Chapter 3 : Pair of Linear Equations in Two Variables – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 3 : Pair of Linear Equations in Two Variables Important Questions with Solutions and Answers

Chapter 3 : Pair of Linear Equations in Two Variables

SECTION = A

Question: If x = 1 and y = 2 is a solution of the pair of linear equations 2x – 3y + a = 0 and 2x + 3y – b = 0, then : [2025 Standard]

(A) a = 2b (B) 2a = b (C) a + 2b = 0 (D) 2a + b = 0

Solution: Given, x = 1 and y = 2 is a solution of the pair of linear equations 2x – 3y + a = 0 and 2x + 3y – b = 0 .

and

]

Question: The value of ‘p’ for which the equations andhas infinitely many solutions is : [2025 Standard]

(a) – 6 (b) 6 only (c) (d) Any real number except .

Solution: (c) .

Given,

]

Question: If the system of equations : 3x+2y = 4 ; 4ax +(a+b)y = 16 has infinitely many solutions, then [2025 Standard]

(a) 5a = 3b (b) 3a = 5b (c) a + b = 15 (d) a – b = 2

Solution: (a) 5a = 3b

[ Given ,

Here,

]

Question: The value of k, if (6,k) lies on the line represented by x – 3y + 6 = 0 , is [2023 Basic]

(a) – 4 (b) 12 (c) – 12 (d) 4

Solution: (d) 4

[ Here,

]

Question: The pair of linear equations has : [2023 basic]

(a) a unique solution (b) exactly two solutions (c) infinitely many solutions (d) no solution

Solution: (d) no solution

[ Given,

Here,

So, the pair of linear equations has no solution. ]

Question: Consider the following pairs of linear equations : [SEBA 2020]

(i) ;

(ii) ;

Choose the correct alternative :

(a) The pair in (i) has no solution, whereas the pair in (ii) has unique solution .

(b) The pair in (i) has infinitely many solutions, whereas the pair in (ii) has no solution .

(d) The pair in (i) has no solution, whereas the pair in (ii) has infinitely many solutions .

Solution : (d) The pair in (i) has no solution, whereas the pair in (ii) has infinitely many solutions .

[ We have , ]

Question: If the point lies on the lines represented by both the equations and , then the lines is :

(a) intersecting (b) coincident (c) Parallel (d) None of these

Solution: (a) intersecting .

[ We have ,

Therefore, the lines are intersecting. ]

Question: The value of for which the pair of linear equations and represents parallel lines is :

(a) (b) (c) (d)

Solution: (a) .

[ We have,

]

Question: Consider the following pairs of linear equations :[SEBA 2019]

(i) ,

(ii) ,

Choose the correct alternative .

(a) The pairs in (i) and (ii) are consistent .

(b) The pairs in (i) and (ii) are inconsistent .

(d) The pair in (i) is consistent, whereas the pair in (ii) is inconsistent .

Solution: (d) The pair in (i) is consistent, whereas the pair in (ii) is inconsistent .

[ We have , ]

Question: If the lines and are coincident , then the value of is :

(a) (b) (c) – 11 (d) – 7

Solution: (a)

[ We have , ]

Question: If , is the solution of the equations and , then the values of and are respectively :

(a) 6 , – 1 (b) 2 , 3 (c) 4 , 1 (d)

Solution: (c) 4 , 1

[ Here , and

We have ,

and

from

From , we get ]

Question: A pair of linear equations ; is said to be inconsistent, if

(a) (b) (c) (d)

Solution: (a)

Question: The graph of is a line parallel to the -

(a) – axis (b) – axis (c) both – axis and – axis (d) none of these

Solution: (b) – axis .

Question: The pair of linear equations and is : [CBSE 2020 standard]

(a) consistent (b) inconsistent (c) consistent with one solution (d) consistent with many solutions

Solution: (b) inconsistent .

[ We have ,

and

]

Question: The graph of is a line :

(a) parallel to – axis (b) perpendicular to – axis (c) parallel to – axis (d) passing through the origin .

Solution: (d) passing through the origin .

[ If x = 0 , then y = 4 × 0 = 0 . So, the line passes through (0,0), the origin. ]

Question: The lines representing the linear equations and are :

(a) intersect at a point (b) parallel (c) coincident (d) intersect at exactly two points .

Solution: (b) parallel .

[ We have , ]

Question: The pair of equations and graphically represents lines which are :

(a) Coincident (b) parallel (c) intersecting at (3,4) (d) intersecting at (4,3)

Solution: (d) intersecting at (4 , 3) .

Question: If pair of linear equations is consistent , then the lines represented by them are :[CBSE 2020 (Basic)]

(a) always coincident (b) parallel (c) always intersecting (d) intersecting or coincident.

Solution: (d) intersecting or coincident.

Question: Which of the following pair of linear equations is intersect at a point ?

(a) , (b) ,

Solution: (d) , .

[ (a) ,

(b) ,

(d) ,

So, the pair of linear equations is intersect at a point . ]

Question: If in the equation , the value of is 6, then the value of will be

(a) 2 (b) 6 (c) – 2 (d) 4

Solution: (c) – 2

[ Given, y = 6 ; We have, ]

Question: The solution of the pair of linear equations and are :

(a) (2 , 3) (b) (3 , 2) (c) (3 , – 2) (d) (2 , – 3)

Solution: (d) (2 , – 3)

[ We have,

and

{from (i)}

Putting in equation , we get ]

Question: If and is a solution of a pair of equations and , then the value of and are :

(a) 5 , 15 (b) 15 , 5 (c) – 5 , 15 (d) – 15 , 5

Solution: 5 and 15

[ Given , and

So,

and ]

Question: 10 students of class X took part in a Mathematics quiz . If the number of girls is 4 more than the number of boys , then the number of boys and girls who took part in the quiz are :

(a) 7 and 3 (b) 3 and 7 (c) 6 and 4 (a) 4 and 6

Solution: (b) 3 and 7 .

[ Let and be number of girls an boys respectively .

A/Q ,

And From

From , we get ]

Question: The value of for which the given pair of linear equations has infinitely many solutions is :

;

(a) 12 (b) – 12 (c) – 6 (d) 6

Solution: (a) 12

[ We have , and

From part and part , we get ]

Question: The value of so that the point ,lie on the line represented by .

(a) 2 (b) 6 (c) – 2 (d) – 6

Solution: (c) – 2

[ Here , ,

We have , ]

Question: Aruna has only Re 1 and Rs 2 coins with her. If the total number of coins that she has is 50 and the amount of money with her is Rs 75, then the number of Re 1 and Rs 2 coins are, respectively

(a) 35 and 15 (b) 35 and 20 (c) 15 and 35 (d) 25 and 25

Solution: (d) 25 and 25

[ Let the number of Rs 1 coins be x and Rs 2 coins be y.

A/Q,

and

From (i) we get, ]

Question: The father’s age is six times his son’s age. Four years hence, the age of the father will be four times his son’s age. The present ages, in years, of the son and the father are, respectively

(a) 4 and 24 (b) 5 and 30 (c) 6 and 36 (d) 3 and 24

Solution: (c) 6 and 36

[ Let, the son's present age be x years and the father's present age is 6x years.

Four years hence, Son's age and father's age

A/Q,

Therefore, the son's present age is 6 years and the father's present age is 6 × 6 = 36 years . ]

SECTION = B

Question: Solve the following pair of linear equations: 21x + 47y = 110 ; 47x + 21y = 162

Solution: We have,

and

From (iii) we get

Question: Find the number of solutions of the following pair of linear equations : [CBSE 2009]

and

Solution: We have , and

Here , , , , , ,

Thus, the pairs of linear equations are infinitely many solutions .

Qustion: Write whether the following pair of linear equations is consistent or inconsistent :

and

Solution: We have , ;

Here , , , c1=6 , , ,

Thus, the pairs of linear equations is consistent .

Question: Solve for and ( Using elimination method) :

;

Solution: We have ,

and

[ From (i)]

From we get ,

Therefore, x = 4 and y = 1 .

Question: Which of the following pairs of linear equations has unique solution , no solution , or infinitely many solutions ?

;

Solution: We have, and

Here , , , , , ,

Thus, the pairs of linear equations has infinitely many solutions .

Question: For what value of does the pair of equations given below has a unique solution ?

;

Solution: We have , and

Here , , , , , ,

Therefore, for all values of , except , the given pair of equations will have a unique solution .

Question: The cost of 2 kg apples and 1 kg of grapes on a day was found to be Rs 320 . The cost of 4 kg apples and 2 kg grapes was found to be Rs 600. If cost of 1 kg of apples and 1kg of grapes is Rs x and Rs y respectively , represent the given situation algebraically as a system of equations and check whether the system so obtained is consistent or not . [2025 Standard]

Solution: Given, the cost of 1 kg of apples and 1kg of grapes is Rs x and Rs y respectively .

A/Q,

and

, which is a false statement.

Therefore, the pair of equations has no solution (inconsistent).

Question: Five years ago, Nuri was thrice as old as sonu . Ten years later, Nuri will be twice as old as Sonu . How old are Nuri and Sonu ?

Solution: Let and be the age of Sonu and Nuri respectively .

Five years ago , the age of Sonu and Nuri will be and years respectively .

And Ten years later , the age of Sonu and Nuri will be and years respectively .

A/Q ,

and

[ From ]

From we get ,

Therefore , 20 years and 50 years are the age of Sonu and Nuri respectively .

Question: Solve : ; andhence find the value of for which .

Solution: We have ,

And

From we get

Therefore, the value of m is – 1 .

Question: In a , . Find the three angles .

Solution: Given,

and

In , we have

, ,

Question: The difference between two numbers is 26 and one number is three times the other . Find them.

Solution: Let and be the two number .

A/Q ,

And

[ From ]

From we get ,

Therefore , the two numbers are 39 and 13 respectively .

Question: Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m . Find the dimensions of the garden . [SEBA 2019]

Solution: Let and (in metres) are the length and width of the rectangular garden respectively .

A/Q ,

And

[ From ]

From we get ,

Therefore, 20 m and 16 m are the length and width of the rectangular garden respectively .

Question: For what value of will the following pair of linear equations has infinitely many solutions :

;

Solution: We have , ;

Here , , , , , ,

From 1st part and 2nd part , we get

From 2nd part and 3rd part , we get

Therefore , the value of is 6 .

Question: If the sum of two positive numbers is 44 and one number is three times the other number, then find the numbers.

Solution : Let and be the first and second numbers respectively .

AQ ,

and

Putting in equation , we get

Therefore, the two positive number are 33 and 11 respectively .

Question: Solve : ; [CBSE2011]

Solution: We have ,

and

Therefore, the value of and are 3 and 2 respectively .

Question: Solve the following pair of equations by substitution method :

;

Solution: We have ,

and

Putting in equation we get ,

Therefore , the solution are and .

Question: Use elimination method to find all possible solution of the following pair of linear equations :

;

Solution: We have ,

and

, which is a false statement .

Therefore , the pair of linear equations has no solution .

Question: Given the linear equation , write another linear equation in these two variables such that the geometrical representation of the pair so formed is : (i) intersecting lines (ii) parallel lines

Solution: Given, the linear equation is

Here,

(i) For intersecting lines:

We consider,

Now,

So, the another linear equation is .

(ii) Given, the linear equation is

Here,

For parallel lines:

We consider,

Now,

So, the linear equation is .

Question: Graphically , find whether the following pair of equations has no solution, unique solution or infinitely many solutions :

;

Solution: We have ,

and

Equation and are same . Hence , the lines represented by equation and are coincident .

Therefore , equation and have infinitely many solutions.

Question: Solve : ;

Solution: We have,

and

Therefore, the value of and are 2 and 1 respectively .

Question: 5 pencils and 7 pens together cost Rs. 50 , whereas 7 pencils and 5 pens together cost Rs. 46 .Find the cost of one pencil and that of one pen . [SEBA 2020]

Solution: let and be the cost of one pencil and one pen respectively .

A/Q ,

From we get ,

Therefore , the cost of one pencil and one pen are Rs. 3 and Rs. 5 respectively .

SECTION = C

Question: The perimeter of a rectangle is 70 cm . The length of the rectangle is 5 cm more than twice is breadth . Express the given situation as a system of linear equations in two variables and hence solve it . [2025 Standard]

Solution: Let, x and y be the length and breadth of the rectangle respectively (in cm) .

A/Q,

and

Putting in (i) we get

Therefore, the length and breadth of the rectangle are 25 cm and 10 cm respectively.

Question: The monthly incomes of two persons are in the ratio 9 : 7 and their monthly expenditures are in the ratio 4 : 3 . If each saved Rs 5000, express situation algebraically as a system of linear equations in two variables . Hence, find their respective monthly incomes . [2025 Standard]

Solution: Let, the incomes of the two person by Rs 9x and Rs 7x and their expenditures by Rs 4y and Rs 3y respectively.

A/Q,

and

From (i) we get

Therefore, the monthly incomes of the persons are Rs 9 × 5000 = Rs 45000 and Rs 7 × 5000 = Rs 35000 respectively.

Question: For what values of m and n , does the following pair of linear equations have infinitely many solutions ? [2025 Standard]

2x + 3y = 7 ; m( x + 2y ) + n( x – y ) = 21

Solution: Given, and

Here,

From (i) we get

Therefore, the value of .

Question: The sum of the numerator and the denominator of a fraction is 4 more than twice the numerator . If the numerator and denominator are increased by 3, they are in the ratio 2 : 3 . Determine the fraction.

Solution: Let, x and y be the numerator and denominator of the fraction respectively.

The fraction

A/Q,

and

[ from (i)]

From (i) we get

So, the fraction is .

Question: For which values of and does the following pair of linear equations have an infinitely number of solutions ?

;

Solution: We have,

Here,

Since, the pair of linear equation have an infinitely number of solutions .

So ,

and

From and we get,

From , we get

Therefore, the value of is and the value of is .

Question: Solve for and : ;

Solution: We have,

and

From we get ,

and .

Question: A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is a added to its denominator . Find the fraction . [CBSE 2020]

Solution : Let and be the numerator and denominator of the fraction respectively .

So, the fraction .

A/Q,

and

Putting in equation , we get

Therefore, the fraction is .

Question: A fraction becomes , if 2 is added to both the numerator and the denominator . If 3 is added to both the numerator and denominator it becomes . Find the fraction . [ SEBA 2016 ,20]

Solution : Let, and are the numerator and the denominator of the fraction respectively .

The fraction .

A/Q ,

and

Putting in equation , we get

Required the fraction is .

Question: Solve for and : [CBSE 2004 , 07C , 08]

;

Solution: We have,

and

Putting in equation , we have

Therefore, the solutions are and .

Question: There are some students in the two examination halls A and B. To make the number of students equal in each hall, 10 students are sent from A to B. But if 20 students are sent from B to A, the number of students in A becomes double the number of students in B. Find the number of students in the two halls.

Solution: Let x and y be the number of students in hall A and hall B respectively .

A/Q,

and

From (i) , we get

Therefore, the number of students in hall A is 100 and the number of students in hall B is 80 .

Question: A shopkeeper gives books on rent for reading. She takes a fixed charge for the first two days, and an additional charge for each day thereafter. Latika paid Rs 22 for a book kept for six days, while Anand paid Rs 16 for the book kept for four days. Find the fixed charges and the charge for each extra day.

Solution: Let x and y be the fixed charges and the charge for each extra day respectively (in Rs) .

A/Q,

and

From (i) we get ,

Therefore, the fixed charges is Rs 10 and the charge for each extra day is Rs 3 .

Question: In a competitive examination, one mark is awarded for each correct answer while mark is deducted for every wrong answer. Jayanti answered 120 questions and got 90 marks. How many questions did she answer correctly?

Solution: Let x and y be the number of correct answers and wrong answers respectively .

A/Q,

and

[From (i)]

From (i) we get

Jayanti answered 100 questions correctly.

SECTION = D

Question: Vijay had some bananas, and he divided them into two lots A and B. He sold the first lot at the rate of Rs 2 for 3 bananas and the second lot at the rate of Re 1 per banana, and got a total of Rs 400. If he had sold the first lot at the rate of Re 1 per banana, and the second lot at the rate of Rs 4 for 5 bananas, his total collection would have been Rs 460. Find the total number of bananas he had.

Solution: Let x and y be the number of bananas in lot A and B respectively.

Total bananas

A/Q,

and

From (i) we get

Therefore, the total number of bananas = x + y = 300 + 200 = 500 .

Question: Find the solution of the pair of equations and .Hence, find , if .

Solution: We have,

and

From (i) we get ,

Now,

Therefore, the value of

Question: The sum of the digits of a two-digit number is 9 . Also, nine times this number is twice the number obtained by reversing the order of the digits . Find the number .

Solution: Let and be the ten’s and the unit’s digits of the number respectively.

Therefore, the first number is and when the digits are reversed , then the number is .

A/Q,

And

Putting in equation , we get

Thus , the number .

Question: The taxi charges in a city consist of a fixed charge together with the charge for the distance covered . For a distance of 10 km , the charge paid is Rs. 105 and for a journey of 15 km, the charge paid is Rs. 155 . What are the fixed charges and the charge per km ? How much does a person have to pay for travelling a distance of 25 km ?

Solution: Let, and be the fixed charge and the charge per km respectively .

A/Q,

and

Putting in equation we get

Therefore, a person have to pay for travelling a distance of 25 km Rs. ( ) Rs.( ) Rs.( ) Rs.

Question: A two-digit number is obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digits by 16 and then adding 3. Find the number.

Solution: Let the ten’s and the unit’s digits in the number be x and y , respectively.

The number is

A/Q,

and

From (i) we get

Therefore, the number

Question: A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from the station A to B costs Rs 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B costs Rs 3810. Find the full first class fare from station A to B, and also the reservation charges for a ticket.

Solution: Let x be the full first class fare from station A to B and Rs y be the reservation charges per ticket .

A/Q,

and

From (i) we get

Therefore, the full first class fare = Rs 2500 and the reservation charges = Rs 30 .

SECTION = E

Question: Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received Rs 1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received Rs 20 more as annual interest. How much money did he invest in each scheme ? [2025 Standard]

Solution: Let, the amounts invested in schemes A and B be x and y respectively (in Rs).

A/Q,

and

Putting in (i) , we get

Therefore, the amounts invested in schemes A and B be Rs 12000 and Rs 10000 respectively .

Question: Jamila sold a table and a chair for Rs 1050, thereby making a profit of 10% on the table and 25% on the chair. If she had taken a profit of 25% on the table and 10% on the chair she would have got Rs 1065. Find the cost price of each.

Solution: Let x and y be the cost price of the table and the chair respectively .

A/Q,

and

Therefore, the cost price of the table is Rs 500 and the cost price of the chair is Rs 400.

Question: Draw the graphs of the equation and . Determine the coordinate of the vertices of the triangle formed by these lines and the x-axis, and shaded the triangular region.

Solution: We have ,

	– 1	0	1
	0	1	2

and

	4	0	2
	0	6	3

Plot the points A( – 1,0) , B(0,1) , C(1,2) ,D(4,0) , E(0,6) and F(2,3) on graph paper, and join the points to form the lines PQ and RS as shown in figure . We get the shaded triangle AFD with vertices A(– 1, 0) , F(2,3) and D(4,0) .

Question: The area of a rectangle gets reduces by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units . If we increase length by 3 units and breadth by 2 units , the area increases by 67 square units . Find the dimensions of the rectangle . [SEBA 2015 , 18]

Solution: let, and are the length and breadth of the rectangle respectively.

Therefore , the area of rectangle

A/Q ,

and

From , we get

Required the length and breadth are 17 unit and 9 unit respectively.

Question: Draw the graphs of the pair of linear equations x – y + 2 = 0 and 4x – y – 4 = 0. Calculate the area of the triangle formed by the lines so drawn and the x-axis.

Solution: We have,

	– 2	0	2
	0	2	4

and

	0	1	2
	– 4	0	4

Plot the points A (– 2 ,0), B (0 , 2), C (2 , 4) , D(0 , – 4) , E(1 , 0) and F(2, 4) on the graph paper and join them.

The triangle formed by these lines and the x- axis is ACE or AFE .

The vertices of this triangle are A (–2, 0), C (2, 4) and E (1, 0) .

Here, AE = 3 units , FM = 4 units .

Posted 5 years ago

Chapter 3 : Pair of Linear Equations in Two Variables – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 3 : Pair of Linear Equations in Two Variables Important Questions with Solutions and Answers

Chapter 3 : Pair of Linear Equations in Two Variables

SECTION = A

SECTION = B

SECTION = C

SECTION = D

SECTION = E

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