Pair of linear equations in two variables — Question Bank

Q1 1 mark MCQ p. 104

Given below is a graph showing two lines that never intersect. These are represented by two linear equations.Which of these can be said about the number of solution(s) of the above pair of linear equations?

Reveal official answer

Correct option: 3 — They do not have a solution.

Q2 1 mark MCQ p. 104

Given below is a pair of linear equations in two variables.4x + 2y = 183x − 6y = 6Which of the following pairs of equations have the same number of solution(s) as the given pair?

Reveal official answer

Correct option: 3 — 6a − 2b = 10; 3a + b = 5

Q3 1 mark MCQ p. 105

If a pair of linear equations given by l₁x + m₁y + n₁ = 0 and l₂x + m₂y + n₂ = 0 has infinitely many solutions, then which of the following is DEFINITELY true?

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Correct option: 4 — l₁m₂ = l₂m₁

Q4 1 mark MCQ p. 105

A gardener bought a mix of 100 flower and vegetable seeds for a total of Rs 1350. Each flower seed costs Rs 12, and each vegetable seed costs Rs 11.Which of the following pairs of linear equations can be used to determine f, the number of flower seeds purchased, and v, the number of vegetable seeds purchased?

Reveal official answer

Correct option: 3 — f + v = 100; 12f + 11v = 1350

Q5 1 mark MCQ p. 105

Sara collected a total of Rs 1800 in a fundraising event. She knew that the event had a mix of Rs 10 and Rs 50 notes, but not sure how many of each. She counted the total number of notes as 60.Which of the following pairs of linear equations can be used to find the number of 10-rupee and 50-rupee notes?(Note: x represents the number of 10-rupee note and y represents the number of 50-rupee note.)

Reveal official answer

Correct option: 4 — x + y = 60; 10x + 50y = 1800

Q6 1 mark MCQ p. 105

Tanisha and Aditya have some chocolates with them such that:♦ if Tanisha were to give 6 chocolates to Aditya, the new quantity of chocolates with each of them would be equal.♦ instead, if Aditya were to give 3 chocolates to Tanisha, then Tanisha would have four times as many chocolates as Aditya initially had.Which of these pairs of equations would help us find the number of chocolates that they have?(Note: Assume the initial number of chocolates with Tanisha as 'x' and that with Aditya as 'y'.)

Reveal official answer

Correct option: 2 — x − 6 = y + 6; x + 3 = 4y

Q7 1 mark MCQ p. 106

For the given pair of linear equations, two statements are given below - one labelled Assertion (A) and the other labelled Reason (R). Read the statements carefully and choose the option that correctly describes statements (A) and (R).x − 2y + 3 = 03x + 4y − 11 = 0Assertion (A): The pair of linear equations has a unique solution.Reason (R): The pair of linear equations represents a pair of coincident lines.

Reveal official answer

Correct option: 3 — (A) is true but (B) is false.

Q8 1 mark Written p. 106

Given below is a pair of linear equations in two variables:6y + 7z = 12; 12y − 12z = 24Which variable's coefficient can be changed such that the given pair has infinitely many solutions? What should it be changed to? Show your work.

Reveal official answer

• Identifies that either of the coefficients of z can be changed. [0.5]

• Changes either 7 to (−6) in first equation or (−12) to 14 in second equation so that all ratios are equal to ½, making it a pair with infinitely many solutions. [0.5]

Q9 1 mark Written p. 106

On a specific day, a budget-friendly restaurant managed to sell 1000 vegetarian meals. These vegetarian meals were priced at Rs 85 per adult and Rs 45 per child. A total of Rs 73000 was collected from these sales.If p represents the number of adult veg meals sold and q represents the number of child veg meals sold, write a pair of equations to find out how many meals of each kind were sold.

Reveal official answer

• Expresses equations in this or any equivalent form:

p + q = 1000

85p + 45q = 73000 [1]

Q10 1 mark Written p. 106

A pair of linear equations is shown below, where b is an integer.2x − y = b + 1x + (b − 1)y = 3bFor any given value of b, how many solution(s) does this pair of equations have? Justify your answer.

Reveal official answer

• Identifies that for any integer b:

2/1 ≠ −1/(b − 1)

This satisfies the condition for intersecting lines, hence there is a unique solution. [1]

Q11 2 marks Written p. 107

Shown below is an image where the lines represent the paths of sight of two people standing at different heights and looking at the bottom level of the buildings. Both lines of sight can be represented by the corresponding linear equations.5x + 2y = 143x − y = 4(Note: The figure is not to scale.)Find the ordered pair that will represent the intersecting point of their lines of sight. Show your work.

Reveal official answer

• Solves the following pair of linear equations to get x = 2 and y = 2:

5x + 2y = 14

3x − y = 4 [1.5]

• Writes that (2, 2) represents the intersecting point of both lines of sight. [0.5]

Q12 2 marks Written p. 107

Nisha and Samarth are preparing a cup of coffee each by mixing two ingredients, milk and brewed coffee in different quantities satisfying the following conditions:♦ The quantity of milk in Nisha's cup is twice the quantity of brewed coffee in her cup.♦ The quantity of milk in Samarth's cup is four times the quantity of brewed coffee in his cup.♦ The quantity of milk in Nisha's cup is 40 ml less than what is found in Samarth's cup.♦ The quantity of brewed coffee in Nisha's cup exceeds Samarth's by 30 ml.Represent the above situation in the form of a pair of linear equations in two variables. Show your work.

Reveal official answer

• Writes that if quantity of brewed coffee = x ml for Nisha, then quantity of milk = 2x ml for Nisha.

Writes that if quantity of brewed coffee = y ml for Samarth, then quantity of milk = 4y ml for Samarth. [1]

• Represents both the equations in this or an equivalent form:

4y − 2x = 40

x − y = 30 [1]

Q13 2 marks Written p. 108

The delivery fees of a delivery service company consists of a fixed fee in addition to a fee based on the distance travelled. For a delivery covering 20 kms, the total fee (fixed + variable fee) is Rs 300, and for a delivery spanning 25 kms, the total fee is Rs 350.How much total amount would a customer need to pay for a delivery that covers a distance of 49 kms? Show your work.

Reveal official answer

• Assumes the fixed fee as Rs x and the variable fee as Rs y, then formulates the following pair of linear equations:

x + 20y = 300

x + 25y = 350 [1]

• Solves the pair of linear equations to find the value of x as Rs 100 and y as Rs 10. [0.5]

• Finds the final total amount as:

100 + (10 × 49) = Rs 590 [0.5]

Q14 2 marks Written p. 108

Different shades of purple are obtained by mixing different quantities of red and blue colours.An artist combined 5 litres of red paint with 7 litres of blue paint to achieve a shade of purple, incurring a cost of Rs 5000. To achieve a different shade of purple, she mixed 7 litres of red paint and 5 litres of blue paint, incurring a cost of Rs 4600.Calculate the price of red and blue paint per litre. Show your work.

Reveal official answer

• Translates the given information to frame a pair of linear equations in two variables as:

5x + 7y = 5000

7x + 5y = 4600

where x = price of red paint per litre; y = price of blue paint per litre [1]

• Solves the pair of equations obtained in the above step to find the price of red and blue paints per litre as Rs 300 and Rs 500 respectively. [1]

Q15 2 marks Written p. 108

In a chemistry lab, scientists are studying a chemical reaction between two substances, Substance A and Substance B. The following was known:♦ The total mass of the substances before the reaction was 9 grams.♦ The difference in mass between Substance A and Substance B before the reaction was 6 grams. Determine the mass of Substances A and B before the reaction. Show your work.

Reveal official answer

• Assumes the mass of Substance A as x grams and that of Substance B as y grams. Represents the given situation in linear equations as:

x + y = 9

x − y = 6 [1]

• Solves the above pair of equations and finds the mass of Substance A as 15/2 grams and that of Substance B as 3/2 grams.

(Award full marks if the mass of Substance A is 3/2 grams and that of Substance B is 15/2 grams.) [1]

Q16 2 marks Written p. 108

Tanvi and her friend Vanshika both made purchases from a local fruit seller on a specific day. Tanvi bought 3 kgs of grapes and 4 kgs of oranges, spending a total of Rs 680. Meanwhile, Vanshika bought 4 kgs of grapes and 2 kgs of oranges from the same fruit seller, spending a total of Rs 640.What is the price of grapes and oranges per kg? Show your work.

Reveal official answer

• Assumes the price of grapes per kg as Rs x and that of oranges as Rs y. Formulates the following pair of linear equations:

3x + 4y = 680

4x + 2y = 640 [1]

• Solves the above pair of linear equations to find the price of grapes and oranges per kg as Rs 120 and Rs 80 respectively. [1]

Q17 3 marks Written p. 108

A two-digit number is such that the sum of its digits is 11. When the digits are reversed, the resulting number increases by 27.Determine the original two-digit number. Show your work.

Reveal official answer

• Assumes the digit at the tens place and ones place as x and y respectively.

Writes the equation as:

x + y = 11

10y + x = 10x + y + 27 [1]

• Solves the above equations correctly to find the values of x and y as 4 and 7 respectively. [1]

• Finds the original two-digit number as 47. [1]

Q18 3 marks Written p. 108

Two real numbers c and d satisfy the following equations:2c − 3d = 74c + d = 1Find the product of c and d. Show your work.

Reveal official answer

• Solves the given pair of linear equation to find the values of c and d as 5/7 and −13/7 respectively. [2]

• Finds the product of c and d as:

5/7 × (−13/7) = −65/49 [1]

Q19 3 marks Written p. 109

Solve the following pair of linear equations in two variables graphically.x + 3y = 62x − 3y = 12Identify the shape resulting from the intersection of the pair of equations with the y-axis and write its vertex coordinates.

Reveal official answer

• Finds at least two points that satisfy the linear equation, x + 3y = 6. For example,

x: 0, 6

y: 2, 0

Finds at least two points that satisfy the linear equation, 2x − 3y = 12. For example,

x: 0, 6

y: −4, 0 [1.5]

• Draws the graph for each equation using the coordinates found in the above two steps and thus finds the solution as x = 6 and y = 0. [1]

• Identifies the shape as a triangle and write its vertices as (0, 2), (6, 0) and (0, −4). [0.5]

Q20 3 marks Written p. 109

The length of a rectangle is 3 cm less than five times the width. The sum of six times the length and two times the width is equal to 46 cm.What is the width of the rectangle? Show your work.

Reveal official answer

• Assumes the width and length of the rectangle as w and x respectively. Expresses the above statement as a pair of linear equations:

x = 5w − 3

6x + 2w = 46 [1]

• Solves the above pair of linear equations to find the width of the rectangle as 2 cm. [2]

Q21 3 marks Written p. 109

In the competitive world of smartphone data plans, two leading telecom companies, TechConnect and SwiftLink offer distinct pricing structures. TechConnect charges a base monthly fee of Rs 300, along with an additional Rs 15 for each gigabyte (GB) of data used beyond the initial 5 GB included in the plan. In contrast, SwiftLink offers a different pricing model with a flat monthly fee of Rs 600 for unlimited data usage.i) Express the data plan structure for TechConnect in the form of linear equation. Use c as the total cost (in Rs) and d as the data usage (in GB).ii) At what point of data usage the cost of a data plan with TechConnect becomes equal to the cost of the data plan with SwiftLink?Show your steps.

Reveal official answer

• i) Expresses the data plan structure for TechConnect in the form of an equation as:

c = 300 + 15(d − 5)

=> c = 15d + 225

(Award full marks if any other correct variation of the equation is written.) [1]

• ii) Identifies that the cost of both plans will become equal when c becomes 600:

600 = 15d + 225 [1]

• Solves the above equation and concludes that at data usage of 25 GB, the cost for both the plans will become equal. [1]

Q22 3 marks Written p. 109

Rahul rode his bike initially at an average speed of 40 km/h. Upon noticing a road sign indicating a speed limit of 35 km/h, he slowed down and rode at an average speed of 35 km/h for the remainder of his journey. He covered a total of 190 km in 5 hours.For how long did Rahul maintain an average speed of 40 km/h? Show your work.

Reveal official answer

• Assumes the time duration for which Rahul rode his bike at an average speed of 40 km/h to be x and at 35 km/h to be y.

Frames the pair of linear equations in two variables as:

x + y = 5

40x + 35y = 190 [1]

• Solves the above pair of linear equations to find x = 3 and y = 2. [1.5]

• Writes that Rahul rode his bike at an average speed of 40 km/h for 3 hours. [0.5]

Q23 5 marks Written p. 109

A geometric shape is formed by the equation 2y + x = 8 and the coordinate axes. For the resulting shape,i) Identify the shape and find its vertices.ii) Find the perimeter and area of the shape.Solve graphically.

Reveal official answer

• Finds at least two points that satisfy the linear equation, 2y + x = 8. For example,

x: 0, 8

y: 4, 0 [2]

• i) Identifies the shape as a triangle and writes the vertices as (0, 4), (0, 0) and (8, 0). [1]

• ii) Finds the perimeter of triangle using pythagoras theorem as 4 + 8 + √(4² + 8²) = 12 + 4√5 units. [1]

• Finds the area of the triangle as ½ × 4 × 8 = 16 sq units. [1]

Q24 5 marks Written p. 110

Muskan lives 12 km away from her college. She walks to the metro station and takes a metro to college everyday. If she goes to the nearest metro station, she needs to walk for 2 km and cover the rest by metro. This takes her 1 hour. If she walks to a metro station farther away, she needs to walk for 4 km and cover the rest by metro. This takes her 1.5 hours.Find the average speeds of Muskan's walking and the metro. Show your work.

Reveal official answer

• Takes average speed while walking and average speed of the metro as x km/h and y km/h respectively. [0.5]

• Uses speed = distance/time to frame equations 2/x + 10/y = 1 and 4/x + 8/y = 1.5 respectively. [1.5]

• Substitutes 1/x = m and 1/y = n, where m and n are also variables. Rewrites the equations as:

2m + 10n = 1

4m + 8n = 1.5 [1]

• Solves the above equations to find the values of m and n as 7/24 and 1/24 respectively.

Expresses the average speed of Muskan as x = 24/7 km/h and average speed of the metro as y = 24 km/h. [2]

Q25 1 mark Written p. 110

Answer the questions based on the given information.Reena and Sonia went to a mall on 14th November 2022. On the occasion of Children's Day, the mall was offering movie tickets and bowling alley tickets at discounted prices. The cost of a movie ticket was Rs 70 more than twice the cost of a bowling alley ticket. Sonia purchased five bowling alley tickets and three movie tickets for a total of Rs 870.Represent the given situation with a pair of linear equations.

Reveal official answer

• Assumes the price of one bowling alley ticket as Rs x and price of one movie ticket as Rs y.

Frames the pair of linear equations as:

y − 2x = 70

5x + 3y = 870 [1]

Q26 2 marks Written p. 110

Answer the questions based on the given information.Reena and Sonia went to a mall on 14th November 2022. On the occasion of Children's Day, the mall was offering movie tickets and bowling alley tickets at discounted prices. The cost of a movie ticket was Rs 70 more than twice the cost of a bowling alley ticket. Sonia purchased five bowling alley tickets and three movie tickets for a total of Rs 870.Find the price of a movie ticket and a bowling alley ticket. Show your work.

Reveal official answer

• Solves the equations simultaneously to find the price of a movie ticket as Rs 190 and a bowling alley ticket as Rs 60. [2]

Q27 2 marks Written p. 110

Answer the questions based on the given information.Reena and Sonia went to a mall on 14th November 2022. On the occasion of Children's Day, the mall was offering movie tickets and bowling alley tickets at discounted prices. The cost of a movie ticket was Rs 70 more than twice the cost of a bowling alley ticket. Sonia purchased five bowling alley tickets and three movie tickets for a total of Rs 870.On the next day, Reena took her siblings to the mall. She observed that prices of the movie ticket and the bowling alley ticket had come back to their standard rates of Rs 220 and Rs 90, respectively. She bought a total of 10 tickets, costing her Rs 1420.Find the number of movie tickets and bowling alley tickets she bought. Show your work.

Reveal official answer

• Assumes the number of movie tickets and bowling alley tickets to be m and b respectively.

Frames the pair of linear equations as:

m + b = 10

220m + 90b = 1420 [1]

• Solves the equations to find the number of movie tickets (m) and bowling alley tickets (b) to be 4 and 6 respectively. [1]