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Worksheet: Identifying Linear Relationships

In this worksheet, we will practice solving problems by identifying linear patterns in concepts.

Q1:

The graph shows the relationship between distance and time in a 160-meter race between a hare and a tortoise. Determine the winner, and then find the average speeds of the hare and the tortoise.

  • Athe hare, 6 / m m i n , 2 2 7 / m m i n
  • Bthe hare, 1 7 9 / m m i n , 2 2 7 / m m i n
  • Cthe tortoise, 1 1 5 / m m i n , 2 2 7 / m m i n
  • Dthe tortoise, 1 7 9 / m m i n , 2 2 7 / m m i n

Q2:

A school organised a trip where there is a fixed cost of 3 1 7 p o u n d s and the rest of the cost is dependent on the number of students attending. The total cost for a trip with 3 5 p o u n d s students is 5 9 7 p o u n d s . Find the total cost of the trip for 60 students given the relationship between the total cost and the number of students is linear.

  • A 1 3 4 0 p o u n d s
  • B 4 8 0 p o u n d s
  • C 8 6 0 p o u n d s
  • D 7 9 7 p o u n d s

Q3:

A school organised a trip where there is a fixed cost of 7 8 p o u n d s and the rest of the cost is dependent on the number of students attending. The total cost for a trip with 1 0 p o u n d s students is 4 2 8 p o u n d s . Find the total cost of the trip for 25 students given the relationship between the total cost and the number of students is linear.

  • A 1 1 4 8 p o u n d s
  • B 8 7 5 p o u n d s
  • C 2 7 3 p o u n d s
  • D 9 5 3 p o u n d s

Q4:

The given figure shows the relation between the covered distance 𝑑 in kilometers and the elapsed time 𝑡 in hours for two trains, A and B, between two railroad stations. Find the average speed of each train.

  • Aaverage speed of train A = 6 0 / k m h , average speed of train B = 4 5 / k m h
  • Baverage speed of train A = 4 5 / k m h , average speed of train B = 3 3 3 4 / k m h
  • Caverage speed of train A = 4 0 / k m h , average speed of train B = 4 5 / k m h
  • Daverage speed of train A = 4 5 / k m h , average speed of train B = 4 5 / k m h
  • Eaverage speed of train A = 5 4 / k m h , average speed of train B = 3 3 3 4 / k m h

Q5:

The given figure shows the relation between the covered distance 𝑑 in kilometers and the elapsed time 𝑡 in hours for two trains, A and B, between two railroad stations. Find the average speed of each train.

  • Aaverage speed of train A = 4 0 / k m h , average speed of train B = 6 0 / k m h
  • Baverage speed of train A = 6 0 / k m h , average speed of train B = 2 6 2 3 / k m h
  • Caverage speed of train A = 8 0 / k m h , average speed of train B = 2 6 2 3 / k m h
  • Daverage speed of train A = 6 0 / k m h , average speed of train B = 6 0 / k m h

Q6:

A boy was reading a book and found out that after reading for 2 hours he still had 344 pages left and after reading for 3 hours he had 43 pages left to read. Find the linear equation between the time 𝑡 and the number of pages left to read 𝑏 .

  • A 𝑏 = 3 0 1 𝑡 − 6 4 5
  • B 𝑏 = 3 0 1 𝑡 + 6 4 5
  • C 𝑏 = − 𝑡 − 6 4 5
  • D 𝑏 = − 3 0 1 𝑡 + 9 4 6

Q7:

A boy was reading a book and found out that after reading for 7 hours he still had 256 pages left and after reading for 8 hours he had 57 pages left to read. Find the linear equation between the time 𝑡 and the number of pages left to read 𝑏 .

  • A 𝑏 = 1 9 9 𝑡 − 4 5 5
  • B 𝑏 = 1 9 9 𝑡 + 4 5 5
  • C 𝑏 = − 𝑡 − 4 5 5
  • D 𝑏 = − 1 9 9 𝑡 + 1 6 4 9

Q8:

Which of the following is the same as the function 𝑔 ( 𝑥 ) = 2 𝑥 − 3 ?

  • AThis is not a function.
  • B 𝑥 = 2 𝑥 − 3
  • C 𝑦 = 2 𝑦 − 3
  • D 𝑦 = 2 𝑥 − 3
  • E 𝑧 = 2 𝑧 − 3

Q9:

Daniel’s favorite drink is hot chocolate; he drinks it whenever it rains. As shown in the table, the number of cups of hot chocolate he drinks in a week is twice the number of rainy days that week. Which of the following statements is true?

Number of Rainy Days 1 2 3 4
Number of Cups 2 4 6 8
  • AThe dependent variables are the number of cups of hot chocolate he drinks and the number of rainy days.
  • BThe dependent variable is the number of rainy days, and the independent variable is the number cups of hot chocolate that he drinks.
  • CThe independent variables are the number of cups of hot chocolate he drinks and the number of rainy days.
  • DThe dependent variable is the number of cups of hot chocolate that he drinks, and the independent variable is the number of rainy days.

Q10:

If 𝑓 ( 𝑥 ) is a linear function, find an equation for it given 𝑓 ( 4 ) = − 1 and 𝑓 ( 9 ) = 2 .

  • A 𝑓 ( 𝑥 ) = − 1 7 5 + 5 3 𝑥
  • B 𝑓 ( 𝑥 ) = − 1 7 5 𝑥 + 3 5
  • C 𝑓 ( 𝑥 ) = − 1 7 5 − 3 5 𝑥
  • D 𝑓 ( 𝑥 ) = − 1 7 5 + 3 5 𝑥

Q11:

Which of the following functions increases at a higher rate?

  • A(a)
  • B(b)

Q12:

Find the correct rule for the numbers in the table.

𝑥 2 4 8 10
𝑦 8 16 32 40
  • A Divide 𝑥 by 4 to find 𝑦 .
  • B Add 2 to 𝑥 to find 𝑦 .
  • C Subtract 2 from 𝑥 to find 𝑦 .
  • D Multiply 𝑥 by 4 to find 𝑦 .
  • E Double 𝑦 to find 𝑥 .

Q13:

Chloe and Noah are professional cyclists training for an upcoming race. The given graph represents the distance Chloe traveled during one training session. The distance traveled by Noah is given by the function 𝑑 = 1 8 𝑡 , where 𝑑 represents the distance in miles and 𝑡 represents the time in hours. Which of them is traveling faster?

  • AChloe
  • BNoah

Q14:

David is going to do a sponsored bike ride to raise money for charity. His uncle has promised to donate $12 more than the total amount that David raises. The graph and table show the relationship between the amount of money his uncle will donate ( 𝑑 ) and the amount of money that he raises ( 𝑚 ) . Which of the following statements is true?

Amount raised ( 𝑚 ) 12 24 36 48
Amount Donated ( 𝑑 ) 24 36 48 60
  • AThe independent variable is the amount of money that his uncle donates.
  • BThe dependent variable is the amount of money that David raises.
  • CThe independent variables are the amount of money that his uncle donates and the amount of money that David raises.
  • DThe dependent variable is the amount of money that his uncle donates.
  • EThe dependent variables are the amount of money that his uncle donates and the amount of money that David raises.

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