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How do you simplify algebraic expressions?

Quick answer

To simplify an expression, expand any brackets, then collect like terms (terms with exactly the same letters and powers). For example, 3x + 2y + 5x − y simplifies to 8x + y.

Ages 11–15Secondary

In one sentence

Simplifying an expression means rewriting it in the shortest, tidiest form that still has exactly the same value, mainly by expanding brackets and collecting like terms.

Explained

An expression is a mix of numbers, letters and operations with no equals sign, such as 3x + 2y + 5x − y. (With an equals sign it would be an equation, which you solve; an expression you only simplify.) Each part separated by + or − is a term: here the terms are 3x, 2y, 5x and −y. The number in front of a letter is its coefficient: in 5x the coefficient is 5, and in −y it is −1.

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3x + 2y + 5x − y = 8x + y: same value, fewer terms.

Like terms have exactly the same letters raised to exactly the same powers. 3x and 5x are like terms. So are 4ab and −ba, because ab and ba mean the same product. But 2x and 2x² are not like terms, and neither are 6 and 6x. You can only add or subtract like terms, by adding or subtracting their coefficients:

3x + 5x = 8x and 2y − y = y, so 3x + 2y + 5x − y = 8x + y.

Think of it as counting the same thing: 3 lots of x plus 5 lots of x is 8 lots of x. But x and y are different unknown numbers, so 8x + y cannot be squeezed into one term. Writing 9xy would be wrong.

The sign belongs to the term after it. In 7a − 2b + a, the terms are 7a, −2b and +a. Move terms around together with their signs: 7a + a − 2b = 8a − 2b.

Multiplying terms is different from adding them. You multiply the numbers and then the letters, and you may multiply unlike terms: 3a × 4b = 12ab, 2x × 5x = 10x². When the same letter is multiplied, add its powers: x³ × x² = x⁵ (see powers and indices). Dividing works the same way in reverse: 12ab ÷ 4a = 3b and x⁵ ÷ x² = x³.

Brackets come first. Multiply the number outside by every term inside (the distributive property): 3(x + 4) = 3x + 12. A minus sign in front of a bracket changes the sign of every term inside: −(2y − 5) = −2y + 5. After expanding, collect like terms. Expressions with only numbers, like 4 + 3 × (6 − 2), are simplified with the order of operations.

Examples

Expression What to do Simplified
2x + 5x add coefficients 7x
8a − 3a + a 8 − 3 + 1 = 6 6a
4x + 2y − x + 3y x-terms and y-terms separately 3x + 5y
3 × 4n multiply the numbers 12n
5p × 2q numbers, then letters 10pq
x² + 3x + 2x² − x x² and x are not like terms 3x² + 2x
15x ÷ 5 divide the number 3x
4(2x − 3) multiply each term by 4 8x − 12

Worked example: simplify 3(2x + 4) − 2(x − 5). Expand the first bracket: 6x + 12. Expand the second, taking the −2 into both terms: −2x + 10 (because −2 × −5 = +10). Now collect: 6x − 2x = 4x and 12 + 10 = 22. Answer: 4x + 22. Check by putting in x = 1: the original gives 3(6) − 2(−4) = 18 + 8 = 26, and 4(1) + 22 = 26. Same value, so the simplification is right.

Common mistakes

  • Adding unlike terms. 2a + 3b is not 5ab. Fix: only terms with identical letters and powers can be combined.
  • Mixing up x², 2x and x + x. x + x = 2x, but x × x = x². Fix: adding counts the letters; multiplying raises the power.
  • Forgetting to multiply every term in the bracket. 4(x + 3) is 4x + 12, not 4x + 3. Fix: draw an arrow from the outside number to each term inside.
  • Dropping the minus sign. 5 − (x − 2) is 5 − x + 2 = 7 − x, not 3 − x. Fix: a minus in front of a bracket flips every sign inside.

Step by step

Step by step

  1. Expand the bracketsMultiply the term outside by every term inside. Watch the signs.
  2. Find the like termsUnderline or colour terms with the same letters and powers (x terms, x² terms, numbers…).
  3. Collect themAdd or subtract the coefficients of each group. Keep each term's sign with it.
  4. Write the answer neatlyUsually the highest power first, then lower powers, then the plain number: 3x² + 2x − 5.
  5. CheckSubstitute an easy value, like x = 2, into the original and your answer. Both must give the same number.

"Simplify" never means "find x". If there is no equals sign, your answer is still an expression.

Practice questions

Simplify the expressions

Try each one, then check your answer.

  1. 4x + 3x
    Show answer
    4 + 3 = 7: 7x
  2. 9y − 5y
    Show answer
    9 − 5 = 4: 4y
  3. 3a + 2b + 5a
    Show answer
    3a + 5a = 8a: 8a + 2b
  4. 6m − m + 2
    Show answer
    6m − 1m = 5m: 5m + 2
  5. 2x + 7 − 5x + 1
    Show answer
    2x − 5x = −3x and 7 + 1 = 8: −3x + 8 (or 8 − 3x)
  6. 4p + 3q − p − 5q
    Show answer
    4p − p = 3p and 3q − 5q = −2q: 3p − 2q
  7. 3x² + 5x − x² + 2x
    Show answer
    3x² − x² = 2x² and 5x + 2x = 7x: 2x² + 7x
  8. 5 × 3y
    Show answer
    5 × 3 = 15: 15y
  9. 2a × 4b
    Show answer
    2 × 4 = 8 and a × b = ab: 8ab
  10. 3x × 2x
    Show answer
    3 × 2 = 6 and x × x = x²: 6x²
  11. 12ab ÷ 4a
    Show answer
    12 ÷ 4 = 3 and ab ÷ a = b: 3b
  12. 3(x + 4)
    Show answer
    3 × x + 3 × 4: 3x + 12
  13. 2(3y − 5) + 4y
    Show answer
    6y − 10 + 4y: 10y − 10
  14. 5(a + 2) − 3(a − 1)
    Show answer
    5a + 10 − 3a + 3: 2a + 13
  15. 2x³ × 5x⁴
    Show answer
    2 × 5 = 10 and x³ × x⁴ = x⁷: 10x⁷
  16. 4(2x − 1) − (x − 6)
    Show answer
    8x − 4 − x + 6: 7x + 2

Word problems

Simplifying in real life

Try each one, then check your answer.

  1. A rectangle is 3x cm long and (x + 2) cm wide. Write its perimeter as a simplified expression.
    Show answer
    3x + 3x + (x + 2) + (x + 2) = 8x + 4 cm
  2. A triangle has sides 2a, 3a + 1 and a + 4. Write its perimeter in its simplest form.
    Show answer
    2a + 3a + a = 6a and 1 + 4 = 5: 6a + 5
  3. You buy 3 notebooks at n dollars each and 2 pens at p dollars each, then 2 more notebooks. Write the total cost.
    Show answer
    3n + 2p + 2n = 5n + 2p dollars
  4. A family buys 2 adult tickets (a) and 3 child tickets (c) on Saturday, and 1 adult and 2 child tickets on Sunday. Write the total, then work it out for a = $12 and c = $7.
    Show answer
    2a + 3c + a + 2c = 3a + 5c; 3 × 12 + 5 × 7 = 36 + 35 = $71
  5. A square has sides of (x + 3) cm. Write its perimeter, then find it when x = 5.
    Show answer
    4(x + 3) = 4x + 12; 4 × 5 + 12 = 32 cm

Questions people ask

What does simplifying an expression mean?

It means writing the same expression in a shorter form with the same value, for example 2x + 3x + 4 becomes 5x + 4. You do not find the value of x.

How do you simplify expressions by combining like terms?

Group the terms that have the same letters and powers, then add or subtract their coefficients. In 6a + 2 − 4a + 5, the a-terms give 2a and the numbers give 7, so the answer is 2a + 7.

How do you simplify expressions with brackets?

Use the distributive property: multiply the term outside by each term inside, then collect like terms. 2(x + 5) + 3x = 2x + 10 + 3x = 5x + 10.

How do you simplify expressions with exponents?

When multiplying the same letter, add the powers: a² × a³ = a⁵. When dividing, subtract them: a⁶ ÷ a² = a⁴. A power of a power multiplies: (a²)³ = a⁶.

How do you simplify expressions with fractions?

Divide every term on top by the number underneath: (6x + 9) ÷ 3 = 2x + 3. You cannot cancel just one term, so (x + 6) ÷ 3 is x/3 + 2, not x + 2.

What is the difference between simplifying and solving?

You simplify an expression (no equals sign) and your answer is another expression. You solve an equation (with an equals sign) and your answer is a value, such as x = 4.

Is 2x + 3 already simplified?

Yes. 2x and 3 are not like terms, so nothing more can be combined.

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