Linear Equations in One Variable Worksheet

Fill in the blanks Type Questions

1. Aarti’s mother recently suffered from a backache and so she visited an orthopaedic doctor. The distance between the hospital and her house is given by . When Aarti calculated, she is getting the same distance but she represents it as . So, the value of t is ____________.
   
   
2. It's Divya’s parents' wedding anniversary tomorrow and so she prepares a party with the help of a few family friends. She ordered the materials required for decoration. Now, the total price of materials is given by the equation “800x + 400”. However, Divya describes the same price as “300(x – 1) + 700”. So, the actual cost of the materials is Rs. _________
   
   Multiple Choice Questions
   
   
3. Amit’s sister Leena is playing with cube blocks having alphabets or numbers written on them.
   
   Leena placed 4 cubes in the order as given below:
   
   Amit put minus sign between 3y and 4. What must be put between 4 and 5 to make this a linear equation?
   (a) Plus sign
   (b) Minus sign
   (c) Equal to sign
   (d) Number 8
   
   
4. To make the lecture interesting, a teacher divides the class into 4 groups and make chits of three categories: numbers, letters and operators. Every group has to pick three chits from each category and form a linear equation where it is not necessary to use all the chits. The four equations formed by the four groups are:
   Group 1: 2x² + z = 8, Group 2: x + 4 = 10, Group 3: 9t⁵ – 16 = 0,
   Group 4:  + 7 = 21
   Which of the following group has correctly formed the linear equation?
   (a) Group 1
   (b) Group 2
   (c) Group 3
   (d) Group 4
   
   
5. A shopping mall has a wide selection of clothing and footwear. When a discount is available, the total number of shoes sold and the number of garments sold are equal. If 900 clothes were sold and the number of shoes sold is given by the expression “7y – 3”, which of the following value of y, correctly gives the number of shoes sold?
   
   (a) 100
   (b) 113
   (c) 120
   (d) 129
   
   
6. Jay recently bought a smartwatch and set a target of walking 1000 steps every day.
   He completes 2x steps till 2 pm and remaining 400 steps after 2 pm. If we represent this situation in the form of an equation, which of the following value(s) of x fits in the equation formed?
   (a) 200
   (b) 250
   (c) 300
   (d) 350
   
   
7. Ansh’s father is an electrician and he has got a contract to install 10 bulbs in each flat and 2 bulbs in each lift of a building. There are 19 floors (excluding the ground floor) and ‘z’ flats on each floor. The ground floor has 4 flats and 3 lifts. If the linear equation 190z + 46 = 806 correctly explains this situation, then which of the following value of z fits in the equation properly?
   (a) 0
   (b) 2
   (c) 3
   (d) 4
   
   
8. A chocolate box has a capacity of holding 250 chocolates in it. Akash ate 15 chocolates in 5 days and there are x chocolates left in the box.
   
   Which of following the linear equation correctly explains the given situation?
   (a) x² – 75 = 250
   (b) 250 – 2x = 75
   (c) 75 + x = 250
   (d) 250 + 75 = x
   
   
9. Atharva has this habit of twisting around the things. So, when Vishal asked him his height, he said is can be expressed as . Vishal copied Atharva and said his height can be expressed as . But when measured, both of them were of same height.
   What could be their common height?
   (a) 4.5 feet
   (b) 5 feet
   (c) 5.5 feet
   (d) 6 feet
   
   
10. The same size shoes are worn by Meera and her mother. One her mother’s birthday, Meera went to buy a pair of shoes for her mother without telling her.
    
    Now, the size of the shoe Meera purchased is listed as  but she is actually wearing a shoe of size . What is the actual size of the shoes that both Meera and her mother wear?
    (a) 6
    (b) 5
    (c) 4
    (d) 3
    
    
11. Let there be ‘x’ plums in 1 kg. Aryan ordered  kg of plum and Ritika ordered  kg of plum.
    
    When measured, the quantity of plums bought by Ritika is the same as that of Aryan. How many plums will be there in 1 kg?
    (a) 18
    (b) 16
    (c) 12
    (c) 4
    
    
12. Anita walks for around half an hour every day and covers a distance of x km. But when asked by her mother, Anita said it’s the value of the unknown used in the equation . How many km does she walk for every day?
    (a) 1 km
    (b) 1.4 km
    (c) 2 km
    (d) 2.6 km
    
    Subjective Questions
    
    
13. Meesha, who is five years old, weighs only 11 kg, which is incredibly low for her age.
    
    Her weight is given by the expression “2x – 3”. However, according to her mother, if we substitute x as 8, in the given expression, we obtain Meesha’s precise weight. Is she correct? Explain your answer.
    
    
14. Neha is in 8th standard and the number of periods conducted in her school is constant for each day. So, she has to take the same number of books each day.
    
    The number of books she has to take can be obtained given by the equation “23 – 2t = 3“. Here, ‘t’ represents the number of books she carries every day. Can the equation above correctly calculate Neha's book count if she carries 10 books? Justify your answer.
    
    
15. Three friends, Shanaya, Anwi, and Ketan, are playing a game in which two of them use an equation to explain their ages and the third person must determine what the unknown is.
    
    Now that Shanaya and Anwi are of the same age, they refer to them as 3x – 2 and 2x + 3 accordingly. Ketan then determines that the value of the unknown, x, is 5. Check to see if Ketan is right.
    
    
16. A stationary shop has two pencil boxes of different colors having the same number of pencils in them.
    
    The number of pencils in the red box is  and that in the yellow box is . How many pencils will each box have?
    
    
17. Being a weekend, a bank has 250 more footfalls compared to weekdays. There are “4z + 2” people in the bank whereas on weekdays, the average number of people in the bank is “6 + 2z”. What is value of the unknown used here?
    
    
    
18. Arpit and Suresh work in different companies but have the same salary. As they didn’t want each other to know about their salaries, so describe it as:
    Arpit says it is Rs. 1000(m + 1)
    Suresh says it is Rs. (m + 10)
    Find their salaries.
    
    
19. Leena has only ‘y’ days left for the exam. The number of hours she studies on the 1^st day is  and that on the 2nd day is .
    
    When looked back, she found that the number of hours spent on day 1 and day 2 are same. Find the number of days left for the exam.
    
    
20. The dimensions of Arushi's home are 'f' metres in both length and breadth. However, when the boundaries are taken into account, the length is calculated to be 0.25 (4f – 7) and the breadth, which is equal to the length, to be 0.05(10f + 15).
    
    Then, find the length of the house.

Fill in the blanks Type Questions

1. Aarti’s mother recently suffered from a backache and so she visited an orthopaedic doctor. The distance between the hospital and her house is given by . When Aarti calculated, she is getting the same distance but she represents it as . So, the value of t is ____________.
   
   Correct Answer: 2
   Solution:
   
   (t – 18) = 4(2 – 3t)
   ⇒ t – 18 = 8 – 12t
   ⇒ 13t = 26
   ⇒ t = 2
   
   
2. It's Divya’s parents' wedding anniversary tomorrow and so she prepares a party with the help of a few family friends. She ordered the materials required for decoration. Now, the total price of materials is given by the equation “800x + 400”. However, Divya describes the same price as “300(x – 1) + 700”. So, the actual cost of the materials is Rs. _________
   
   Correct Answer: 400
   Solution:
   Since, the price of materials given by the equation “800x + 400” is same as described by Divya.
   ⇒ 800x + 400 = 300(x – 1) + 700
   ⇒ 8x + 4 = 3(x – 1) + 7
   ⇒ 8x + 4 = 3x – 3 + 7
   ⇒ 5x = 0
   ⇒ x = 0
   ⇒ 800x + 400 = 400
   So, the actual cost of the materials is Rs. 400.
   
   Multiple Choice Questions
   
   
3. Amit’s sister Leena is playing with cube blocks having alphabets or numbers written on them.
   
   Leena placed 4 cubes in the order as given below:
   
   Amit put minus sign between 3y and 4. What must be put between 4 and 5 to make this a linear equation?
   (a) Plus sign
   (b) Minus sign
   (c) Equal to sign
   (d) Number 8
   
   Correct Option: (c) Equal to Sign
   Solution:
   When Amit put minus sign, it is given by:
   3y – 4   5
   An equation must have an equality i.e. an equal to sign.
   So, it must be 3y – 4 = 5
   Hence, equal to sign must be put between 4 and 5 to make this a linear equation.
   
   
4. To make the lecture interesting, a teacher divides the class into 4 groups and make chits of three categories: numbers, letters and operators. Every group has to pick three chits from each category and form a linear equation where it is not necessary to use all the chits. The four equations formed by the four groups are:
   Group 1: 2x² + z = 8, Group 2:  x + 4 = 10, Group 3: 9t⁵ – 16 = 0,
   Group 4:  + 7 = 21
   Which of the following group has correctly formed the linear equation?
   (a) Group 1
   (b) Group 2
   (c) Group 3
   (d) Group 4
   
   Correct Option: (b) Group 2
   Solution:
   A linear equation has highest power of the variable as 1.
   So, the equation formed by group 2 i.e.  x + 4 = 10 is a linear equation.
   
   
5. A shopping mall has a wide selection of clothing and footwear. When a discount is available, the total number of shoes sold and the number of garments sold are equal. If 900 clothes were sold and the number of shoes sold is given by the expression “7y – 3”, which of the following value of y, correctly gives the number of shoes sold?
   
   (a) 100
   (b) 113
   (c) 120
   (d) 129
   
   Correct Option: (d) 129
   Solution:
   Since, the total number of shoes sold and the number of garments sold are equal.
   ⇒ 7y – 3 = 900
   When y = 100 ⇒ LHS = 7×100 – 3 = 697 ≠ RHS
   When y = 113 ⇒ LHS = 7×113 – 3 = 791 – 3 = 788 ≠ RHS
   When y = 120 ⇒ LHS = 7×120 – 3 = 840 – 3 = 837 ≠ RHS
   When y = 129 ⇒ LHS = 7×129 – 3 = 903 – 3 = 900 = RHS
   Hence, y = 129 correctly gives the number of shoes sold.
   It can be solved in this way as well:
   7y – 3 = 900
   ⇒ 7y = 900 + 3
   ⇒ 7y = 903
   ⇒ y = 129
   
   
6. Jay recently bought a smartwatch and set a target of walking 1000 steps every day.
   He completes 2x steps till 2 pm and remaining 400 steps after 2 pm. If we represent this situation in the form of an equation, which of the following value(s) of x fits in the equation formed?
   (a) 200
   (b) 250
   (c) 300
   (d) 350
   
   Correct Option: (c) 300
   Solution:
   Target set is to complete 1000 steps.
   He completes 2x steps and then the remaining 400 steps.
   ⇒ 2x + 400 = 1000
   When x = 200 ⇒ LHS = 2×200 + 400 = 800 ≠ RHS
   When x = 250 ⇒ LHS = 2×250 + 400 = 900 ≠ RHS
   When x = 300 ⇒ LHS = 2×300 + 400 = 1000 = RHS
   Hence, x = 300 fits in the equation formed.
   
   
7. Ansh’s father is an electrician and he has got a contract to install 10 bulbs in each flat and 2 bulbs in each lift of a building. There are 19 floors (excluding the ground floor) and ‘z’ flats on each floor. The ground floor has 4 flats and 3 lifts. If the linear equation 190z + 46 = 806 correctly explains this situation, then which of the following value of z fits in the equation properly?
   (a) 0
   (b) 2
   (c) 3
   (d) 4
   
   Correct Option: (d) 4
   Solution:
   190z + 46 = 806
   When z = 0 ⇒ LHS = 0 + 46 ≠ RHS
   When z = 2 ⇒ LHS = 190×2 + 46 = 380 + 46 ≠ RHS
   When z = 3 ⇒ LHS = 190×3 + 46 = 570 + 46 ≠ RHS
   When z = 4 ⇒ LHS = 190×4 + 46 = 760 + 46 = 806 = RHS
   Hence, z = 4 fits in the equation properly.
   
   
8. A chocolate box has a capacity of holding 250 chocolates in it. Akash ate 15 chocolates in 5 days and there are x chocolates left in the box.
   
   Which of following the linear equation correctly explains the given situation?
   (a) x² – 75 = 250
   (b) 250 – 2x = 75
   (c) 75 + x = 250
   (d) 250 + 75 = x
   
   Correct Option: (c) 75 + x = 250
   Solution:
   Total number of chocolates = 250
   Akash ate 75 and so x chocolates are left.
   ⇒ 75 + x = 250
   
   
9. Atharva has this habit of twisting around the things. So, when Vishal asked him his height, he said is can be expressed as . Vishal copied Atharva and said his height can be expressed as . But when measured, both of them were of same height.
   What could be their common height?
   (a) 4.5 feet
   (b) 5 feet
   (c) 5.5 feet
   (d) 6 feet
   
   Correct Option: (b) 5 feet
   Solution:
   Atharva and Vishal are of same height.
   
   ⇒ 5(2x + 6) = 7x + 45
   ⇒ 10x + 30 = 7x + 45
   ⇒ 3x = 15
   ⇒ x = 5
   Hence, both Atharva and Vishal are 5 feet tall.
   
   
10. The same size shoes are worn by Meera and her mother. One her mother’s birthday, Meera went to buy a pair of shoes for her mother without telling her.
    
    Now, the size of the shoe Meera purchased is listed as  but she is actually wearing a shoe of size . What is the actual size of the shoes that both Meera and her mother wear?
    (a) 6
    (b) 5
    (c) 4
    (d) 3
    
    Correct Option: (a) 6
    Solution:
    Since, the same size shoes are worn by Meera and her mother.
    
    ⇒ 6x - 8 = 3x + 10
    ⇒ 6x – 3x = 10 + 8
    ⇒ 3x = 18
    ⇒ x = 6
    
    
11. Let there be ‘x’ plums in 1 kg. Aryan ordered  kg of plum and Ritika ordered  kg of plum.
    
    When measured, the quantity of plums bought by Ritika is the same as that of Aryan. How many plums will be there in 1 kg?
    (a) 18
    (b) 16
    (c) 12
    (c) 4
    
    Correct Option: (b) 16
    Solution:
    As the quantity of plums bought by Ritika is the same as that of Aryan.
    
    
    ⇒ 5(5x – 8) = 2(11x + 4)
    ⇒ 25x – 40 = 22x + 8
    ⇒ 25x – 22x = 40 + 8
    ⇒ 3x = 48
    ⇒ x = 16
    So, there will be 16 plums in 1 kg.
    
    
12. Anita walks for around half an hour every day and covers a distance of x km. But when asked by her mother, Anita said it’s the value of the unknown used in the equation . How many km does she walk for every day?
    (a) 1 km
    (b) 1.4 km
    (c) 2 km
    (d) 2.6 km
    
    Correct Option: (b) 1.4 km
    Solution:
    
    
    
    
    ⇒ 3(m + 1) = 2(5 – m)
    ⇒ 3m + 3 = 10 – 2m
    ⇒ 3m + 2m = 10 – 3
    ⇒ 5m = 7
    ⇒ m = 1.4
    Anita said the distance she covered is the value of the unknown used in the above equation.
    ⇒ x = m
    ⇒ x = 1.4
    Hence, she walk for 1.4 km every day.
    
    Subjective Questions
    
    
13. Meesha, who is five years old, weighs only 11 kg, which is incredibly low for her age.
    
    Her weight is given by the expression “2x – 3”. However, according to her mother, if we substitute x as 8, in the given expression, we obtain Meesha’s precise weight. Is she correct? Explain your answer.
    
    Solution:
    Meesha’s weight is 11 kg which is expressed as “2x – 3”
    ⇒ 2x – 3 = 11
    Take x = 8
    ⇒ LHS = 2×8 – 3 = 16 – 3 = 13 ≠ 11
    So, Meesha’s mother is not correct.
    
    
14. Neha is in 8th standard and the number of periods conducted in her school is constant for each day. So, she has to take the same number of books each day.
    
    The number of books she has to take can be obtained given by the equation “23 – 2t = 3“. Here, ‘t’ represents the number of books she carries every day. Can the equation above correctly calculate Neha's book count if she carries 10 books? Justify your answer.
    
    Solution:
    Neha carry 10 books each day.
    Also, number of books can be obtained by the equation “23 – 2t = 3”.
    Take t = 10
    ⇒ LHS = 23 – 2×10 = 23 – 20 = 3 = RHS
    So, the equation given above correctly explains the number of books.
    
    
15. Three friends, Shanaya, Anwi, and Ketan, are playing a game in which two of them use an equation to explain their ages and the third person must determine what the unknown is.
    
    Now that Shanaya and Anwi are of the same age, they refer to them as 3x – 2 and 2x + 3 accordingly. Ketan then determines that the value of the unknown, x, is 5. Check to see if Ketan is right.
    
    Solution:
    Since, Shanaya and Anwi are of the same age.
    ⇒ 3x – 2 = 2x + 3
    ⇒ 3x – 2x = 3 + 2
    ⇒ x = 5
    Hence, Ketan is right.
    
    
16. A stationary shop has two pencil boxes of different colors having the same number of pencils in them.
    
    The number of pencils in the red box is  and that in the yellow box is . How many pencils will each box have?
    
    Solution:
    Number of pencils in the two boxes are same.
    
    
    
    ⇒ 3y = 
    ⇒ 3y = 21
    ⇒ y = 7
    
    
17. Being a weekend, a bank has 250 more footfalls compared to weekdays. There are “4z + 2” people in the bank whereas on weekdays, the average number of people in the bank is “6 + 2z”. What is value of the unknown used here?
    
    
    Solution:
    Since, the number of people in the bank on weekends is 250 more than on weekdays.
    ⇒ 4z + 2 = 6 + 2z + 250
    ⇒ 4z + 2 = 2z + 256
    ⇒ 4z – 2z = 256 – 2
    ⇒ 2z = 254
    ⇒ z = 127
    
    
18. Arpit and Suresh work in different companies but have the same salary. As they didn’t want each other to know about their salaries, so describe it as:
    Arpit says it is Rs. 1000(m + 1)
    Suresh says it is Rs. (m + 10)
    Find their salaries.
    
    Solution:
    As Arpit and Suresh have same salary.
    ⇒ 1000(m + 1) = (m + 10)
    ⇒ m + 1 =  (m + 10)
    ⇒ 5(m + 1) = 4(m + 10)
    ⇒ 5m + 5 = 4m + 40
    ⇒ 5m – 4m = 40 – 5
    ⇒ m = 35
    ⇒ 1000(35 + 1) = Rs. 36000
    Hence, Arpit’s salary is Rs. 36,000 and that of Surresh is also Rs. 36,000.
    
    
19. Leena has only ‘y’ days left for the exam. The number of hours she studies on the 1^st day is  and that on the 2nd day is .
    
    When looked back, she found that the number of hours spent on day 1 and day 2 are same. Find the number of days left for the exam.
    
    Solution:
    As per Leena, the number of hours spent on day 1 and day 2 are same.
    
    
    
    
    ⇒ 2(21 – 10y) = 17 – 15y
    ⇒ 42 – 20y = 17 – 15y
    ⇒ 42 – 17 = 20y – 15y 
    ⇒ 25 = 5y
    ⇒ y = 5
    So, 5 days are left for the exam.
    
    
20. The dimensions of Arushi's home are 'f' metres in both length and breadth. However, when the boundaries are taken into account, the length is calculated to be 0.25 (4f – 7) and the breadth, which is equal to the length, to be 0.05(10f + 15).
    
    Then, find the length of the house.
    
    Solution:
    As the length and breadth are equal even when the boundaries are considered.
    ⇒ 0.25 (4f – 7) = 0.05 (10f + 15)
    ⇒  (4f-7)=10f+15
    ⇒ 5(4f – 7) = 10f + 15
    ⇒ 20f – 35 = 10f + 15
    ⇒ 20f – 10f = 15 + 35
    ⇒ 10f = 50
    ⇒ f = 5
    As the length and breadth are ‘f’ metres each.
    So, the length of the house is 5 metre.