Combining like terms | Year 8

01 / LEARN

Understand the idea.

Which terms belong together?

Like terms have the same variable factors with the same powers. The terms $5 x$ and $- 2 x$ can be combined, but $5 x$ , $5 y$ and $5 x^{2}$ describe different variable parts. Constants form their own group.

A coefficient counts copies of its variable part. The supplied tiles show $3 x + 2 x = 5 x$ : adding copies changes how many there are, not the variable itself. Multiplying $x$ by $x$ , in contrast, produces $x^{2}$ .

Keep the sign with the term

Keep each sign attached to its term when rearranging. In $6 a - 4 + 2 a + 7$ , the constant terms are $- 4$ and $+ 7$ . Group the matching variable parts, add their coefficients, and collect the constants separately.

After collecting, unlike terms stay separate. Write $x$ for $1 x$ and omit a term with coefficient zero. The simplified expression must keep the original value for every allowed input; you are changing its form, not choosing a value for the variable.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Collect variables and constants

Simplify $4 a + 3 + 2 a - 5$ .

Worked solution

1
Group like terms with their signs.
$4 a + 3 + 2 a - 5 = (4 a + 2 a) + (3 - 5)$
The two a terms share a variable part; the remaining terms are constants.
2
Combine the a terms.
$(4 + 2) a = 6 a$
Adding coefficients counts the total number of copies of a.
3
Combine constants and write the result.
$\begin{aligned} 3 - 5 & = - 2 \\ 4 a + 3 + 2 a - 5 & = 6 a - 2 \end{aligned}$
The constant minus two cannot be combined further with the variable term.

Answer

$6 a - 2$

EXAMPLE 02

Two item types

Simplify the cost expression $2 p + 3 q + p - q$ , in dollars.

Worked solution

1
Group each item type.
$2 p + 3 q + p - q = (2 p + p) + (3 q - q)$
Terms involving p and q represent different parts of the cost.
2
Collect each group.
$\begin{aligned} 2 p + p & = (2 + 1) p = 3 p \\ 3 q - q & = (3 - 1) q = 2 q \end{aligned}$
An unwritten coefficient is one, including the subtracted q term.
3
Retain the two different variable parts.
$Cost = 3 p + 2 q dollars$
The p and q terms are unlike, so the cost expression is now simplified.

Answer

$3 p + 2 q$ dollars

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $8 m - 3 + 2 m + 5$ .

Show answer & reasoning

$10 m + 2$

Group the m terms and the signed constants: $8 m + 2 m = 10 m$ and $- 3 + 5 = 2$ . The simplified expression is $10 m + 2$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Simplify $6 x + 2 x$ .
2 marks
Simplify $9 a - 4 a$ .
2 marks
Simplify $3 p + 7 + 2 p - 4$ .
2 marks
Simplify $5 x + 2 y - 3 x + y$ .
2 marks
Simplify $4 t - 6 t + 9 - 2$ .
2 marks
Sam says $2 a + 3 b = 5 a b$ . Explain the error.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Simplify $6 x + 2 x$ .
2 marks
Add coefficients: $6 + 2 = 8$ .
$8 x$
Simplify $9 a - 4 a$ .
2 marks
Subtract coefficients: $9 - 4 = 5$ .
$5 a$
Simplify $3 p + 7 + 2 p - 4$ .
2 marks
$3 p + 2 p = 5 p$ and $7 - 4 = 3$ .
$5 p + 3$
Simplify $5 x + 2 y - 3 x + y$ .
2 marks
The $x$ coefficients give $5 - 3 = 2$ .
The $y$ coefficients give $2 + 1 = 3$ .
$2 x + 3 y$
Simplify $4 t - 6 t + 9 - 2$ .
2 marks
$4 - 6 = - 2$ and $9 - 2 = 7$ .
$- 2 t + 7$
Sam says $2 a + 3 b = 5 a b$ . Explain the error.
3 marks
The variable parts $a$ and $b$ differ.
With $a = 1, b = 2$ , the original is $8$ but $5 a b = 10$ .
Unlike terms cannot be collected; the expression remains $2 a + 3 b$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 Mathematics with SA curriculum context

Combining like terms; substitution with signed/rational values; multiplying/dividing algebraic terms; expanding a single bracket; factorising common factors; equivalent linear expressions; rearranging simple formulas.

Official curriculum source ↗

In this lesson

Understand the idea.

Which terms belong together?

Keep the sign with the term

See the method in action.

Collect variables and constants

Group like terms with their signs.

Combine the a terms.

Combine constants and write the result.

Two item types

Group each item type.

Collect each group.

Retain the two different variable parts.

Try it yourself.

Check your understanding

Keep practising