Linear Equations

Question 1

Sophie bought 3 chocolate bars and 2 packets of crisps for a total of £5.50. A chocolate bar costs £1.20 more than a packet of crisps. Let 𝑥 be the cost of a packet of crisps. Write an equation to represent this situation and solve it to find the cost of a chocolate bar and a packet of crisps.

Answer 1

2𝑥 + 3( 𝑥 + 1.2 ) = 5.5
3x + 3.6 + 2x = 5.5
5𝑥 + 3.6 = 5.5
5𝑥 = 1.9
𝑥 = 0.38
A chocolate bar costs £1.58
A packet of crisps costs £0.38

Question 2

Tom and Jerry are painting a wall. Tom can paint the wall in 4 hours, while Jerry can paint the same wall in 6 hours. Together, they painted the wall in 2 hours. Let 𝑥 be the fraction of the wall Tom paints per hour, and 𝑦 the fraction Jerry paints per hour. Write an equation to represent the work done and solve for their individual rates of painting.

Question 2

Lisa went shopping and bought 5 pens and 3 notebooks. The total cost was £9.50. The cost of each pen is £1.50 more than the cost of each notebook. Let 𝑥 represent the cost of a notebook. Write an equation for this situation and solve to find the cost of a pen and a notebook.

Answer 2

3𝑥 + 5( 𝑥 + 1.5 ) = 9.5
3𝑥 + 5𝑥 + 7.5 = 9.5
8𝑥 + 7.5 = 9.5
8𝑥 = 2
𝑥 = 0.25
A notebook costs £0.25
A pen costs £1.75

Question 3

A theater sold 250 tickets for a show. Adult tickets cost £8, and children’s tickets cost £4. The total revenue from ticket sales was £1,800. Let 𝑥 represent the number of adult tickets sold. Write a linear equation to represent this situation and solve it to find how many adult and children’s tickets were sold.

Answer 3

8𝑥 + 4( 250 - 𝑥 ) = 1800
8𝑥 + 1000 -4𝑥 = 1800
4𝑥 + 1000 = 1800
4𝑥 = 800
𝑥 = 200
200 adult tickets were sold
50 children’s tickets were sold

Question 4

In 10 years, Kate will be twice as old as Sarah. Currently, Kate is 5 years older than Sarah. Let 𝑥 represent Sarah’s current age. Write an equation to represent the situation and solve to find their current ages.

Answer 4

𝑥 + 5 + 10 = 2 ( 𝑥 + 10 )
𝑥 + 15 = 2𝑥 +20
15 - 20 = 2𝑥 - 𝑥
-5 = 𝑥
Sarah is currently -5 years old
Kate is currently 5 years old
(The result is impossible in this context)

Question 5

There is a two-digit natural number where the ones place is 3. If the tens place and ones place are switched, it is 27 units smaller than the original natural number. Find the original natural number.

Answer 5

The original number is: 10𝑥 + 3
The natural number whose tens and one place are switched is: 30 + 𝑥
30 + 𝑥 = 10𝑥 + 3 -27
30 + 𝑥 = 10𝑥 - 24
30 + 24 = 10𝑥 - 𝑥
54 = 9𝑥
𝑥 = 6
10 x 6 + 3 = 63
The original natural number is 63

Question 6

There is a two-digit natural number where the tens place is 5. If the tens place and ones place are switched, it is 18 units smaller than the original natural number. Find the original natural number.

Answer 6

The original number is: 50 + 𝑥
The natural number whose tens and one place are switched is: 10𝑥 + 5
50 + 𝑥 + 18 = 10𝑥 + 5
68 + 𝑥 = 10𝑥 + 5
68 - 5 = 10𝑥 - 𝑥
63 = 9𝑥
𝑥 = 7
50 + 7 = 57
The original natural number is 57

Question 7

Qibli departed from his house and walked to school on foot. 12 minutes later, his elder brother noticed Qibli had frogotten his homework on the kitchen table and chased him by bike. If Qibli walks at the speed of 80m/min and the speed of his brother’s bike is 200m/min, how many minutes does it take for his brother to catch up to him after he leaves home?

Answer 7

When Qibli’s brother catched up with Qibli after his departure, the distance he has travelled is: 80 ( 12 + 𝑥 ) m
The distance his brother travelled is: 200𝑥
80 (12 + 𝑥 ) = 200𝑥
960 + 80𝑥 = 200𝑥
960 = 200𝑥 - 80𝑥
960 = 120𝑥
𝑥 = 8
His Qibli’s brother catches up with Qibli 8 minuter later

Question 8

Bianca left home and headed to the station. 2 minutes later, her younger brother, Nico, left from the same house and ran after her. If Bianca walks at the speed of 60m/min and Nico runs after her at a speed of 70m/min, how many minutes does it take for him to catch up to her after she leaves home?

Answer 8

When Nico catched up with Bianca after her departure, the distance she has travelled is: 60 ( 2 + 𝑥 ) m
The distance Nico travelled is: 70𝑥
60 (2 + 𝑥 ) = 70𝑥
120 + 60𝑥 = 70𝑥
120 = 70𝑥 - 60𝑥
120 = 10𝑥
𝑥 = 12
Nico catches up with Bianca 12 minutes later

Question 9

To travel from Qibli’s home Athens to Thessaloniki at a speed of 80km/h and to go back from Thessaloniki to Qibli’s house at a speed of 100km/h created a time difference of 2 hours in the length of time it takes to travel between places. How many kilometers is it from Qibli’s house to Thessalonki?

Answer 9

The length of time from Qibli’s house to Thessaloniki is: 𝑥 / 80 hours
The length of time from Thessaloniki to Qibli’s house is: 𝑥 / 100 hours
𝑥 / 80 - 𝑥 / 100 = 2 hours
400 𝑥 / 80 - 400 𝑥 / 100 = 400 x 2
5𝑥 - 4𝑥 = 800
𝑥 = 800
It is 800 km from Qibli’s house to Thessaloniki