Worked Example
Two market stalls sell handmade bracelets.
Stall A earns 5 € per bracelet but paid 3 € to set up.
Stall B earns 2 € per bracelet and paid 9 € to set up.
After how many bracelets do both stalls have the same profit?
Setting up the equation
Profit of Stall A: $5x - 3$ · Profit of Stall B: $2x + 9$ · Set them equal:
Solving with the balance method
$$\begin{align*}
5x - 3 &= 2x + 9 &\quad\mid& -2x \\
3x - 3 &= 9 &\quad\mid& +3 \\
3x &= 12 &\quad\mid& \div 3 \\
x &= 4
\end{align*}$$
The | shows what you do to both sides at once. Every term on both sides is affected — not just the x.
The rule: Whatever you write after the |, you apply equally to every term on both sides.
Subtraction removes one term. Division or multiplication changes all of them.
Your Turn — solve on paper
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Problem 1
Lena downloads some songs. Then she buys 7 more. She now has 15 songs.
How many did she start with?
Equation: $x + 7 = 15$
solve on paper using the | method →
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Problem 2
Two friends split a pizza bill equally. The total was 14 €.
How much does each person pay?
Equation: $2x = 14$
solve on paper using the | method →
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Problem 3
After spending 4 € on a bus ticket, Noah has 9 € left.
How much money did he start with?
Equation: $x - 4 = 9$
solve on paper using the | method →
From here — write the equation yourself, then solve
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Problem 4
A smoothie bar charges 2 € per smoothie plus a one-time 3 € reusable cup deposit.
Emma's total bill was 11 €. How many smoothies did she buy?
write your equation and steps on paper →
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Problem 5
Two streaming plans. Plan A: 5 € base fee, then 3 € per extra screen.
Plan B: 13 € base fee, then 1 € per extra screen.
How many extra screens make both plans cost the same?
write your equation and steps on paper →
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Problem 6
Rosa charges 5 € per hour of babysitting but gives a 2 € discount to regular customers.
Kate charges 2 € per hour plus a 10 € booking fee.
After how many hours do they charge the same total amount?
write your equation and steps on paper →
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Problem 7
A road trip is split into two equal halves. Each half is $x$ km of motorway
followed by 3 km through town. The whole trip is 14 km.
How long is the motorway section in each half?
write your equation and steps on paper →
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Problem 8
Sofia packs 3 bags for a trip, each with $x$ books. At the airport
she removes 2 books from each bag to save weight. She now has exactly
as many books as her friend, who is carrying $x + 4$ books.
How many books were in each bag originally?
write your equation and steps on paper →
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Problem 9 Challenge
A paddling pool contains 10 litres. It loses 2 litres per hour through a slow leak.
After $x$ hours there are 4 litres left. How many hours have passed?
write your equation and steps on paper →
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Problem 10 Write your own
Two friends are saving money.
Anna has 5 € and saves 3 € every week.
Beth has 17 € and spends 1 € every week on snacks.
First, write an equation for when they have the same amount.
Then solve it. After how many weeks will they have equal savings?
write your equation and solution on paper →
Show answers
Problem 1
$x + 7 = 15$
$x = 8$ songs
Problem 2
$2x = 14$
$x = 7$ €
Problem 3
$x - 4 = 9$
$x = 13$ €
Problem 4
$2x + 3 = 11$
$x = 4$ smoothies
Problem 5
$3x + 5 = x + 13$
$x = 4$ screens
Problem 6
$5x - 2 = 2x + 10$
$x = 4$ hours
Problem 7
$2(x + 3) = 14$
$x = 4$ km
Problem 8
$3(x - 2) = x + 4$
$x = 5$ books
Problem 9
$10 - 2x = 4$
$x = 3$ hours
Problem 10
$5 + 3x = 17 - x$
$x = 3$ weeks
Coming up next →
You've isolated x when one side is simple. Next we'll look at what happens
when the coefficient of x is a fraction — and see why dividing by $\frac{2}{3}$
is the same as multiplying by $\frac{3}{2}$.