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Worksheet: Applications of Linear Equations

In this worksheet, we will practice describing a real-world situation involving a linear relationship using a straight line equation.

Q1:

If the top speed of a car is 231 miles per hour, write an equation using two variables that shows the relationship between the number of miles π‘š the car can travel in β„Ž hours, and then find the distance, in miles, the car would travel in 2 hours.

  • A π‘š = 2 3 1 β„Ž , 233 mi
  • B β„Ž = 2 3 1 π‘š , 462 mi
  • C β„Ž = 2 3 1 π‘š , 233 mi
  • D π‘š = 2 3 1 β„Ž , 462 mi
  • E π‘š = 2 3 1 + β„Ž , 233 mi

Q2:

A cell phone provider charges a customer $74 for each month of service. Write a formula for the total cost 𝑐 after π‘š months of cell phone service. Then find the total cost for 3 months of cell phone service.

  • A 𝑐 = 7 4 π‘š , $77
  • B π‘š = 7 4 𝑐 , $222
  • C π‘š = 7 4 𝑐 , $77
  • D 𝑐 = 7 4 π‘š , $222
  • E 𝑐 = 7 4 + π‘š , $77

Q3:

In 1995, music stores sold cassette tapes for $2. Write an equation to find 𝑑 , the total cost in dollars for buying 𝑐 cassette tapes, and then find how much it would cost to buy 3 cassette tapes.

  • A 𝑑 = 2 + 𝑐 , $5
  • B 𝑐 = 2 𝑑 , $6
  • C 𝑐 = 2 + 𝑑 , $5
  • D 𝑑 = 2 𝑐 , $6
  • E 𝑑 = 2 βˆ’ 𝑐 , $3

Q4:

In the equations below, 𝑀 represents the width of a rectangle and 𝑙 represents the length.

Which equation represents the statement β€œThe width of a rectangle is 12 less than twice its length”?

  • A 𝑀 = 𝑙 2 βˆ’ 1 2
  • B 𝑀 = 1 2 βˆ’ 2 𝑙
  • C 𝑀 = 1 2 βˆ’ 𝑙 2
  • D 𝑀 = 2 𝑙 βˆ’ 1 2
  • E 𝑀 = 2 ( 𝑙 βˆ’ 1 2 )

Q5:

A plumber charges $ 4 0 for a call-out and $ 6 0 per hour of labor.

Write an equation for 𝐢 , the cost of a job, in dollars, which takes 𝑑 hours.

  • A 𝐢 = 1 0 0 𝑑
  • B 𝐢 = 4 0 𝑑 + 6 0
  • C 𝐢 = 6 0 𝑑 βˆ’ 4 0
  • D 𝐢 = 6 0 𝑑 + 4 0
  • E 𝐢 = 4 0 𝑑 βˆ’ 6 0

What is the cost of a job which takes 3 hours?

Q6:

A pet rescue center relies on donations to look after the cats and dogs. It costs $ π‘₯ per week to feed a cat and $ 𝑦 per week to feed a dog.

There are 65 cats and 45 dogs in the center, and the total weekly food bill for them is $ 6 8 5 .

Write an equation connecting π‘₯ and 𝑦 .

  • A 4 5 π‘₯ + 6 5 𝑦 = 6 8 5
  • B 6 5 π‘₯ βˆ’ 4 5 𝑦 = 6 8 5
  • C 4 5 π‘₯ βˆ’ 6 5 𝑦 = 6 8 5
  • D 6 5 π‘₯ + 4 5 𝑦 = 6 8 5
  • E 1 1 0 π‘₯ 𝑦 = 6 8 5

Use the equation to find the weekly cost of feeding a dog, given that it costs $ 5 per week to feed a cat.

Q7:

Write an equation that describes the statement β€œThe value of 𝑦 is the same as when five times π‘₯ is added to 14.”

  • A 𝑦 = 5 + 1 4 π‘₯
  • B 𝑦 = 1 4 βˆ’ 5 π‘₯
  • C 𝑦 = 5 βˆ’ 1 4 π‘₯
  • D 𝑦 = 1 4 + 5 π‘₯
  • E 𝑦 = 5 π‘₯ βˆ’ 1 4

Q8:

There were 𝑝 passengers on a bus. At a stop, 𝑙 people got off the bus, and nobody got on. The bus continued on its journey with 27 passengers.

Write an equation connecting 𝑝 and 𝑙 .

  • A 𝑝 βˆ’ 2 𝑙 = 2 7
  • B 𝑝 + 𝑙 = 2 7
  • C 𝑝 + 2 𝑙 = 2 7
  • D 𝑝 βˆ’ 𝑙 = 2 7
  • E 𝑝 𝑙 = 2 7

If there were originally 31 people on the bus, what is the value of 𝑙 ?

Q9:

A group of friends are going to a theme park. An adult ticket costs $70 and a child ticket costs $50. There are π‘₯ adults and 𝑦 children in the group, and the total cost for the group’s tickets is $710.

Write an equation connecting π‘₯ and 𝑦 .

  • A 5 0 π‘₯ + 7 0 𝑦 = 7 1 0
  • B 7 0 π‘₯ βˆ’ 5 0 𝑦 = 7 1 0
  • C 5 0 π‘₯ βˆ’ 7 0 𝑦 = 7 1 0
  • D 7 0 π‘₯ + 5 0 𝑦 = 7 1 0
  • E 1 2 0 π‘₯ 𝑦 = 7 1 0

Q10:

Amelia is a distance runner and she is helping Mason, her nephew, to train for a race. Amelia can maintain a speed of π‘₯ m/s and Mason can maintain a speed of 𝑦 m/s.

Amelia gives Mason a 100-meter head start. They start running at the same time and she overtakes Mason after 50 seconds. Write an equation for 𝑦 in terms of π‘₯ .

  • A 𝑦 = π‘₯ + 1 2
  • B 𝑦 = π‘₯ + 2
  • C 𝑦 = π‘₯ βˆ’ 1 2
  • D 𝑦 = π‘₯ βˆ’ 2
  • E 𝑦 = 2 βˆ’ π‘₯

Q11:

Write an equation that describes the statement β€œThe value of 𝑦 is the same as when 7 is added to four times π‘₯ and the result is multiplied by 4.”

  • A 𝑦 = 4 ( 7 π‘₯ + 4 )
  • B 𝑦 = 4 Γ— 4 π‘₯ + 7
  • C 𝑦 = 4 Γ— 7 π‘₯ + 4
  • D 𝑦 = 4 ( 4 π‘₯ + 7 )
  • E 𝑦 = 4 π‘₯ + 7

Q12:

Sophia has $ 1 0 in her bank account. Every week she will deposit $ 2 0 into the account. Write an equation that represents this situation, where 𝑇 is the total money in her account after 𝑀 weeks.

  • A 𝑇 = 2 0 𝑀 βˆ’ 1 0
  • B 𝑇 = 1 0 𝑀 + 2 0
  • C 𝑇 = 1 0 𝑀 βˆ’ 2 0
  • D 𝑇 = 2 0 𝑀 + 1 0
  • E 𝑇 = 2 0 𝑀 + 3 0

Q13:

Write an equation to represent the statement β€œWhen four times 𝑦 is added to seven times π‘₯ , the result is 14.”

  • A 7 π‘₯ βˆ’ 4 𝑦 = 1 4
  • B 4 π‘₯ + 7 𝑦 = 1 4
  • C 4 π‘₯ βˆ’ 7 𝑦 = 1 4
  • D 7 π‘₯ + 4 𝑦 = 1 4
  • E 1 1 ( π‘₯ + 𝑦 ) = 1 4

Q14:

Jacob read 9 pages of a novel each hour. Write an equation using two variables that can determine how many pages 𝑝 he read in β„Ž hours, and then find how many pages Jacob read in 9 hours.

  • A 𝑝 = 9 β„Ž , 18 pages
  • B β„Ž = 9 𝑝 , 81 pages
  • C β„Ž = 9 𝑝 , 18 pages
  • D 𝑝 = 9 β„Ž , 81 pages
  • E β„Ž = 9 + 𝑝 , 18 pages

Q15:

Chloe and Amelia are baking cupcakes for an event. So far, Chloe has baked 24 and plans to bake 6 per hour, while Amelia has baked 4 and plans to bake 11 per hour. Write and solve an equation that can determine how many hours will have passed before Chloe and Amelia have made the same number of cupcakes.

  • A 6 π‘₯ βˆ’ 2 4 = 1 1 π‘₯ + 4 , 4 hours
  • B 6 π‘₯ + 2 4 = 1 1 π‘₯ + 4 , 5 hours
  • C 6 π‘₯ βˆ’ 2 4 = 1 1 π‘₯ + 4 , 5 hours
  • D 6 π‘₯ + 2 4 = 1 1 π‘₯ + 4 , 4 hours
  • E 6 π‘₯ + 2 4 = 1 1 π‘₯ , 4 hours

Q16:

Write an equation that describes the statement β€œThe value of 𝑦 is 4 less than three times the value of π‘₯ .”

  • A 𝑦 = 3 π‘₯ + 4
  • B 𝑦 = 4 π‘₯ βˆ’ 3
  • C 𝑦 = 4 π‘₯ + 3
  • D 𝑦 = 3 π‘₯ βˆ’ 4
  • E 𝑦 = π‘₯ βˆ’ 4

Q17:

Write an equation that describes the statement β€œWhen 2 is subtracted from 𝑦 , the result is the same as when 4 is added to π‘₯ , and the result is multiplied by 3.”

  • A 𝑦 = 3 ( π‘₯ + 4 ) βˆ’ 2
  • B 𝑦 βˆ’ 2 = 3 π‘₯ + 4
  • C 𝑦 = ( 3 π‘₯ + 4 ) βˆ’ 2
  • D 𝑦 βˆ’ 2 = 3 ( π‘₯ + 4 )
  • E 𝑦 + 2 = 3 ( π‘₯ + 4 )

Q18:

Some final-year students are planning a leaving party. They will hire a disco and provide refreshments for each student attending the party. The cost of the disco is Β£ 3 5 0 , and refreshments will cost Β£ 3 per student. Write a formula for the total cost 𝐢 in pounds, if 𝑛 students attend the party.

  • A 𝐢 = 3 5 0 𝑛 + 3
  • B 𝐢 = 3 5 0 + 𝑛
  • C 𝐢 = 3 𝑛
  • D 𝐢 = 3 5 0 + 3 𝑛
  • E 𝐢 = 1 0 5 0 𝑛

Q19:

In a coffee shop, Jason was charged Β£ 1 7 . 1 0 for 4 coffees and 3 sandwiches. Let the cost of a coffee be Β£ 𝑐 and the cost of a sandwich be Β£ 𝑠 . Write an equation in terms of 𝑐 and 𝑠 for the order.

  • A 1 2 𝑐 𝑠 = 1 7 . 1
  • B 𝑐 + 𝑠 = 1 7 . 1
  • C 7 𝑐 𝑠 = 1 7 . 1
  • D 4 𝑐 + 3 𝑠 = 1 7 . 1
  • E 3 𝑐 + 4 𝑠 = 1 7 . 1

Q20:

Suppose that the average annual income, in dollars, for the years 1990 through 1999 is given by the linear function 𝐼 ( π‘₯ ) = 1 0 5 4 π‘₯ + 2 3 2 8 6 , where π‘₯ is the number of years after 1990. Which of the following interprets the slope in the context of the problem?

  • AIn the ten-year period from 1990–1999, the average annual income increased by a total of $ 1 0 5 4 .
  • BThe average annual income rose to a level of $ 2 3 2 8 6 by the end of 1999.
  • CAs of 1990, the average annual income was $ 2 3 2 8 6 .
  • DEach year in the decade of the 1990s, the average annual income increased by $ 1 0 5 4 .

Q21:

An electrician charges a call-out fee and an hourly labor charge.

The graph represents what the electrician charges in dollars for jobs of different durations.

What is the electrician’s call-out fee?

What is the hourly labor charge?

Let 𝑦 be the cost in dollars for a job that takes π‘₯ hours. Write an equation for 𝑦 in terms of π‘₯ .

  • A 𝑦 = 6 0 π‘₯ + 1 0 0
  • B 𝑦 = 4 0 π‘₯ + 6 0
  • C 𝑦 = 6 0 π‘₯ + 4 0
  • D 𝑦 = 6 0 π‘₯ βˆ’ 4 0
  • E 𝑦 = 4 0 π‘₯ βˆ’ 6 0

Q22:

In the game of musical chairs, players walk around a group of chairs while music plays. When the music stops, the players sit on the chairs. As there is one fewer chair than the number of players, one player cannot find a chair and is then eliminated from the game. At the next round, a chair is removed and so on until there is only one player left. What is the formula for the number 𝑁 of the remaining players after π‘Ÿ rounds with 20 players in total?

  • A 𝑁 = 2 0 βˆ’ 2 π‘Ÿ
  • B 𝑁 = 1 0 βˆ’ 2 π‘Ÿ
  • C 𝑁 = 2 0 βˆ’ 3 π‘Ÿ
  • D 𝑁 = 2 0 βˆ’ π‘Ÿ
  • E 𝑁 = 1 0 βˆ’ π‘Ÿ

Q23:

For journeys between 10 pm and 7 am, a cab company charges a fixed fare of $4.50 plus 50 cents per fifth of a mile.

Let 𝑓 represent the fare in dollars for a journey of 𝑑 miles.

Write an equation for 𝑓 in terms of 𝑑 .

  • A 𝑓 = 2 . 5 + 4 . 5 𝑑
  • B 𝑓 = 4 . 5 + 0 . 5 𝑑
  • C 𝑓 = 0 . 5 + 4 . 5 𝑑
  • D 𝑓 = 4 . 5 + 2 . 5 𝑑
  • E 𝑓 = ( 4 . 5 + 0 . 5 ) 𝑑

Q24:

There are crayons and pens in a pencil case. In total, there are 20 items in the pencil case. Let 𝑐 represent the number of crayons and 𝑝 the number of pens. Write an equation connecting 𝑐 and 𝑝 .

  • A 𝑐 𝑝 = 2 0
  • B 𝑐 βˆ’ 𝑝 = 2 0
  • C 𝑐 𝑝 = 2 0
  • D 𝑐 + 𝑝 = 2 0
  • E 𝑝 𝑐 = 2 0

Q25:

The given graphs represent Matthew’s bike rides last week, where 𝑑 represents the distance from his home and 𝑑 the duration in hours. Select a suitable graph for each of the following scenarios. Note that some graphs may apply to more than one scenario.

  1. Starts 5 km from home and rides 5 km per hour away from home.
  2. Starts 5 km from home and rides 10 km per hour away from home.
  3. Starts 10 km from home and arrives home one hour later.
  4. Starts 10 km from home and is halfway home after one hour.
  5. Starts 5 km from home and is 10 miles from home after one hour.
  • A(1)-(ii), (2)-(i), (3)-(iv), (4)-(v), (5)-(ii)
  • B(1)-(ii), (2)-(v), (3)-(i), (4)-(iv), (5)-(ii)
  • C(1)-(iv), (2)-(i), (3)-(v), (4)-(ii), (5)-(ii)
  • D(1)-(ii), (2)-(i), (3)-(v), (4)-(iv), (5)-(ii)
  • E(1)-(ii), (2)-(i), (3)-(v), (4)-(iv), (5)-(i)

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