Writing an equation from a problem | Year 7

01 / LEARN

The idea, step by step

Say what the variable represents

Begin with a sentence such as: let $n$ be the number of tickets. Then translate the situation into operations on $n$ . Tickets at 4 dollars each cost $4 n$ dollars. Adding a booking fee of 5 dollars gives the expression $4 n + 5$ .

An equation also needs a known total. If the total is 29 dollars, write $4 n + 5 = 29$ . The booking fee is paid once, so it is added once. Defining the variable helps you keep amounts and units in the correct places.

Solve, check and interpret

Solve the equation using inverse operations, but do not stop at an unexplained number. In $4 n + 5 = 29$ , subtract 5 then divide by 4 to obtain $n = 6$ . This means six tickets, not six dollars.

Check the answer in the situation: six tickets cost 24 dollars, and the single fee of 5 dollars makes 29 dollars. Also check that the answer is reasonable: a negative number of tickets or a fraction of an indivisible ticket would signal a problem in these tasks.

A common trap

A fixed fee must be multiplied by the number of items as well.

Instead: Read whether a cost is per item or paid once. A fee of 5 dollars plus 3 dollars per item is $3 n + 5$ , not $8 n$ .

02 / WORKED EXAMPLES

See the thinking

Example 1

A starting cost and equal extras

A swimming session costs 3 dollars for entry plus 2 dollars for each equipment item hired. The total is 17 dollars. How many items were hired?

Let $h$ be the number of equipment items.
The total cost gives $2 h + 3 = 17$ .
Subtract 3: $2 h = 14$ . Divide by 2: $h = 7$ .
Check the context: $2 \times 7 + 3 = 17$ .

7 equipment items

Example 2

Use a perimeter equation

A rectangle is 5 cm long and $w$ cm wide. Its perimeter is 26 cm. Find its width.

Two lengths and two widths give $2 w + 10 = 26$ .
Subtract 10 from both sides: $2 w = 16$ .
Divide by 2: $w = 8$ .
Check: $5 + 8 + 5 + 8 = 26$ cm.

8 cm

Check your understanding

A club charges 4 dollars for entry and 3 dollars per ride. A visitor pays 25 dollars. Write an equation and find the number of rides.

Show answer

$3 r + 4 = 25$ ; 7 rides.

Subtract the entry fee of 4 dollars: $3 r = 21$ . Divide by the cost of 3 dollars per ride: $r = 7$ . The check is $3 \times 7 + 4 = 25$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Mia has $s$ stickers. She receives 9 more and now has 28. Write an equation and find her starting number.
3 marks
A rope was $l$ metres long. Cutting off 7 m leaves 15 m. Write an equation and find its original length.
3 marks
Five equal boxes hold 45 books. Let $b$ be the number of books in one box. Write and solve an equation.
3 marks
Some counters are shared equally among 4 pupils. Each pupil receives 8 counters. Use $c$ for the total number. Write and solve an equation.
3 marks
A game costs 5 dollars for entry plus 2 dollars per round. A player pays 19 dollars. Write an equation and find the number of rounds.
3 marks
A rectangle is 6 cm long and $w$ cm wide. Its perimeter is 30 cm. Form an equation and find the width.
3 marks

Read the worked solutions

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

Mia has $s$ stickers. She receives 9 more and now has 28. Write an equation and find her starting number.
3 marks
Let $s$ be the starting number.
Subtract 9 from $s + 9 = 28$ : $s = 19$ .
Check: $19 + 9 = 28$ .
$s + 9 = 28$ ; 19 stickers.
A rope was $l$ metres long. Cutting off 7 m leaves 15 m. Write an equation and find its original length.
3 marks
Let $l$ be the original length in metres.
Add 7 to both sides: $l = 22$ .
Check: $22 - 7 = 15$ .
$l - 7 = 15$ ; 22 m.
Five equal boxes hold 45 books. Let $b$ be the number of books in one box. Write and solve an equation.
3 marks
Five equal groups give $5 b = 45$ .
Divide by 5: $b = 9$ .
Check: $5 \times 9 = 45$ .
$5 b = 45$ ; 9 books per box.
Some counters are shared equally among 4 pupils. Each pupil receives 8 counters. Use $c$ for the total number. Write and solve an equation.
3 marks
The total divided into 4 groups gives $c \div 4 = 8$ .
Multiply by 4: $c = 32$ .
Check: $32 \div 4 = 8$ .
$c \div 4 = 8$ ; 32 counters.
A game costs 5 dollars for entry plus 2 dollars per round. A player pays 19 dollars. Write an equation and find the number of rounds.
3 marks
Let $r$ be the number of rounds.
Subtract 5: $2 r = 14$ . Divide by 2: $r = 7$ .
Check: $2 \times 7 + 5 = 19$ .
$2 r + 5 = 19$ ; 7 rounds.
A rectangle is 6 cm long and $w$ cm wide. Its perimeter is 30 cm. Form an equation and find the width.
3 marks
The two lengths total 12 cm, so $2 w + 12 = 30$ .
Subtract 12: $2 w = 18$ . Divide by 2: $w = 9$ .
Check: $6 + 9 + 6 + 9 = 30$ cm.
$2 w + 12 = 30$ ; 9 cm.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Year 7 linear equations with natural-number solutions: equality, inverse operations, substitution checks and familiar models. Formal variables-on-both-sides techniques are not required here.

Official curriculum source ↗

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