CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables is to check in this article. The main purpose of this article is to provide a CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables. Thisinformation is completely related to the Central Board Of Secondary Education. Visit once in this article and start your exam preparation. You can download the NCERT Solutions for Maths CBSE Class 10 Chapter 3 (सीबीएसई कक्षा 10 एनसीईआरटी मैथ्स अध्याय 3 के लिए समाधान), Pair of Linear Equations in Two Variables. You can download the CBSE Class 10 NCERT Solutions for Maths Chapter 3 PDF is also available here.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables

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CBSE Class 10 NCERT Solutions for Maths Chapter 3 Problems & Solutions

Question 1:

Aftab tells his daughter, “Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be.” (Isn’t this interesting?) Represent this situation algebraically and graphically.

ANSWER:

Let the present age of Aftab be x.

And, present age of his daughter = y

Seven years ago,

Age of Aftab = x − 7

Age of his daughter = y − 7

According to the question,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 1

Three years hence,

Age of Aftab = x + 3

Age of his daughter = y + 3

According to the question,

Therefore, the algebraic representation is

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 2

For ,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 3

The solution table is

x

− 7

0

7

y

5

6

7

For ,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 4

The solution table is

x

6

3

0

y

0

− 1

− 2

The graphical representation is as follows.

PAGE NO 44:

Question 3:

The cost of 2 kg of apples and 1 kg of grapes on a day was found to be Rs 160. After a month, the cost of 4 kg of apples and 2 kg of grapes is Rs 300. Represent the situation algebraically and geometrically.

ANSWER:

Let the cost of 1 kg of apples be Rs x.

And, cost of 1 kg of grapes = Rs y

According to the question, the algebraic representation is

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 5

For ,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 6

The solution table is

x

50

60

70

y

60

40

20

For 4x + 2y = 300,

The solution table is

x

70

80

75

y

10

−10

0

The graphical representation is as follows.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 7

PAGE NO 49:

Question 1:

Form the pair of linear equations in the following problems, and find their solutions graphically.

(i) 10 students of Class X took part in a Mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

(ii) 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.

ANSWER:

(i) Let the number of girls be and the number of boys be y.

According to the question, the algebraic representation is

x + y = 10

x − y = 4

For x + y = 10,

x = 10 − y

x

5

4

6

y

5

6

4

For x − y = 4,

x = 4 + y

x

5

4

3

y

1

0

−1

Hence, the graphic representation is as follows.

From the figure, it can be observed that these lines intersect each other at point (7, 3).

Therefore, the number of girls and boys in the class are 7 and 3 respectively.

(ii) Let the cost of 1 pencil be Rs and the cost of 1 pen be Rs y.

According to the question, the algebraic representation is

5x + 7y = 50

7x + 5y = 46

For 5x + 7y = 50,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 8

x

3

10

− 4

y

5

0

10

7x + 5y = 46

x

8

3

− 2

y

− 2

5

12

Hence, the graphic representation is as follows.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 9

From the figure, it can be observed that these lines intersect each other at point (3, 5).

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

PAGE NO 49:

Question 2:

On comparing the ratios, find out whether the lines representing the following pairs of linear equations at a point, are parallel or coincident:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 10  CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 11

ANSWER:

(i) 5x − 4y + 8 = 0

7x + 6y − 9 = 0

Comparing these equations with 

and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 12, we obtain

Since CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 13,

Hence, the lines representing the given pair of equations have a unique solution and the pair of lines intersects at exactly one point.

(ii) 9x + 3y + 12 = 0

18x + 6y + 24 = 0

Comparing these equations with 

and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 12, we obtain

Since CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 15,

Hence, the lines representing the given pair of equations are coincident and there are infinite possible solutions for the given pair of equations.

(iii)6x − 3y + 10 = 0

2x − y + 9 = 0

Comparing these equations with 

and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 12, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 17

Since ,

Hence, the lines representing the given pair of equations are parallel to each other and hence, these lines will never intersect each other at any point or there is no possible solution for the given pair of equations.

PAGE NO 49:

Question 3:

On comparing the ratiosCBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 18, find out whether the following pair of linear equations are consistent, or inconsistent.

ANSWER:

(i) 3x + 2y = 5

2x − 3y = 7

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 19

These linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(ii)2x − 3y = 8

4x − 6y = 9

Since CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 20,

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii) 

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 21

Since ,

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

(iv)5x − 3 y = 11

− 10x + 6y = − 22

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 22

Since ,

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

(v) CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 23

Since CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 24

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

PAGE NO 49:

Question 4:

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically:

ANSWER:

(i)y = 5

2x + 2y = 10

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 25

Since ,

Therefore, these linear equations are coincident pair of lines and thus have infinite number of possible solutions. Hence, the pair of linear equations is consistent.

x + y = 5

x = 5 − y

x

4

3

2

y

1

2

3

And, 2x + 2y = 10

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 26

x

4

3

2

y

1

2

3

Hence, the graphic representation is as follows.

From the figure, it can be observed that these lines are overlapping each other. Therefore, infinite solutions are possible for the given pair of equations.

(ii)x − y = 8

3x − 3y = 16

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 27

Since ,

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

(iii)2x + y − 6 = 0

4x − 2y − 4 = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 28

Since ,

Therefore, these linear equations are intersecting each other at one point and thus have only one possible solution. Hence, the pair of linear equations is consistent.

2x + y − 6 = 0

y = 6 − 2x

x

0

1

2

y

6

4

2

And 4− 2y − 4 = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 29

x

1

2

3

y

0

2

4

Hence, the graphic representation is as follows.

From the figure, it can be observed that these lines are intersecting each other at the only point i.e., (2, 2) and it is the solution for the given pair of equations.

(iv)2x − 2y − 2 = 0

4x − 4− 5 = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 30

Since ,

Therefore, these linear equations are parallel to each other and thus have no possible solution. Hence, the pair of linear equations is inconsistent.

PAGE NO 50:

Question 5:

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.

ANSWER:

Let the width of the garden be x and length be y.

According to the question,

y − x = 4 (1)

y + x = 36 (2)

y − x = 4

y = x + 4

x

0

8

12

y

4

12

16

y + x = 36

x

0

36

16

y

36

0

20

Hence, the graphic representation is as follows.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 31

From the figure, it can be observed that these lines are intersecting each other at only point i.e., (16, 20). Therefore, the length and width of the given garden is 20 m and 16 m respectively.

Question 6:

Given the linear equation 2x + 3y − 8 = 0, write another linear equations in two variables such that the geometrical representation of the pair so formed is:

(i) intersecting lines (ii) parallel lines

(iii) coincident lines

ANSWER:

(i)Intersecting lines:

For this condition,

The second line such that it is intersecting the given line is CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 32.

(ii) Parallel lines:

For this condition,

Hence, the second line can be

4x + 6y − 8 = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 33

(iii)Coincident lines:

For coincident lines,

Hence, the second line can be

6x + 9y − 24 = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 34

PAGE NO 50:

Question 7:

Draw the graphs of the equations x − y + 1 = 0 and 3x + 2y − 12 = 0. Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

ANSWER:

x − y + 1 = 0

x = y − 1

x

0

1

2

y

1

2

3

3x + 2y − 12 = 0

x

4

2

0

y

0

3

6

Hence, the graphic representation is as follows.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 35

From the figure, it can be observed that these lines are intersecting each other at point (2, 3) and x-axis at (−1, 0) and (4, 0). Therefore, the vertices of the triangle are (2, 3), (−1, 0), and (4, 0).

PAGE NO 53:

Question 1:

Solve the following pair of linear equations by the substitution method.

ANSWER:

(i) x + y = 14 (1)

x − y = 4 (2)

From (1), we obtain

x = 14 − y (3)

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 36

Substituting this in equation (3), we obtain

(ii) CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 37

From (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 38

Substituting this value in equation (2), we obtain

Substituting in equation (3), we obtain

s = 9

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 39s = 9, t = 6

(iii)3x − y = 3 (1)

9x − 3y = 9 (2)

From (1), we obtain

y = 3x − 3 (3)

Substituting this value in equation (2), we obtain

9 = 9

This is always true.

Hence, the given pair of equations has infinite possible solutions and the relation between these variables can be given by

y = 3x − 3

Therefore, one of its possible solutions is x = 1, y = 0.

(iv) CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 40

From equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 41

Substituting this value in equation (2), we obtain

Substituting this value in equation (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 42

(v) 

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 43

From equation (1), we obtain

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 44

Substituting this value in equation (3), we obtain

x = 0

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 39x = 0, y = 0

(vi) 

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 46

From equation (1), we obtain

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 47

Substituting this value in equation (3), we obtain

Hence, x = 2, y = 3

PAGE NO 53:

Question 2:

Solve 2x + 3y = 11 and 2− 4y = − 24 and hence find the value of ‘m’ for which y = mx + 3.

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 48

From equation (1), we obtain

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 49

Putting this value in equation (3), we obtain

Hence, x = −2, y = 5

Also,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 50

PAGE NO 53:

ANSWER:

(i) Let the first number be x and the other number be y such that y > x.

According to the given information,

On substituting the value of y from equation (1) into equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 51

Substituting this in equation (1), we obtain

y = 39

Hence, the numbers are 13 and 39.

(ii) Let the larger angle be x and smaller angle be y.

We know that the sum of the measures of angles of a supplementary pair is always 180º.

According to the given information,

From (1), we obtain

x = 180º − y (3)

Substituting this in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 52

Putting this in equation (3), we obtain

x = 180º − 81º

= 99º

Hence, the angles are 99º and 81º.

(iii) Let the cost of a bat and a ball be x and y respectively.

According to the given information,

From (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 53

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 54

Substituting this in equation (3), we obtain

Hence, the cost of a bat is Rs 500 and that of a ball is Rs 50.

(iv) Let the fixed charge be Rs and per km charge be Rs y.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 55

From (1), we obtain

Substituting this in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 56

Putting this in equation (3), we obtain

Hence, fixed charge = Rs 5

And per km charge = Rs 10

Charge for 25 km = x + 25y

= 5 + 250 = Rs 255

(v) Let the fraction be CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 57.

According to the given information,

From equation (1), we obtain CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 58

Substituting this in equation (2), we obtain

Substituting this in equation (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 59

Hence, the fraction is .

(vi) Let the age of Jacob be and the age of his son be y.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 60

From (1), we obtain

Substituting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 61

Substituting this value in equation (3), we obtain

Hence, the present age of Jacob is 40 years whereas the present age of his son is 10 years.

PAGE NO 56:

Question 1:

Solve the following pair of linear equations by the elimination method and the substitution method:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 62 

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 63

ANSWER:

(i) By elimination method

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 64

Multiplying equation (1) by 2, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 65

Subtracting equation (2) from equation (3), we obtain

Substituting the value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 66

By substitution method

From equation (1), we obtain

 (5)

Putting this value in equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 67

−5y = −6

Substituting the value in equation (5), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 66

(ii) By elimination method

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 69

Multiplying equation (2) by 2, we obtain

Adding equation (1) and (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 70

Substituting in equation (1), we obtain

Hence, x = 2, y = 1

By substitution method

From equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 71 (5)

Putting this value in equation (1), we obtain

7y = 7

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 72

Substituting the value in equation (5), we obtain

(iii) By elimination method

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 73

Multiplying equation (1) by 3, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 74

Subtracting equation (3) from equation (2), we obtain

Substituting in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 75

∴ 

By substitution method

From equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 76(5)

Putting this value in equation (2), we obtain

Substituting the value in equation (5), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 77

∴ CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 78

(iv)By elimination method

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 79

Subtracting equation (2) from equation (1), we obtain

Substituting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 80

Hence, x = 2, y = −3

By substitution method

From equation (2), we obtain

(5)

Putting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 81

5y = −15

Substituting the value in equation (5), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 82

∴ x = 2, y = −3

PAGE NO 57:

Question 2:

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes if we only add 1 to the denominator. What is the fraction?

(ii) 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?

(iii) 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.

(iv) Meena went to bank to withdraw Rs 2000. She asked the cashier to give her Rs 50 and Rs 100 notes only. Meena got 25 notes in all. Find how many notes of Rs 50 and Rs 100 she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

ANSWER:

(i)Let the fraction be CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 83.

According to the given information,

Subtracting equation (1) from equation (2), we obtain

= 3 (3)

Substituting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 84

Hence, the fraction is .

(ii)Let present age of Nuri = x

and present age of Sonu = y

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 85

Subtracting equation (1) from equation (2), we obtain

y = 20 (3)

Substituting it in equation (1), we obtain

Hence, age of Nuri = 50 years

And, age of Sonu = 20 years

(iii)Let the unit digit and tens digits of the number be and y respectively. Then, number = 10y + x

Number after reversing the digits = 10x + y

According to the given information,

x + = 9 (1)

9(10y + x) = 2(10x + y)

88y − 11= 0

− x + 8=0 (2)

Adding equation (1) and (2), we obtain

9y = 9

y = 1 (3)

Substituting the value in equation (1), we obtain

x = 8

Hence, the number is 10y + x = 10 × 1 + 8 = 18

(iv)Let the number of Rs 50 notes and Rs 100 notes be x and respectively.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 86

Multiplying equation (1) by 50, we obtain

Subtracting equation (3) from equation (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 87

Substituting in equation (1), we have x = 10

Hence, Meena has 10 notes of Rs 50 and 15 notes of Rs 100.

(v)Let the fixed charge for first three days and each day charge thereafter be Rs and Rs respectively.

According to the given information,

Subtracting equation (2) from equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 88

Substituting in equation (1), we obtain

Hence, fixed charge = Rs 15

And Charge per day = Rs 3

PAGE NO 62:

Question 1:

Which of the following pairs of linear equations has unique solution, no solution or infinitely many solutions? In case there is a unique solution, find it by using cross multiplication method.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 89

ANSWER:

Therefore, the given sets of lines are parallel to each other. Therefore, they will not intersect each other and thus, there will not be any solution for these equations.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 90

Therefore, they will intersect each other at a unique point and thus, there will be a unique solution for these equations.

By cross-multiplication method,

∴ x = 2, y = 1

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 91

Therefore, the given sets of lines will be overlapping each other i.e., the lines will be coincident to each other and thus, there are infinite solutions possible for these equations.

Therefore, they will intersect each other at a unique point and thus, there will be a unique solution for these equations.

By cross-multiplication,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 92

∴ 

PAGE NO 62:

Question 2:

(i) For which values of and b will the following pair of linear equations have an

infinite number of solutions?

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 93

(ii) For which value of k will the following pair of linear equations have no solution?

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 94

For infinitely many solutions,

Subtracting (1) from (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 95

Substituting this in equation (2), we obtain

Hence, a = 5 and b = 1 are the values for which the given equations give infinitely many solutions.

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 96

For no solution,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 97

Hence, for k = 2, the given equation has no solution.

PAGE NO 62:

Question 3:

Solve the following pair of linear equations by the substitution and cross-multiplication methods:

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 98

From equation (ii), we obtain

Substituting this value in equation (i), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 99

Substituting this value in equation (ii), we obtain

Hence, CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 100

Again, by cross-multiplication method, we obtain

PAGE NO 62:

Question 4:

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method:

(i)A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.

(ii)A fraction becomesCBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 101when 1 is subtracted from the numerator and it becomeswhen 8 is added to its denominator. Find the fraction.

(iii)Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?

(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?

(v)The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.

ANSWER:

(i)Let x be the fixed charge of the food and y be the charge for food per day.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 102

Subtracting equation (1) from equation (2), we obtain

Substituting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 103

Hence, fixed charge = Rs 400

And charge per day = Rs 30

(ii)Let the fraction be .

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 104

Subtracting equation (1) from equation (2), we obtain

Putting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 105

Hence, the fraction is .

(iii)Let the number of right answers and wrong answers be and y

respectively.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 106

Subtracting equation (2) from equation (1), we obtain

x = 15 (3)

Substituting this in equation (2), we obtain

Therefore, number of right answers = 15

And number of wrong answers = 5

Total number of questions = 20

(iv)Let the speed of 1st car and 2nd car be u km/h and v km/h.

Respective speed of both cars while they are travelling in same direction = (CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 107) km/h

Respective speed of both cars while they are travelling in opposite directions i.e., travelling towards each other = () km/h

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 108

Adding both the equations, we obtain

Substituting this value in equation (2), we obtain

v = 40 km/h

Hence, speed of one car = 60 km/h and speed of other car = 40 km/h

(v) Let length and breadth of rectangle be x unit and unit respectively.

Area = xy

According to the question,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 109

By cross-multiplication method, we obtain

Hence, the length and breadth of the rectangle are 17 units and 9 units respectively.

PAGE NO 67:

Question 1:

Solve the following pairs of equations by reducing them to a pair of linear equations:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 110
CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 111 

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 112

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 113

ANSWER:

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 114

Let  andCBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 115, then the equations change as follows.

Using cross-multiplication method, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 116

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 117

Putting and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 118in the given equations, we obtain

Multiplying equation (1) by 3, we obtain

6p + 9q = 6 (3)

Adding equation (2) and (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 119

Putting in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 120

Hence,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 111

Substituting  in the given equations, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 122

By cross-multiplication, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 123

Putting  and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 124in the given equation, we obtain

Multiplying equation (1) by 3, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 125

Adding (2) an (3), we obtain

Putting this value in equation (1), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 126

Putting CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 127and in the given equation, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 128

By cross-multiplication method, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 129

Putting and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 115in these equations, we obtain

By cross-multiplication method, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 131

Hence, x = 1, y = 2

Putting CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 132and in the given equations, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 133

Using cross-multiplication method, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 134

Adding equation (3) and (4), we obtain

Substituting in equation (3), we obtain

y = 2

Hence, x = 3, y = 2

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 135

Putting in these equations, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 136

Adding (1) and (2), we obtain

Substituting in (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 137

Adding equations (3) and (4), we obtain

Substituting in (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 138

Hence, x = 1, y = 1

PAGE NO 67:

Question 2:

Formulate the following problems as a pair of equations, and hence find their solutions:

(i) Ritu can row downstream 20 km in 2 hours, and upstream 4 km in 2 hours. Find her speed of rowing in still water and the speed of the current.

(ii) 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone to finish the work, and also that taken by 1 man alone.

(iii) Roohi travels 300 km to her home partly by train and partly by bus. She takes 4 hours if she travels 60 km by train and remaining by bus. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus separately.

ANSWER:

(i)Let the speed of Ritu in still water and the speed of stream be x km/h

and y km/h respectively.

Speed of Ritu while rowing

Upstream = km/h

Downstream = CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 139km/h

According to question,

Adding equation (1) and (2), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 140

Putting this in equation (1), we obtain

y = 4

Hence, Ritu’s speed in still water is 6 km/h and the speed of the current is 4 km/h.

(ii)Let the number of days taken by a woman and a man be and y respectively.

Therefore, work done by a woman in 1 day = 

Work done by a man in 1 day = CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 141

According to the question,

Putting CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 142in these equations, we obtain

By cross-multiplication, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 143

Hence, number of days taken by a woman = 18

Number of days taken by a man = 36

(iii) Let the speed of train and bus be u km/h and v km/h respectively.

According to the given information,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 144

Putting  and CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 145 in these equations, we obtain

Multiplying equation (3) by 10, we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 146

Subtracting equation (4) from (5), we obtain

Substituting in equation (3), we obtain

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 147

Hence, speed of train = 60 km/h

Speed of bus = 80 km/h

PAGE NO 68:

Question 1:

The ages of two friends Ani and Biju differ by 3 years. Ani’s father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differs by 30 years. Find the ages of Ani and Biju.

ANSWER:

The difference between the ages of Biju and Ani is 3 years. Either Biju is 3 years older than Ani or Ani is 3 years older than Biju. However, it is obvious that in both cases, Ani’s father’s age will be 30 years more than that of Cathy’s age.

Let the age of Ani and Biju be x and y years respectively.

Therefore, age of Ani’s father, Dharam = 2 × x = 2x years

And age of Biju’s sister Cathy years

By using the information given in the question,

Case (I) When Ani is older than Biju by 3 years,

x − y = 3 (i)

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 148

4x − y = 60 (ii)

Subtracting (i) from (ii), we obtain

3x = 60 − 3 = 57

Therefore, age of Ani = 19 years

And age of Biju = 19 − 3 = 16 years

Case (II) When Biju is older than Ani,

y − x = 3 (i)

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 148

4x − y = 60 (ii)

Adding (i) and (ii), we obtain

3x = 63

x = 21

Therefore, age of Ani = 21 years

And age of Biju = 21 + 3 = 24 years

PAGE NO 68:

Question 2:

One says, “Give me a hundred, friend! I shall then become twice as rich as you”. The other replies, “If you give me ten, I shall be six times as rich as you”. Tell me what is the amount of their (respective) capital? [From the Bijaganita of Bhaskara II)

[Hint: x + 100 = 2 (y − 100), y + 10 = 6(x − 10)]

ANSWER:

Let those friends were having Rs x and y with them.

Using the information given in the question, we obtain

x + 100 = 2(y − 100)

x + 100 = 2y − 200

x − 2y = −300 (i)

And, 6(x − 10) = (y + 10)

6x − 60 = y + 10

6x − y = 70 (ii)

Multiplying equation (ii) by 2, we obtain

12x − 2y = 140 (iii)

Subtracting equation (i) from equation (iii), we obtain

11x = 140 + 300

11x = 440

x = 40

Using this in equation (i), we obtain

40 − 2y = −300

40 + 300 = 2y

2y = 340

y = 170

Therefore, those friends had Rs 40 and Rs 170 with them respectively.

PAGE NO 68:

Question 3:

A train covered a certain distance at a uniform speed. If the train would have been 10 km/h faster, it would have taken 2 hours less than the scheduled time. And if the train were slower by 10 km/h; it would have taken 3 hours more than the scheduled time. Find the distance covered by the train.

ANSWER:

Let the speed of the train be x km/h and the time taken by train to travel the given distance be t hours and the distance to travel was d km. We know that,

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 150

Or, d = xt (i)

Using the information given in the question, we obtain

By using equation (i), we obtain

− 2x + 10t = 20 (ii)

CBSE Class 10 NCERT Solutions for Maths Chapter 3, Pair of Linear Equations in Two Variables 151

By using equation (i), we obtain

3x − 10t = 30 (iii)

Adding equations (ii) and (iii), we obtain

x = 50

Using equation (ii), we obtain

(−2) × (50) + 10t = 20

−100 + 10t = 20

10t = 120

= 12 hours

From equation (i), we obtain

Distance to travel = d = xt

= 50 × 12

= 600 km

Hence, the distance covered by the train is 600 km.

PAGE NO 68:

Question 4:

The students of a class are made to stand in rows. If 3 students are extra in a row, there would be 1 row less. If 3 students are less in a row, there would be 2 rows more. Find the number of students in the class.

ANSWER:

Let the number of rows be x and number of students in a row be y.

Total students of the class

= Number of rows × Number of students in a row

xy

Using the information given in the question,

Condition 1

Total number of students = (x − 1) (y + 3)

xy = (− 1) (y + 3) = xy − y + 3x − 3

3x − y − 3 = 0

3x − y = 3 (i)

Condition 2

Total number of students = (x + 2) (y − 3)

xy = xy + 2y − 3x − 6

3x − 2y = −6 (ii)

Subtracting equation (ii) from (i),

(3x − y) − (3x − 2y) = 3 − (−6)

− y + 2y = 3 + 6

y = 9

By using equation (i), we obtain

3x − 9 = 3

3x = 9 + 3 = 12

x = 4

Number of rows = x = 4

Number of students in a row = y = 9

Number of total students in a class = xy = 4 × 9 = 36

CBSE Class 10 NCERT Solutions for Maths Chapter 3

PAGE NO 68:

Question 5:

In a ΔABC, ∠C = 3 ∠B = 2 (∠A + ∠B). Find the three angles.

ANSWER:

Given that,

∠C = 3∠B = 2(∠A + ∠B)

3∠B = 2(∠A + ∠B)

3∠B = 2∠A + 2∠B

∠B = 2∠A

2 ∠A − ∠B = 0 … (i)

We know that the sum of the measures of all angles of a triangle is 180°. Therefore,

∠A + ∠B + ∠C = 180°

∠A + ∠B + 3 ∠B = 180°

∠A + 4 ∠B = 180° … (ii)

Multiplying equation (i) by 4, we obtain

8 ∠A − 4 ∠B = 0 … (iii)

Adding equations (ii) and (iii), we obtain

9 ∠A = 180°

∠A = 20°

From equation (ii), we obtain

20° + 4 ∠B = 180°

4 ∠B = 160°

∠B = 40°

∠C = 3 ∠B

= 3 × 40° = 120°

Therefore, ∠A, ∠B, ∠C are 20°, 40°, and 120° respectively.

PAGE NO 68:

Question 6:

Draw the graphs of the equations 5x − y = 5 and 3x − y = 3. Determine the co-ordinates of the vertices of the triangle formed by these lines and the y axis.

ANSWER:

5x − y = 5

Or, y = 5x − 5

The solution table will be as follows.

x

0

1

2

y

−5

0

5

3x − y = 3

Or, y = 3x − 3

The solution table will be as follows.

x

0

1

2

y

− 3

0

3

The graphical representation of these lines will be as follows.

It can be observed that the required triangle is ΔABC formed by these lines and y-axis.

The coordinates of vertices are A (1, 0), B (0, − 3), C (0, − 5).

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