YEAR 7 · ALGEBRAIC EXPRESSIONS AND FORMULAS · 12 MIN

Simple equivalent expressions

Combine like terms and use the structure of an expression to explain equivalence.

In this lesson

Combine copies of one variable
Explain small grouped sums
Disprove a false rule with a counterexample

Before you start

Terms and coefficients
Multiplication notation
Substitution

01 / LEARN

Understand the idea.

Count copies of the same quantity

Equivalent expressions have equal values for every allowed value of their variables. Like terms count copies of the same quantity: $3 x + 2 x = (3 + 2) x = 5 x$ . Combine their coefficients while keeping the variable factor unchanged.

Unlike terms must stay separate. In $3 x + 2$ , the constant $2$ does not count copies of $x$ . Likewise, $3 x + 2 y$ adds two different quantities; changing it to $5 x y$ would introduce a multiplication the original expression never contained.

Explain a general rule

Explain an equivalence using groups, not just a few examples. Three groups of $x + 2$ contain three copies of $x$ and six units, so $3 (x + 2) = 3 x + 6$ for every $x$ . This is the distributive property made visible by counting.

Substitution can disprove a proposed equivalence: one input giving unequal results is a counterexample. Matching results for one or several inputs do not prove that all inputs match. To establish equivalence, use a general operation or grouping argument.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine lengths

A path has sections $4 x$ metres and $3 x$ metres long. Give one expression for the total.

Worked solution

1
Add the two section lengths.
$4 x + 3 x metres$
The complete path contains both sections, so their lengths are added.
2
Combine the like terms.
$4 x + 3 x = (4 + 3) x = 7 x$
Four copies and three copies of the same length make seven copies.
3
State the total length.
$7 x metres$
Combining coefficients preserves the length unit and works for every permitted value of $x$ .

Answer

$7 x$ metres.

EXAMPLE 02

Test a claim

Leah says $2 (x + 3) = 2 x + 3$ for every $x$ . Check $x = 4$ .

Worked solution

1
Evaluate the left side at the given input.
$2 (4 + 3) = 2 \times 7 = 14$
Substitution preserves the brackets, so the addition is performed first.
2
Evaluate the right side at the same input.
$2 \times 4 + 3 = 8 + 3 = 11$
The proposed right side doubles only $x$ before adding three.
3
Compare the two results.
$14 \neq 11$
A single counterexample disproves a claim made for every value of $x$ .
4
Explain the correct expansion.
$2 (x + 3) = 2 x + 2 \times 3 = 2 x + 6$
Both terms in each of the two groups must be counted, including both copies of three.

Answer

The claim is false.

03 / YOUR TURN

Try it yourself.

Check your understanding

Simplify $6 m + m$ .

Show answer & reasoning

$7 m$

The unwritten coefficient of $m$ is one. Thus $6 m + m = (6 + 1) m = 7 m$ : six copies and one more copy make seven.

Keep practising

6 worksheet questions · 9 marks

Open worksheet Worked solutions

Read the questions online

Simplify $2 x + 5 x$ .
1 mark
Simplify $8 a - 3 a$ .
1 mark
Write $p + p + p + p$ more briefly.
1 mark
Simplify $3 b + 2 + b + 5$ .
2 marks
Write $2 (n + 4)$ without brackets.
2 marks
Is $4 x + 2$ equivalent to $6 x$ ? Use $x = 3$ to explain.
2 marks

Read the worked solutions

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

Simplify $2 x + 5 x$ .
1 mark
Add the coefficients: $2 + 5 = 7$ .
$7 x$
Simplify $8 a - 3 a$ .
1 mark
Eight copies minus three leave five copies.
$5 a$
Write $p + p + p + p$ more briefly.
1 mark
There are four identical terms.
$4 p$
Simplify $3 b + 2 + b + 5$ .
2 marks
$3 b + b = 4 b$ .
$2 + 5 = 7$ .
$4 b + 7$
Write $2 (n + 4)$ without brackets.
2 marks
There are two copies of $n$ and two copies of 4.
$2 n + 8$
Is $4 x + 2$ equivalent to $6 x$ ? Use $x = 3$ to explain.
2 marks
$4 \times 3 + 2 = 14$ .
$6 \times 3 = 18$ .
A counterexample disproves equivalence.
No: the values are 14 and 18.

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Curriculum reference & lesson scope

Australian Curriculum v9.0 / SA Curriculum Mathematics

Year 7 variables, expression notation, substitution, simple equivalence and formula tables. Digital methods are explained with supplied data; the topic test works by hand.

Official curriculum source ↗

In this lesson

Understand the idea.

Count copies of the same quantity

Explain a general rule

See the method in action.

Combine lengths

Add the two section lengths.

Combine the like terms.

State the total length.

Test a claim

Evaluate the left side at the given input.

Evaluate the right side at the same input.

Compare the two results.

Explain the correct expansion.

Try it yourself.

Check your understanding

Keep practising