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Simplifying expressions

We simplify an expression by collecting like terms: x terms on their own and numbers on their own. In 3x + 5 − x + 2 we get 3x − x = 2x and 5 + 2 = 7, so the expression becomes 2x + 7.

Years 7–9 and upper secondary · Updated

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What is a term?

Plus and minus signs split the expression into terms. The sign in front of a term belongs to that term.

3x + 5 − x + 2

The terms are 3x, 5, −x and 2. x means one x, and −x means minus one x.

Collect like terms

We want to collect the x terms on their own and the numbers on their own.

3x is three x's and 2x is two x's. Together that makes five x's.

3x + 2x = (x + x + x) + (x + x) = 5x

2x + 3 cannot be combined, because x terms and numbers are different kinds. x² and x are different kinds too.

Plus or times?

When we add, we count how many x's we have. When we multiply, we multiply the numbers on their own and the x's on their own.

3x + 2x = 5x

3x · 2x = 3 · 2 · x · x = 6x²

Check with x = 4: 12 + 8 = 20 and 5 · 4 = 20. 12 · 8 = 96 and 6 · 16 = 96.

Example

Example 1: 3x + 5 − x + 2

We want to collect the x terms on their own and the numbers on their own.

  1. The terms with their signs are 3x, +5, −x and +2.

    3x + 5 − x + 2

  2. We put like terms next to each other. The minus sign stays with the x.

    3x − x + 5 + 2

  3. Three x's minus one x is two x's. Five plus two is seven.

    2x + 7

We put in x = 2. At the start: 6 + 5 − 2 + 2 = 11. In the answer: 4 + 7 = 11. The same value, so it is correct.

Example 2: 2x² + 3x − x² + 4x − 6

We want to collect the x² terms, the x terms and the numbers separately.

  1. We sort the terms by kind.

    2x² − x² + 3x + 4x − 6

  2. Two x² minus one x² is x². Three x's plus four x's is seven x's.

    x² + 7x − 6

  3. x² and 7x are different kinds, so we are done.

    x² + 7x − 6

We put in x = 2. At the start: 8 + 6 − 4 + 8 − 6 = 12. In the answer: 4 + 14 − 6 = 12. That is correct.

Pitfalls

  • Adding an x term and a number together, so that 2x + 3 becomes 5x.

    x terms and numbers are different kinds. With x = 2, 2x + 3 = 7 but 5x = 10. 2x + 3 is already simplified.

  • Losing a minus sign, so that 7x − 2x + 3x becomes 7x − 5x = 2x.

    The terms are 7x, −2x and +3x. 7 − 2 + 3 = 8, so the answer is 8x.

Practise on your own

0 of 4 correct

  1. Simplify 5x + 3 + 2x + 4.

  2. Simplify 6y − 2 − 4y + 9.

  3. Simplify 4x · 3x.

  4. Simplify 3a + 2b − 5a + b − 4.

Common questions

What is the difference between simplifying an expression and solving an equation?

An expression has no equals sign. We write it in a shorter way without changing its value. An equation has an equals sign, and we find out which number x is.

Can you add x and x² together?

No. x² is x · x, a different kind from x. x² + 3x cannot be simplified, but 4x² − x² = 3x².

Does the order of the terms matter?

No, 2x + 7 and 7 + 2x have the same value. We usually write the x term first and the number last.

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