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In algebra we use letters to represent unknown values or variables. This lesson covers the foundations of algebra for Edexcel GCSE Mathematics (1MA1): writing algebraic expressions, collecting like terms, substitution, and using the laws of indices in algebraic contexts.
| Term | Meaning | Example |
|---|---|---|
| Variable | A letter representing an unknown or changing value | x, y, n |
| Expression | A collection of terms (no equals sign) | 3x + 2y − 7 |
| Term | A single number, variable, or their product | 5x², −3y, 7 |
| Coefficient | The number in front of a variable | In 5x², the coefficient is 5 |
| Constant | A term with no variable — just a number | 7, −3 |
| Like terms | Terms with exactly the same variable parts | 3x and −5x; 2x²y and 7x²y |
| Equation | A statement that two expressions are equal | 3x + 1 = 10 |
| Formula | An equation showing the relationship between variables | A = πr² |
When translating words into algebra, look for the mathematical operation hidden in the language.
| English phrase | Algebraic expression |
|---|---|
| 5 more than x | x + 5 |
| 3 less than y | y − 3 |
| twice n | 2n |
| the product of a and b | ab |
| x divided by 4 | x/4 |
| the square of p | p² |
| 3 times the sum of x and 2 | 3(x + 2) |
A pen costs p pence and a ruler costs r pence. Write an expression for the total cost of 4 pens and 3 rulers.
Solution: 4p + 3r
Tara is t years old. Her brother is 5 years older. Their mother is three times Tara's age.
(a) Write an expression for her brother's age: t + 5
(b) Write an expression for their mother's age: 3t
(c) Write an expression for the sum of all three ages: t + (t + 5) + 3t = 5t + 5
To simplify an expression, combine terms that have identical variable parts.
Simplify: 7a + 3b − 2a + 5b − 4
Group the like terms:
Answer: 5a + 8b − 4
Simplify: 4x² + 3x − 2x² + 7 − x + 1
Answer: 2x² + 2x + 8
Simplify: 5ab − 3ba + 2a²b
Since ab = ba (multiplication is commutative): 5ab − 3ab + 2a²b = 2ab + 2a²b
Note: 2ab and 2a²b are NOT like terms — they have different variable parts.
When multiplying:
When dividing:
Simplify: 3a × 4ab
3 × 4 = 12; a × a = a²; × b = b
Answer: 12a²b
Simplify: 12x³y² ÷ 4xy
12 ÷ 4 = 3; x³ ÷ x = x²; y² ÷ y = y
Answer: 3x²y
The laws of indices apply to algebraic expressions in exactly the same way as for numbers.
| Law | Rule | Algebraic Example |
|---|---|---|
| Multiply | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x⁵ = x⁸ |
| Divide | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | y⁷ ÷ y² = y⁵ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x⁴)³ = x¹² |
| Zero index | a⁰ = 1 | x⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | x⁻³ = 1/x³ |
Simplify: (2x³)⁴
= 2⁴ × (x³)⁴ = 16 × x¹² = 16x¹²
Simplify: (3a²b)³ ÷ 9a⁴b
Step 1: (3a²b)³ = 27a⁶b³
Step 2: 27a⁶b³ ÷ 9a⁴b = 3a²b²
Answer: 3a²b²
Replace each variable with its given value, then evaluate.
Key rule: Always use brackets when substituting, especially for negative numbers.
Given a = 3, b = −2, c = 5, evaluate:
(a) 4a + 2b = 4(3) + 2(−2) = 12 − 4 = 8
(b) a² − bc = (3)² − (−2)(5) = 9 − (−10) = 9 + 10 = 19
(c) 2(a + c)² = 2(3 + 5)² = 2(8)² = 2 × 64 = 128
The formula for the area of a trapezium is A = ½(a + b)h.
Find A when a = 6, b = 10, h = 4.
A = ½(6 + 10)(4) = ½ × 16 × 4 = 32
| Mistake | Why it's wrong | Correct version |
|---|---|---|
| 3x + 2y = 5xy | You cannot add unlike terms | 3x + 2y (it's already simplified) |
| 2x² means (2x)² | The index applies only to x, not to 2 | 2x² = 2 × x × x; (2x)² = 4x² |
| Forgetting to multiply the sign | −2 × −3 is positive, not negative | (−2)(−3) = +6 |
| x + x = x² | Adding is not the same as multiplying | x + x = 2x; x × x = x² |
| a³ × a² = a⁶ | Indices are added, not multiplied | a³ × a² = a⁵ |