YEAR 10 · ALGEBRA · 15 MIN

Simplifying expressions

Collect like terms and use the structure of an expression to simplify it without changing its value.

In this lesson

Identify terms with identical variable parts
Simplify expressions while keeping signs attached to terms

Before you start

Multiplying signed numbers
Variables and powers

01 / LEARN

Understand the idea.

Which terms can be combined?

Like terms have exactly the same variable part, including every power. In $7 a - 3 b$ , the coefficients are $7$ and $- 3$ : the minus belongs to the second term. Collecting like terms changes their coefficients, while leaving their variable part unchanged.

Simplifying in a reliable order

Expand any brackets first, then organise terms by variable and power. A subtraction before a bracket affects every term inside it. Keep constants in their own group so that none are accidentally combined with variable terms.

Addition and multiplication behave differently: $3 x + 2 x = 5 x$ , whereas $3 x \times 2 x = 6 x^{2}$ . Substituting a number into both expressions can expose an error, but agreement at one input alone does not prove they are identical.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Collect coefficients

$7 a - 3 b - 2 a + 5 b$

Worked solution

1
Group terms with matching variables.
$7 a - 3 b - 2 a + 5 b = (7 a - 2 a) + (- 3 b + 5 b)$
Reordering addition is valid when each sign stays with its term.
2
Add the coefficients in each group.
$\begin{aligned} 7 a - 2 a & = (7 - 2) a = 5 a \\ - 3 b + 5 b & = (- 3 + 5) b = 2 b \end{aligned}$
Each group counts multiples of the same variable.
3
Write the simplified expression.
$5 a + 2 b$
The remaining variable parts differ, so these two terms cannot be collected further.

Answer

$5 a + 2 b$

EXAMPLE 02

Keep powers distinct

Simplify $4 x^{2} - 3 x + 6 - 2 x^{2} + 5 x - 9$ .

Worked solution

1
Separate square terms, linear terms and constants.
$(4 x^{2} - 2 x^{2}) + (- 3 x + 5 x) + (6 - 9)$
The powers of the variable must match before terms can be combined.
2
Combine each group.
$\begin{aligned} (4 - 2) x^{2} & = 2 x^{2} \\ (- 3 + 5) x & = 2 x \\ 6 - 9 & = - 3 \end{aligned}$
Only the coefficients change when like terms are added.
3
Assemble the result in descending powers.
$2 x^{2} + 2 x - 3$
The square term, linear term and constant remain separate.

Answer

$2 x^{2} + 2 x - 3$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $6 p - 2 q - 9 p + 5 q$ .

Show answer & reasoning

$- 3 p + 3 q$

Collect the $p$ terms to get $(6 - 9) p = - 3 p$ and the $q$ terms to get $(- 2 + 5) q = 3 q$ . Therefore the expression is $- 3 p + 3 q$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Simplify $9 x - 4 x + 2 x$ .
2 marks
Simplify $5 a + 3 b - 2 a + 4 b$ .
2 marks
Simplify $8 m^{2} - 3 m + 2 m^{2} + 7 m$ .
2 marks
Simplify $4 p - (3 p - 5)$ .
2 marks
A rectangle has adjacent sides $3 x + 2$ cm and $x + 5$ cm. Write its perimeter in simplified form.
3 marks
Rina says $2 x + 5 y = 7 x y$ . Explain the error and test the claim using $x = 2, y = 1$ .
3 marks

Read the worked solutions

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

Simplify $9 x - 4 x + 2 x$ .
2 marks
Add coefficients: $9 - 4 + 2 = 7$ .
$7 x$
Simplify $5 a + 3 b - 2 a + 4 b$ .
2 marks
Collect $5 a - 2 a$ and $3 b + 4 b$ .
$3 a + 7 b$
Simplify $8 m^{2} - 3 m + 2 m^{2} + 7 m$ .
2 marks
Collect equal powers separately.
$10 m^{2} + 4 m$
Simplify $4 p - (3 p - 5)$ .
2 marks
The subtraction gives $4 p - 3 p + 5$ .
$p + 5$
A rectangle has adjacent sides $3 x + 2$ cm and $x + 5$ cm. Write its perimeter in simplified form.
3 marks
$P = 2 (3 x + 2) + 2 (x + 5)$ .
Expand and collect: $6 x + 4 + 2 x + 10 = 8 x + 14$ .
$8 x + 14$ cm
Rina says $2 x + 5 y = 7 x y$ . Explain the error and test the claim using $x = 2, y = 1$ .
3 marks
The variable parts $x$ and $y$ differ.
Substitution gives $2 (2) + 5 (1) = 9$ , while $7 (2) (1) = 14$ .
Unlike terms cannot be combined. The original equals $9$ ; $7 x y$ equals $14$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian Mathematics Curriculum

Year 10 algebra, including the approved fuller school-course treatment of rational expressions and formula restrictions; quadratic methods are in their own chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Which terms can be combined?

Simplifying in a reliable order

See the method in action.

Collect coefficients

Group terms with matching variables.

Add the coefficients in each group.

Write the simplified expression.

Keep powers distinct

Separate square terms, linear terms and constants.

Combine each group.

Assemble the result in descending powers.

Try it yourself.

Check your understanding

Keep practising