Linear Equations Flashcards
Sophie bought 3 chocolate bars and 2 packets of crisps for a total of Β£5.50. A chocolate bar costs Β£1.20 more than a packet of crisps. Let π₯ 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.
2π₯ + 3( π₯ + 1.2 ) = 5.5
3x + 3.6 + 2x = 5.5
5π₯ + 3.6 = 5.5
5π₯ = 1.9
π₯ = 0.38
A chocolate bar costs Β£1.58
A packet of crisps costs Β£0.38
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 π₯ be the fraction of the wall Tom paints per hour, and π¦ the fraction Jerry paints per hour. Write an equation to represent the work done and solve for their individual rates of painting.
Lisa went shopping and bought 5 pens and 3 notebooks. The total cost was Β£9.50. The cost of each pen is Β£1.50 more than the cost of each notebook. Let π₯ represent the cost of a notebook. Write an equation for this situation and solve to find the cost of a pen and a notebook.
3π₯ + 5( π₯ + 1.5 ) = 9.5
3π₯ + 5π₯ + 7.5 = 9.5
8π₯ + 7.5 = 9.5
8π₯ = 2
π₯ = 0.25
A notebook costs Β£0.25
A pen costs Β£1.75
A theater sold 250 tickets for a show. Adult tickets cost Β£8, and childrenβs tickets cost Β£4. The total revenue from ticket sales was Β£1,800. Let π₯ represent the number of adult tickets sold. Write a linear equation to represent this situation and solve it to find how many adult and childrenβs tickets were sold.
8π₯ + 4( 250 - π₯ ) = 1800
8π₯ + 1000 -4π₯ = 1800
4π₯ + 1000 = 1800
4π₯ = 800
π₯ = 200
200 adult tickets were sold
50 childrenβs tickets were sold
In 10 years, Kate will be twice as old as Sarah. Currently, Kate is 5 years older than Sarah. Let π₯ represent Sarahβs current age. Write an equation to represent the situation and solve to find their current ages.
π₯ + 5 + 10 = 2 ( π₯ + 10 )
π₯ + 15 = 2π₯ +20
15 - 20 = 2π₯ - π₯
-5 = π₯
Sarah is currently -5 years old
Kate is currently 5 years old
(The result is impossible in this context)
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.
The original number is: 10π₯ + 3
The natural number whose tens and one place are switched is: 30 + π₯
30 + π₯ = 10π₯ + 3 -27
30 + π₯ = 10π₯ - 24
30 + 24 = 10π₯ - π₯
54 = 9π₯
π₯ = 6
10 x 6 + 3 = 63
The original natural number is 63
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.
The original number is: 50 + π₯
The natural number whose tens and one place are switched is: 10π₯ + 5
50 + π₯ + 18 = 10π₯ + 5
68 + π₯ = 10π₯ + 5
68 - 5 = 10π₯ - π₯
63 = 9π₯
π₯ = 7
50 + 7 = 57
The original natural number is 57
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 brotherβs bike is 200m/min, how many minutes does it take for his brother to catch up to him after he leaves home?
When Qibliβs brother catched up with Qibli after his departure, the distance he has travelled is: 80 ( 12 + π₯ ) m
The distance his brother travelled is: 200π₯
80 (12 + π₯ ) = 200π₯
960 + 80π₯ = 200π₯
960 = 200π₯ - 80π₯
960 = 120π₯
π₯ = 8
His Qibliβs brother catches up with Qibli 8 minuter later
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?
When Nico catched up with Bianca after her departure, the distance she has travelled is: 60 ( 2 + π₯ ) m
The distance Nico travelled is: 70π₯
60 (2 + π₯ ) = 70π₯
120 + 60π₯ = 70π₯
120 = 70π₯ - 60π₯
120 = 10π₯
π₯ = 12
Nico catches up with Bianca 12 minutes later
To travel from Qibliβs home Athens to Thessaloniki at a speed of 80km/h and to go back from Thessaloniki to Qibliβs 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 Qibliβs house to Thessalonki?
The length of time from Qibliβs house to Thessaloniki is: π₯ / 80 hours
The length of time from Thessaloniki to Qibliβs house is: π₯ / 100 hours
π₯ / 80 - π₯ / 100 = 2 hours
400 π₯ / 80 - 400 π₯ / 100 = 400 x 2
5π₯ - 4π₯ = 800
π₯ = 800
It is 800 km from Qibliβs house to Thessaloniki
How many months would it take Nico to save up enough money to buy a car that costs $7,200 if he has $900 and can save $450 per month?
It will take him14 months
What is the lowest grade Angela can score on the next test to achieve an average of at least 90% if her current scores are 92, 86, and 88?
The lowest score she can score is 94%
What is the original two-digit number if the number in the tens place is 5 and switching the tens and ones places makes it 18 units bigger?
57
How many years will it take for Qibliβs fatherβs age to be 3 times Qibliβs age if Qibli is currently 7 and his father is 41?
10
How many years ago was Nicoβs age a quarter of his motherβs age if Nico is currently 15 and his mother is 45?
5
How many years later will Annaβs mother be 3 times Annaβs age if Anna is currently 12 and her mother is 44?
4
How many years later will Angelaβs sister be twice Angelaβs age if Angela is currently 7 and her sister is 17?
3
What is the original two-digit number if the number in the ones place is 3 and switching the tens and ones places makes it 27 units smaller?
63
What is the original two-digit number if the number in the ones place is 7 and switching the tens and ones places makes it 9 units bigger?
67
What is the original two-digit number if the number in the tens place is 8 and switching the tens and ones places makes it 36 units smaller?
84
What is the value of x if 8 times x divided by 9 gives a quotient of 5 and a remainder of 3?
6
What is the value of x if 7 times x divided by 9 gives an answer of 5 and the remainder is 3 smaller than x?
7
What is the value of x if 12 times x divided by 7 gives an answer of 6 and the remainder is 2 larger than x?
4
How many donuts did Anna buy if they cost $1.50 each and the box cost $0.50, totaling $11?
7
What is the price of 1 box of caramels if Qibli had $5, bought 5 boxes of caramels and 1 box of gum at $0.40, and received $0.60 in change?
$0.80
What is the price of 1 pencil if Angela had $10, bought 7 pencils and 2 pens at $1.30 each, and received $1.10 in change?
$0.90
How many cakes should Nico give to Angela if Nico bought 28 cakes and Angela bought 12 cakes, and he wants her to have 6 more than him?
11
How many apples did Nico buy if he bought 20 bananas and apples for a total of $26, with bananas costing $1.20 each?
10
How many peaches did Anna buy if she bought 10 peaches and pears for a total of $13.80, with peaches costing $1.50 each and pears $1.20 each?
6 peaches
What is the price of 1 pencil if the total price of 3 pencils and 1 protractor at $3 is 4 times the total price of 1 pencil and 1 eraser at $0.60?
0.80
What is the price of 1 cookie if the price of 5 cookies and 1 cupcake at $3 is twice the total price of 3 cookies and 2 chocolates at $0.40 each?
$2.20
What is the price of 1 orange if the total price of 8 oranges and 1 apple at $1.20 is double the total price of 1 orange and 1 pear at $1.80?
$0.60
