Writing an equation from a problem | Year 7

01 / LEARN

Understand the idea.

Say what the variable represents

Define the unknown before writing a model: let $n$ be the number of tickets. At $4$ dollars per ticket, the ticket cost is $4 n$ dollars. A $5$ dollar booking fee paid once makes the total $4 n + 5$ .

An equation sets this expression equal to a known total. A bill of $29$ dollars gives $4 n + 5 = 29$ . Keep each quantity consistent: the fee is added once, while the per-ticket price is multiplied by the ticket count.

Solve, check and interpret

Solve the model and then translate the variable back into the situation. From $4 n + 5 = 29$ , remove the fee to get $4 n = 24$ and divide by the price per ticket to find $n = 6$ tickets.

Check both the arithmetic and the meaning. Six tickets cost $24$ dollars, and the fee brings the bill to $29$ dollars. Counts of whole objects must be whole numbers, and a physical length must be positive in these problems.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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?

Worked solution

1
Define the unknown and model the bill.
$2 h + 3 = 17$
Let h be the number of hired items; each costs 2 dollars and entry adds 3 dollars once.
2
Remove the entry charge from both sides.
$\begin{aligned} 2 h + 3 - 3 & = 17 - 3 \\ 2 h & = 14 \end{aligned}$
The remaining 14 dollars pays only for equipment hire.
3
Divide by the charge per item.
$\begin{aligned} \frac{2 h}{2} & = \frac{14}{2} \\ h & = 7 \end{aligned}$
Each item contributes 2 dollars, so 14 dollars pays for 7 items.
4
Check the complete bill.
$2 \times 7 + 3 = 17$
Seven items plus the single entry charge reproduce the total.

Answer

7 equipment items

EXAMPLE 02

Use a perimeter equation

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

Worked solution

1
Add all four sides to model the perimeter.
$\begin{aligned} w + 5 + w + 5 & = 26 \\ 2 w + 10 & = 26 \end{aligned}$
A rectangle has two sides of each length; every quantity here is measured in centimetres.
2
Subtract the two known lengths.
$\begin{aligned} 2 w + 10 - 10 & = 26 - 10 \\ 2 w & = 16 \end{aligned}$
The two widths account for the remaining 16 cm of perimeter.
3
Divide the remaining total equally.
$w = \frac{16}{2} = 8$
The two widths are equal, so each is 8 cm.
4
Check all four sides.
$5 + 8 + 5 + 8 = 26$
The calculated width gives the stated 26 cm perimeter.

Answer

8 cm

03 / YOUR TURN

Try it yourself.

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 & reasoning

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

Let $r$ count rides. The total is $3 r + 4 = 25$ . Removing the entry fee gives $3 r = 21$ , so $r = 7$ rides. The check $3 \times 7 + 4 = 25$ matches the bill.

Keep practising

6 worksheet questions · 18 marks

Open worksheet 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.

Checking saved progress…

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 ↗

In this lesson

Understand the idea.

Say what the variable represents

Solve, check and interpret

See the method in action.

A starting cost and equal extras

Define the unknown and model the bill.

Remove the entry charge from both sides.

Divide by the charge per item.

Check the complete bill.

Use a perimeter equation

Add all four sides to model the perimeter.

Subtract the two known lengths.

Divide the remaining total equally.

Check all four sides.

Try it yourself.

Check your understanding

Keep practising