4
Group like terms:
Answer: 3x + 10y - 4
Simplify: 4a^2 + 3a - 2a^2 + 5a - 1
Group like terms:
Answer: 2a^2 + 8a - 1
Exam Tip: A very common mistake is to combine terms like 3x and 3x^2 — these are not like terms because the powers of x are different. Always check the power before combining.
When multiplying algebraic terms:
When dividing algebraic terms:
Simplify: 3a x 4b
Multiply coefficients: 3 x 4 = 12
Multiply variables: a x b = ab
Answer: 12ab
Simplify: 2x^3 x 5x^2
Multiply coefficients: 2 x 5 = 10
Multiply variables: x^3 x x^2 = x^(3+2) = x^5
Answer: 10x^5
Simplify: 12x^4 divided by 4x
Divide coefficients: 12 / 4 = 3
Divide variables: x^4 / x = x^(4-1) = x^3
Answer: 3x^3
Index notation (also called powers or exponents) is a shorthand for repeated multiplication.
| Rule | In Words | Example |
|---|---|---|
| a^m x a^n = a^(m+n) | When multiplying, add the powers | x^3 x x^4 = x^7 |
| a^m / a^n = a^(m-n) | When dividing, subtract the powers | x^8 / x^3 = x^5 |
| (a^m)^n = a^(m x n) | When raising a power to a power, multiply | (x^2)^3 = x^6 |
| a^0 = 1 | Anything to the power 0 is 1 | 5^0 = 1 |
| a^1 = a | Anything to the power 1 is itself | x^1 = x |
| a^(-n) = 1/a^n | A negative power means the reciprocal [H] | x^(-2) = 1/x^2 |
| a^(1/n) = the nth root of a | A fractional power means a root [H] | 8^(1/3) = 2 |
Simplify: (3x^2)^3
Cube the coefficient: 3^3 = 27
Multiply the power of x: 2 x 3 = 6
Answer: 27x^6
Exam Tip: When a bracket contains a coefficient, you must raise the coefficient to the power as well as the variable. A common error is to write (3x^2)^3 = 3x^6, forgetting to cube the 3.
Substitution means replacing variables with given numerical values and then calculating the result.
If a = 3, b = -2, and c = 5, find the value of:
(a) 2a + b
2(3) + (-2) = 6 - 2 = 4
(b) a^2 - 3b
(3)^2 - 3(-2) = 9 + 6 = 15
(c) 3c - ab
3(5) - (3)(-2) = 15 + 6 = 21
(d) b^2 + 4c
(-2)^2 + 4(5) = 4 + 20 = 24
| Mistake | Correct Approach |
|---|---|
| Forgetting that (-2)^2 = 4, not -4 | Always put negative values in brackets when squaring |
| Writing 3 x -2 = -5 | 3 x -2 = -6 (multiply, do not add) |
| Ignoring order of operations | Follow BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction |
The following diagram shows how to build and simplify an algebraic expression:
flowchart LR
A[Start with a word problem] --> B[Identify the unknown - assign a letter]
B --> C[Write each operation as an algebraic term]
C --> D[Combine into an expression]
D --> E{Can you collect like terms?}
E -- Yes --> F[Simplify by collecting like terms]
E -- No --> G[Expression is already in simplest form]
Exam Tip: Always show each step of your working when simplifying or substituting. Even if your final answer is wrong, you can still earn method marks for correct intermediate steps.