Algebraic Expressions and Simplifying

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.

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Key Vocabulary

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

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Writing Algebraic Expressions

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)

Worked Example 1

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

Worked Example 2

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

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Collecting Like Terms

To simplify an expression, combine terms that have identical variable parts.

Rules

• Only like terms can be added or subtracted.
• 3x and 5x are like terms (both have x).
• 3x and 3x² are not like terms (different powers).
• 2xy and 5xy are like terms; 2xy and 2x are not.

Worked Example 3

Simplify: 7a + 3b − 2a + 5b − 4

Group the like terms:

• a-terms: 7a − 2a = 5a
• b-terms: 3b + 5b = 8b
• Constants: −4

Answer: 5a + 8b − 4

Worked Example 4

Simplify: 4x² + 3x − 2x² + 7 − x + 1

• x²-terms: 4x² − 2x² = 2x²
• x-terms: 3x − x = 2x
• Constants: 7 + 1 = 8

Answer: 2x² + 2x + 8

Worked Example 5

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.

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Multiplying and Dividing Algebraic Terms

When multiplying:

• Multiply the coefficients.
• Multiply the variables using the index laws.

When dividing:

• Divide the coefficients.
• Divide the variables using the index laws.

Worked Example 6

Simplify: 3a × 4ab

3 × 4 = 12; a × a = a²; × b = b

Answer: 12a²b

Worked Example 7

Simplify: 12x³y² ÷ 4xy

12 ÷ 4 = 3; x³ ÷ x = x²; y² ÷ y = y

Answer: 3x²y

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Index Laws in Algebra

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

Worked Example 8

Simplify: (2x³)⁴

= 2⁴ × (x³)⁴ = 16 × x¹² = 16x¹²

Worked Example 9

Simplify: (3a²b)³ ÷ 9a⁴b

Step 1: (3a²b)³ = 27a⁶b³

Step 2: 27a⁶b³ ÷ 9a⁴b = 3a²b²

Answer: 3a²b²

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Substitution

Replace each variable with its given value, then evaluate.

Key rule: Always use brackets when substituting, especially for negative numbers.

Worked Example 10

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

Worked Example 11

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

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Common Mistakes and Misconceptions

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

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Edexcel Exam Tips

• Paper 1 is non-calculator — practise substitution by hand, especially with negative numbers and fractions.
• On Edexcel papers, "Simplify" means collect like terms and/or use index laws to write in simplest form. Show each step of working.
• "Write an expression for…" means leave it in algebraic form — do NOT try to solve it.
• When a question says "Show that…", you must demonstrate every step of working clearly — the answer is given to you.
• Edexcel GCSE has 3 papers, each 1 hour 30 minutes. Algebra appears heavily across all three papers.
• Look for marks allocated: a 1-mark "simplify" needs one step; a 3-mark "simplify" means there are multiple stages.

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Practice Problems

1. Simplify: 6x + 3y − 2x + 4y − 1
2. Simplify: 5a²b × 3ab²
3. Simplify: 20x⁵y³ ÷ 5x²y
4. Simplify: (4m²)³
5. If p = −3 and q = 4, evaluate: 2p² − 3pq + q
6. Write an expression for: "Five less than twice the square of n."
7. Simplify: 3x + 7 − x² + 2x² − 5x + 1
8. Simplify: (2a³b²)² × 3a²b

Answers

1. 4x + 7y − 1
2. 15a³b³
3. 4x³y²
4. 64m⁶
5. 2(9) − 3(−3)(4) + 4 = 18 + 36 + 4 = 58
6. 2n² − 5
7. x² − 2x + 8
8. 4a⁶b⁴ × 3a²b = 12a⁸b⁵

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Summary

• An expression contains terms but no equals sign.
• Like terms have identical variable parts and can be collected.
• The index laws (multiply → add indices; divide → subtract indices; power of a power → multiply indices) work the same way for algebra as for numbers.
• Substitution means replacing letters with values — always use brackets for negative numbers.
• On Edexcel GCSE, simplifying expressions and substitution appear on every paper series — practise until they are automatic.