YEAR 9 · ALGEBRA · 15 MIN

Simplifying algebraic expressions

Collect like terms and simplify products or quotients without changing an expression.

In this lesson

Identify like variable parts.
Collect signed coefficients.
Simplify products and quotients with restrictions.

Before you start

Variables and coefficients
Integer arithmetic and index laws

01 / LEARN

Understand the idea.

Only like terms can be collected

A term includes its sign: in $5 x - 2 y + 3$ , the terms are $5 x$ , $- 2 y$ and $3$ . Like terms have exactly the same variable part, including every exponent. Their coefficients count copies of that same part.

Collect $x$ terms, $y$ terms and constants separately. For example, $5 x + 2 x = 7 x$ , but $5 x + 2 y$ has no single combined term. When reordering a sum, move the sign with its term so subtraction does not become addition.

Multiplication is different from collection

Multiplication uses a different operation from collecting like terms: $3 x + 4 x = 7 x$ , whereas $3 x \times 4 x = 12 x^{2}$ . In a product, multiply the coefficients and combine powers of matching variable bases.

In a quotient, cancel common factors, rather than pieces of a sum. The equality $\frac{8 x^{3}}{2 x} = 4 x^{2}$ holds for $x \neq 0$ . Keep every restriction imposed by the original denominator, even after a factor disappears.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Collect terms with signs

Simplify $7 x - 3 y - 2 x + 5 y - 4$ .

Worked solution

1
Group the signed like terms.
$7 x - 3 y - 2 x + 5 y - 4 = (7 x - 2 x) + (- 3 y + 5 y) - 4$
Reordering addition preserves value when each sign stays with its term.
2
Combine coefficients within each group.
$\begin{aligned} 7 x - 2 x & = (7 - 2) x = 5 x \\ - 3 y + 5 y & = (- 3 + 5) y = 2 y \end{aligned}$
Only terms with identical variable parts can be collected.
3
Retain the constant term.
$5 x + 2 y - 4$
There is no other constant to combine with -4.

Answer

$5 x + 2 y - 4$

EXAMPLE 02

Simplify a product and quotient

Simplify $\frac{6 a^{2} b \times 2 a b}{3 a}$ for $a \neq 0$ .

Worked solution

1
Multiply the numerator factors.
$6 a^{2} b \times 2 a b = 12 a^{3} b^{2}$
Multiply 6 by 2 and add exponents separately for a and b.
2
Separate the quotient factors.
$\frac{12 a^{3} b^{2}}{3 a} = \frac{12}{3} \frac{a^{3}}{a} b^{2}$
The original condition a nonzero permits cancellation of a.
3
Simplify and retain the domain.
$4 a^{2} b^{2}, a \neq 0$
The coefficient quotient is 4 and the exponent of a decreases from 3 to 2.

Answer

$4 a^{2} b^{2}$

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $4 p - 3 q + 2 p + q$ .

Show answer & reasoning

$6 p - 2 q$

Collect the signed terms: $(4 + 2) p + (- 3 + 1) q = 6 p - 2 q$ . The q coefficient is -2 because $- 3 + 1 = - 2$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Simplify $3 x + 8 x$ .
1 mark
Simplify $9 a - 4 + 2 a + 7$ .
2 marks
Simplify $5 m - 2 n - 8 m + 6 n$ .
2 marks
Simplify $4 x^{2} + 3 x - 2 x^{2} + 5 x$ .
2 marks
Simplify $3 p^{2} \times (- 2 p)$ .
2 marks
Simplify $\frac{18 x^{3} y}{6 x y}$ for $x, y \neq 0$ .
3 marks

Read the worked solutions

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

Simplify $3 x + 8 x$ .
1 mark
$(3 + 8) x = 11 x$ .
$11 x$
Simplify $9 a - 4 + 2 a + 7$ .
2 marks
$9 a + 2 a = 11 a$ and $- 4 + 7 = 3$ .
$11 a + 3$
Simplify $5 m - 2 n - 8 m + 6 n$ .
2 marks
$5 m - 8 m = - 3 m$ ; $- 2 n + 6 n = 4 n$ .
$- 3 m + 4 n$
Simplify $4 x^{2} + 3 x - 2 x^{2} + 5 x$ .
2 marks
Collect squares separately from first powers: $(4 - 2) x^{2} + (3 + 5) x$ .
$2 x^{2} + 8 x$
Simplify $3 p^{2} \times (- 2 p)$ .
2 marks
$3 (- 2) = - 6$ and $p^{2} p = p^{3}$ .
$- 6 p^{3}$
Simplify $\frac{18 x^{3} y}{6 x y}$ for $x, y \neq 0$ .
3 marks
$18 \div 6 = 3$ , $x^{3 - 1} = x^{2}$ , $\frac{y}{y} = 1$ .
The original denominator excludes zero for either variable.
$3 x^{2}$

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 9 algebraic simplification and equivalent expressions, with consolidation of linear formula skills. Binomial expansion and quadratic factorisation/equations are in the separate Quadratics and Parabolas chapter.

Official curriculum source ↗

In this lesson

Understand the idea.

Only like terms can be collected

Multiplication is different from collection

See the method in action.

Collect terms with signs

Group the signed like terms.

Combine coefficients within each group.

Retain the constant term.

Simplify a product and quotient

Multiply the numerator factors.

Separate the quotient factors.

Simplify and retain the domain.

Try it yourself.

Check your understanding

Keep practising