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# NCERT solutions Mathematics Class 10 chapter 3 Pair of Linear Equations in Two Variables

## Chapter 3 - Pair of Linear Equations in Two Variables

#### Page 44

A father 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.” Represent this situation algebraically and graphically.

Q 1 | Page 44

The coach of a cricket team buys 3 bats and 6 balls for Rs 3900. Later, she buys another bat and 3 more balls of the same kind for Rs 1300. Represent this situation algebraically and geometrically.

Q 2 | Page 44

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.

Q 2 | Page 44

#### Pages 49 - 50

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.

Q 1.1 | Page 49

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

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

Q 1.2 | Page 49

On comparing the ratios a_1/a_2,b_1/b_2 and without drawing them, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincide.

(i) 5x – 4y + 8 = 0, 7x + 6y – 9 = 0

(ii) 9x + 3y + 12 = 0, 18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0, 2x – y + 9 = 0

Q 2 | Page 49

On comparing the ratios a_1/a_2, b_1/b_2 and c_1/c_2 ,find out whether the following pair of linear equations are consistent, or inconsistent.

3x + 2y = 5 ; 2x – 3y = 7

Q 3.1 | Page 49

On comparing the ratios a_1/a_2, b_1/b_2 and c_1/c_2, find out whether the following pair of linear equations are consistent, or inconsistent.

2x – 3y = 8 ; 4x – 6y = 9

Q 3.2 | Page 49

On comparing the ratios a_1/a_2, b_1/b_2 and c_1/c_2, find out whether the following pair of linear equations are consistent, or inconsistent. 3/2x + 5/3y = 7 ; 9x - 10y = 14

Q 3.3 | Page 49

On comparing the ratios a_1/a_2, b_1/b_2 and c_1/c_2, find out whether the following pair of linear equations are consistent, or inconsistent.

5x – 3y = 11 ; – 10x + 6y = –22

Q 3.4 | Page 49

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically: x + y = 5, 2x + 2y = 10

Q 4.1 | Page 50

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

x – y = 8, 3x – 3y = 16

Q 4.2 | Page 50

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically: 2x + y – 6 = 0, 4x – 2y – 4 = 0

Q 4.3 | Page 50

Which of the following pairs of linear equations are consistent/ inconsistent? If consistent, obtain the solution graphically 2x – 2y – 2 = 0, 4x – 4y – 5 = 0

Q 4.4 | Page 50

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

Q 5 | Page 50

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

(i) intersecting lines

(ii) parallel lines

(iii) coincident lines

Q 6 | Page 50

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.

Q 7 | Page 50

#### Pages 53 - 54

Solve the following pair of linear equations by the substitution method

x + y = 14

x – y = 4

Q 1.1 | Page 53

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

s – t = 3

s/3 + t/2 = 6

Q 1.2 | Page 53

Solve the following pair of linear equations by the substitution method

3x – y = 3

9x – 3y = 9

Q 1.3 | Page 53

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

0.2x + 0.3y = 1.3

0.4x + 0.5y = 2.3

Q 1.4 | Page 53

Solve the following pair of linear equations by the substitution method

sqrt2x + sqrt3y = 0

sqrt3x - sqrt8y = 0

Q 1.5 | Page 53

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

(3x)/2 - (5y)/3 = -2

x/y+y/2 = 13/6

Q 1.6 | Page 53

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

Q 2 | Page 53

Form the pair of linear equations for the following problems and find their solution by substitution method

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

Q 3.1 | Page 53

Form the pair of linear equations for the following problems and find their solution by substitution method .

The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.

Q 3.2 | Page 53

The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball

Q 3.3 | Page 53

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?

Q 4 | Page 54

A fraction becomes 9/11 if 2 is added to both the numerator and the denominator. If, 3 is added to both the numerator and the denominator it becomes 5/6 . Find the fraction.

Q 5 | Page 54

Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?

Q 6 | Page 54

#### Pages 56 - 57

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

x + y = 5 and 2x – 3y = 4

Q 1.1 | Page 56

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

3x + 4y = 10 and 2x – 2y = 2

Q 1.2 | Page 56

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

3x – 5y – 4 = 0 and 9x = 2y + 7

Q 1.3 | Page 56

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

x/2 + (2y)/3 = -1 and x - y /3 = 3

Q 1.4 | Page 56

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

If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?

Q 2.1 | Page 57

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

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

Q 2.2 | Page 57

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

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.

Q 2.3 | Page 57

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

Meena went to a 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.

Q 2.4 | Page 57

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

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

Q 2.5 | Page 57

#### Pages 62 - 63

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

x – 3y – 3 = 0

3x – 9y – 2 = 0

Q 1.1 | Page 62

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

2x + y = 5

3x + 2y = 8

Q 1.2 | Page 62

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

3x – 5y = 20

6x – 10y = 40

Q 1.3 | Page 62

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.

x – 3y – 7 = 0

3x – 3y – 15 = 0

Q 1.4 | Page 62

For which values of a and b does the following pair of linear equations have an infinite number of solutions?

2x + 3y = 7

(a – b) x + (a + b) y = 3a + b – 2

Q 2.1 | Page 62

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

3x + y = 1

(2k – 1) x + (k – 1) y = 2k + 1

Q 2.2 | Page 62

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

8x + 5y = 9

3x + 2y = 4

Q 3 | Page 62

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

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.

Q 4.1 | Page 63

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

A fraction becomes 1/3 when 1 is subtracted from the numerator and it becomes 1/4 when 8 is added to its denominator. Find the fraction.

Q 4.2 | Page 63

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

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?

Q 4.3 | Page 63

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

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?

Q 4.4 | Page 63

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

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.

Q 4.5 | Page 63

#### Page 67

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

1/(2x) + 1/(3y) = 2

1/(3x) + 1/(2y) = 13/6

Q 1.1 | Page 67

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

2/sqrtx +3/sqrty = 2

4/sqrtx - 9/sqrty = -1

Q 1.1 | Page 67

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

4/x + 3y = 14

3/x - 4y = 23

Q 1.3 | Page 67

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

5/(x-1) + 1/y-2 = 2

6/(x-1) - 3/(y-2) = 1

Q 1.4 | Page 67

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

(7x-2y)/(xy) = 5

(8x + 7y)/(xy) = 15

Q 1.5 | Page 67

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

6x + 3y = 6xy

2x + 4y = 5xy

Q 1.6 | Page 67

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

10/(x+y) + 2/(x-y) = 4

15/(x+y) - 5/(x-y) = -2

Q 1.7 | Page 67

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

1/(3x+y) + 1/(3x-y) = 3/4

1/(2(3x-y)) - 1/(2(3x-y)) = (-1)/8

Q 1.8 | Page 67

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

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

Q 2.1 | Page 67

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

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.

Q 2.2 | Page 67

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

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.

Q 2.3 | Page 67

#### Page 68

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 differ by 30 years. Find the ages of Ani and Biju

Q 1 | Page 68

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)]

Q 2 | Page 68

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.

Q 3 | Page 68

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.

Q 4 | Page 68

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

Q 5 | Page 68

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.

Q 6 | Page 68

Solve the following pair of linear equations: px + qy = p − q, qx − py = p + q

Q 7.1 | Page 68

Solve the following pair of linear equations

ax + by = c

bx + ay = 1 + c

Q 7.2 | Page 68

Solve the following pair of linear equations.

x/a-y/b = 0

ax + by = a2 + b2

Q 7.3 | Page 68

Solve the following pair of linear equations.

(a − b) x + (a + b) y = a2− 2ab − b2

(a + b) (x + y) = a2 + b2

Q 7.4 | Page 68

Solve the following pair of linear equations.

152x − 378y = − 74

− 378x + 152y = − 604

Q 7.5 | Page 68

ABCD is a cyclic quadrilateral finds the angles of the cyclic quadrilateral.

Q 8 | Page 68

#### Extra questions

Solve the following system of equations by cross-multiplication method x + y = a – b; ax – by = a2 + b2

Solve each of the following system of equations by eliminating x (by substitution) :

(i) x + y = 7

2x – 3y = 11

(ii) x + y = 7

12x + 5y = 7

(iii) 2x – 7y = 1

4x + 3y = 15

(iv) 3x – 5y = 1

5x + 2y = 19

(v) 5x + 8y = 9

2x + 3y = 4

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

Solve the following system of equations \frac { 1 }{ 2x } – \frac { 1 }{ y } = – 1; \frac { 1 }{ x } + \frac { 1}{ 2y } = 8

Solve \frac{2}{x+2y}+\frac{6}{2x-y}=4\text{ ;}\frac{5}{2( x+2y)}+\frac{1}{3( 2x-y)}=1 where, x + 2y ≠ 0 and 2x – y ≠ 0

Solve \frac { 2 }{ x } + \frac { 1 }{ 3y } = \frac { 1}{ 5 }; \frac { 3 }{ x } + \frac { 2 }{ 3y } = 2 and also find ‘a’ for which y = ax – 2

Solve \frac{1}{x+y}+\frac{2}{x-y}=2\text{ and }\frac{2}{x+y}-\frac{1}{x-y}=3 where, x + y ≠ 0 and x – y ≠ 0

Solve the following systems of equations

(i)\frac{15}{u} + \frac{2}{v} = 17

\frac{1}{u} + \frac{1}{v} = \frac{36}{5}

(ii)  \frac{11}{v} – \frac{7}{u} = 1

\frac{9}{v} + \frac{4}{u} = 6

Solve the following system of equations by cross-multiplications method.

a(x + y) + b (x – y) = a^2 – ab + b^2

a(x + y) – b (x – y) = a^2 + ab + b^2

Solve the following system of equations by cross-multiplication method.

ax + by = a – b; bx – ay = a + b

Solve the following system of equations by cross-multiplication method ax + by = 1;  bx + ay = \frac{(a+b)^{2}}{a^{2}+b^{2}-1

Solve the following system of equations by the method of cross-multiplication \frac{x}{a}+\frac{y}{b}=a+b ;   \frac{x}{a^{2}}+\frac{y}{b^{2}}=2

Solve the following system of equations by the method of cross-multiplication.

11x + 15y = – 23; 7x – 2y = 20

Solve the following system of equations by the method of cross-multiplication. \frac{a}{x}-\frac{b}{y}=0;\text{}\frac{ab^{2}}{x}+\frac{a^{2}b}{y}=a^{2}+b^{2}; Where x ≠ 0, y ≠ 0

Solve the following system of equations in x and y by cross-multiplication method

(a – b) x + (a + b) y = a^2 – 2ab – b^2

(a + b) (x + y) = a^2 + b^2

Solve the following system of linear equations :

2(ax – by) + (a + 4b) = 0

2(bx + ay) + (b – 4a) = 0

Solve the follownig system of equations by the method of cross-multiplication.

2x – 6y + 10 = 0

3x – 7y + 13 = 0

Solve the following system of equations by cross-multiplication method.

2x + 3y + 8 = 0

4x + 5y + 14 = 0

Show graphically that the system of equations 2x + 5y = 16; 3x+\frac { 15 }{ 2 }=24  has infinitely many solutions.

The path of highway number 1 is given by the equation x + y = 7 and the highway number 2 is given by the equation 5x + 2y = 20. Represent these equations geometrically

Solve the following system of linear equations graphically; 3x + y – 11 = 0 ; x – y – 1 = 0 Shade the region bounded by these lines and also y-axis. Then, determine the areas of the region bounded by these lines and y-axis.

Show graphically that the system of equations 2x + 3y = 10, 4x + 6y = 12 has no solution

Show graphically that the system of equations x – 4y + 14 = 0; 3x + 2y – 14 = 0 is consistent with unique solution.

Draw the graphs of the following equations 2x – 3y = – 6; 2x + 3y = 18; y = 2 Find the vertices of the triangles formed and also find the area of the triangle.

Half the perimeter of a garden, whose length is 4 more than its width is 36m. Find the dimensions of the garden

Draw the graph of the equation y – x = 2

Draw the graph of

(i) x – 7y = – 42

(ii) x – 3y = 6

(iii) x – y + 1 = 0

(iv) 3x + 2y = 12

If (9/2, 6) is lies on graph of 4x + ky = 12 then find value of k

Find five equations of lines which passes through (3, –5).

Draw a graph of the line x – 2y = 3. From the graph, find the coordinates of the point when (i) x = – 5 (ii) y = 0.

Solve (a – b) x + (a + b) y = a^2 – 2ab – b^2 (a + b) (x + y) = a^2 + b^2

Solve the following system of linear equations by using the method of elimination by equating the coefficients: 3x + 4y = 25 ; 5x – 6y = – 9

Solve the following system of linear equations by using the method of elimination by equating the coefficients √3x – √2y = √3 = ; √5x – √3y = √2

Solve for x and y : \frac { ax }{ b } – \frac { by }{ a } = a + b ; ax – by = 2ab

Solve the following system of linear equations by applying the method of elimination by equating the coefficients

(i)4x – 3y = 4

2x + 4y = 3

(ii)5x – 6y = 8

3x + 2y = 6

Solve the following system of equations by using the method of elimination by equating the co-efficients.

\frac { x }{ y } + \frac { 2y }{ 5 } + 2 = 10; \frac { 2x }{ 7 } – \frac { 5 }{ 2 } + 1 = 9

Solve the following system of equations: 15x + 4y = 61; 4x + 15y = 72

Solve the following systems of equations by eliminating ‘y’ (by substitution) :

(i) 3x – y = 3

7x + 2y = 20

(ii) 7x + 11y – 3 = 0

8x + y – 15 = 0

(iii) 2x + y = 0

17x – 11y – 8 = 0

S