# RD Sharma solutions for Class 10 Maths chapter 3 - Pair of Linear Equations in Two Variables [Latest edition]

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

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9
Exercise 3.1 [Page 12]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.1 [Page 12]

Exercise 3.1 | Q 1 | Page 12

Akhila went to a fair in her village. She wanted to enjoy rides in the Giant Wheel and play Hoopla (a game in which you throw a rig on the items kept in the stall, and if the ring covers any object completely you get it.) The number of times she played Hoopla is half the number of rides she had on the Giant Wheel. Each ride costs Rs 3, and a game of Hoopla costs Rs 4. If she spent Rs 20 in the fair, represent this situation algebraically and graphically.

Exercise 3.1 | Q 2 | Page 12

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

Exercise 3.1 | Q 3 | Page 12

The path of a train A is given by the equation 3x + 4y − 12 = 0 and the path of another train B is given by the equation 6x + 8y − 48 = 0. Represent this situation graphically.

Exercise 3.1 | Q 4 | Page 12

Gloria is walking along the path joining (−2, 3) and (2, −2), while Suresh is walking along the path joining (0, 5) and (4, 0). Represent this situation graphically.

Exercise 3.1 | Q 5 | Page 12

On comparing the ratios a_1/a_2,b_1/b_2 and c_1/c_2 without drawing them, find out whether the lines representing the following pair of linear equations intersect at a point, are parallel or coincide.

5x – 4y + 8 = 0, 7x + 6y – 9 = 0

Exercise 3.1 | Q 6 | Page 31

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

Exercise 3.1 | Q 7 | Page 12

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.

Exercise 3.2 [Pages 29 - 31]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.2 [Pages 29 - 31]

Exercise 3.2 | Q 1 | Page 29

Solve the following systems of equations graphically:

x − 2y = 5

2x + 5y = 12

Exercise 3.2 | Q 2 | Page 29

Solve the following systems of equations graphically:

x − 2y = 5

2x + 3y = 10

Exercise 3.2 | Q 3 | Page 29

Solve the following systems of equations graphically:

3x + y + 1 = 0

2x − 3y + 8 = 0

Exercise 3.2 | Q 4 | Page 29

Solve the following systems of equations graphically:

2x + y − 3 = 0

2x − 3y − 7 = 0

Exercise 3.2 | Q 5 | Page 29

Solve the following systems of equations graphically:

x + y = 6

x − y = 2

Exercise 3.2 | Q 6 | Page 29

Solve the following systems of equations graphically:
x − 2y = 6

3x − 6y = 0

Exercise 3.2 | Q 7 | Page 29

Solve the following systems of equations graphically:
x + y = 4

2x − 3y = 3

Exercise 3.2 | Q 8 | Page 29

Solve the following systems of equations graphically:

2x + 3y = 4

x − y + 3 = 0

Exercise 3.2 | Q 9 | Page 29

Solve the following systems of equations graphically:

2x − 3y + 13 = 0

3x − 2y + 12 = 0

Exercise 3.2 | Q 10 | Page 29

Solve the following systems of equations graphically:

2x + 3y + 5 = 0

3x − 2y − 12 = 0

Exercise 3.2 | Q 11 | Page 29

Show graphically that each one of the following systems of equations has infinitely many solutions:

2x + 3y = 6

4x + 6y = 12

Exercise 3.2 | Q 12 | Page 29

Show graphically that each one of the following systems of equations has infinitely many solutions:

x − 2y = 5

3x − 6y = 15

Exercise 3.2 | Q 13 | Page 29

Show graphically that each one of the following systems of equations has infinitely many solutions:

3x + y = 8

6x + 2y = 16

Exercise 3.2 | Q 14 | Page 29

Show graphically that each one of the following systems of equations has infinitely many solutions:

x − 2y + 11 = 0

3x − 6y + 33 = 0

Exercise 3.2 | Q 15 | Page 29

Show graphically that each one of the following systems of equations is inconsistent (i.e. has no solution) :

3x − 5y = 20

6x − 10y = −40

Exercise 3.2 | Q 16 | Page 29

Show graphically that each one of the following systems of equations is inconsistent (i.e. has no solution) :

x − 2y = 6

3x − 6y = 0

Exercise 3.2 | Q 17 | Page 29

Show graphically that each one of the following systems of equations is inconsistent (i.e. has no solution) :

2y − x = 9

6y − 3x = 21

Exercise 3.2 | Q 18 | Page 29

Show graphically that each one of the following systems of equations is inconsistent (i.e. has no solution) :

3x − 4y − 1 = 0

2x - 8/3y + 5 = 0

Exercise 3.2 | Q 19.1 | Page 29

Determine graphically the vertices of the triangle, the equations of whose sides are given below :

2y − = 8, 5y − x = 14 and y − 2x = 1

Exercise 3.2 | Q 19.2

Determine graphically the vertices of the triangle, the equations of whose sides are given below :

y = xy = 0 and 3x + 3y = 10

Exercise 3.2 | Q 20 | Page 29

Determine, graphically whether the system of equations x − 2y = 2, 4x − 2y = 5 is consistent or in-consistent.

Exercise 3.2 | Q 21.1 | Page 29

Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :

2x − 3y = 6, x + y = 1

Exercise 3.2 | Q 21.2 | Page 29

Determine, by drawing graphs, whether the following system of linear equations has a unique solution or not :

2y = 4x − 6, 2x = y + 3

Exercise 3.2 | Q 22.1 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y.

2x − 5y + 4 = 0,

2x + y − 8 = 0

Exercise 3.2 | Q 22.2 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y.

3x + 2y = 12,

5x − 2y = 4

Exercise 3.2 | Q 22.3 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y.

2x + y − 11 = 0,

x − y − 1 = 0

Exercise 3.2 | Q 22.4 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y.

x + 2y − 7 = 0,

2x − y − 4 = 0

Exercise 3.2 | Q 22.5 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y.

3x + y − 5 = 0

2x − y − 5 = 0

Exercise 3.2 | Q 22.6 | Page 29

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet axis of y

2x − y − 5 = 0,

x − y − 3 = 0

Exercise 3.2 | Q 23.1 | Page 31

Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are :

y = x, y = 2x and y + x = 6

Exercise 3.2 | Q 23.2 | Page 31

Determine graphically the coordinates of the vertices of a triangle, the equations of whose sides are :

y = x, 3y = xx + y = 8

Exercise 3.2 | Q 24.1 | Page 30

Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:

2x + 3y = 12

x − y = 1

Exercise 3.2 | Q 24.2 | Page 30

Solve the following system of linear equation graphically and shade the region between the two lines and x-axis:

3x + 2y − 4 = 0, 2x − 3y − 7 = 0

Exercise 3.2 | Q 24.3 | Page 30

Solve the following system of linear equations graphically and shade the region between the two lines and x-axis:

3x + 2y − 11 = 0

2x − 3y + 10 = 0

Exercise 3.2 | Q 25 | Page 30

Draw the graphs of the following equations on the same graph paper:

2x + 3y = 12,

x − y = 1

Exercise 3.2 | Q 26 | Page 30

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

Exercise 3.2 | Q 28 | Page 30

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.

Exercise 3.2 | Q 29.1 | Page 30

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.

2x + = 6
x − 2y = −2

Exercise 3.2 | Q 29.2 | Page 30

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.

2x − = 2
4x − y = 8

Exercise 3.2 | Q 29.3 | Page 30

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.

x + 2y = 5
2x − 3y = −4

Exercise 3.2 | Q 29.4 | Page 30

Solve graphically each of the following systems of linear equations. Also, find the coordinates of the points where the lines meet the axis of x in each system.

2x + 3y = 8
x − 2y = −3

Exercise 3.2 | Q 30 | Page 30

Draw the graphs of the following equations:

2x − 3y + 6 = 0
2x + 3y − 18 = 0
y − 2 = 0

Find the vertices of the triangle so obtained. Also, find the area of the triangle.

Exercise 3.2 | Q 31 | Page 30

Solve the following system of equations graphically.

2x − 3y + 6 = 0
2x + 3y − 18 = 0

Also, find the area of the region bounded by these two lines and y-axis.

Exercise 3.2 | Q 32 | Page 30

Solve the following system of linear equations graphically

4x − 5y − 20 = 0
3x + 5y − 15 = 0

Determine the vertices of the triangle formed by the lines representing the above equation and the y-axis.

Exercise 3.2 | Q 33 | Page 31

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

Exercise 3.2 | Q 34.1 | Page 31

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.

Exercise 3.2 | Q 34.2 | Page 31

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

Exercise 3.2 | Q 34.3 | Page 31

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

Champa went to a 'sale' to purchase some pants and skirts. When her friends asked her how many of each she had bought, she answered, "The number of skirts is two less than twice the number of pants purchased. Also, the number of skirts is four less than four times the number of pants purchased." Help her friends to find how many pants and skirts Champa bought.

Exercise 3.2 | Q 35.1 | Page 31

Solve the following system of equations graphically:
Shade the region between the lines and the y-axis

3x − 4y = 7
5x + 2y = 3

Exercise 3.2 | Q 35.2 | Page 31

Solve the following system of equations graphically:
Shade the region between the lines and the y-axis

4x − y = 4
3x + 2y = 14

Exercise 3.2 | Q 36 | Page 31

Represent the following pair of equations graphically and write the coordinates of points where the lines intersect y-axis.

x + 3y = 6
2x − 3y = 12

Exercise 3.3 [Pages 44 - 46]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.3 [Pages 44 - 46]

Exercise 3.3 | Q 1 | Page 44

Solve the following systems of equations:

11x + 15y + 23 = 0

7x – 2y – 20 = 0

Exercise 3.3 | Q 2 | Page 44

Solve the following systems of equations:

3x − 7y + 10 = 0
y − 2x − 3 = 0

Exercise 3.3 | Q 3 | Page 44

Solve the following systems of equations:

0.4x + 0.3y = 1.7
0.7x − 0.2y = 0.8

Exercise 3.3 | Q 4 | Page 44

Solve the following systems of equations:

x/2 + y = 0.8

7/(x + y/2) = 10

Exercise 3.3 | Q 5 | Page 44

Solve the following systems of equations:

7(y + 3) − 2(x + 2) = 14
4(y − 2) + 3(x − 3) = 2

Exercise 3.3 | Q 6 | Page 44

Solve the following systems of equations:

x/7 + y/3 = 5

x/2 - y/9 = 6

Exercise 3.3 | Q 7 | Page 44

Solve the following systems of equations:

x/3 + y/4 =11

(5x)/6 - y/3 = -7

Exercise 3.3 | Q 8 | Page 44

Solve the following systems of equations:

4u + 3y = 8

6u - 4y = -5

Exercise 3.3 | Q 9 | Page 44

Solve the following systems of equations:

x + y/2 = 4

x/3 + 2y = 5

Exercise 3.3 | Q 10 | Page 44

Solve the following systems of equations:

x + 2y = 3/2

2x + y = 3/2

Exercise 3.3 | Q 11 | Page 44

Solve the following systems of equations:

sqrt2x + sqrt3y = 0

sqrt3x - sqrt8y = 0

Exercise 3.3 | Q 12 | Page 44

Solve the following systems of equations:

3x - (y + 7)/11 + 2 = 10

2y + (x + 10)/7 = 10

Exercise 3.3 | Q 13 | Page 44

Solve the following systems of equations:

2x - 3/y = 9

3x + 7/y = 2,  y != 0

Exercise 3.3 | Q 14 | Page 44

Solve the following systems of equations:

0.5x + 0.7y = 0.74
0.3x + 0.5y = 0.5

Exercise 3.3 | Q 15 | Page 45

Solve the following systems of equations:

1/(7x) + 1/(6y) = 3

1/(2x) - 1/(3y) = 5

Exercise 3.3 | Q 16 | Page 45

Solve the following systems of equations:

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

1/(3x) + 1/(2y) = 13/6

Exercise 3.3 | Q 17 | Page 45

Solve the following systems of equations:

(x + y)/(xy) = 2

(x - y)/(xy) = 6

Exercise 3.3 | Q 18 | Page 45

Solve the following systems of equations:

15/u + 2/v = 17

Exercise 3.3 | Q 19 | Page 45

Solve the following systems of equations:

3/x - 1/y = -9

2/x + 3/y  = 5

Exercise 3.3 | Q 20 | Page 45

Solve the following systems of equations:
2/x + 5/y = 1

60/x + 40/y = 19, x = ! 0, y != 0

Exercise 3.3 | Q 21 | Page 45

Solve the following systems of equations:

1/(5x) + 1/(6y) = 12

1/(3x) - 3/(7y) = 8, x ~= 0, y != 0

Exercise 3.3 | Q 22 | Page 45

Solve the following systems of equations:

2/x + 3/y = 9/(xy)

4/x + 9/y = 21/(xy), where x != 0, y != 0

Exercise 3.3 | Q 24 | Page 45

Solve the following systems of equations:

2/sqrtx + 3/sqrty = 2

4/sqrtx - 9/sqrty = -1

Exercise 3.3 | Q 27 | Page 45

Solve the following systems of equations:

6/(x + y) = 7/(x - y) + 3

1/(2(x + y)) = 1/(3(x - y)), where x + y ≠ 0 and x – y ≠ 0

Exercise 3.3 | Q 28 | Page 45

Solve the following systems of equations:

"xy"/(x + y) = 6/5

"xy"/(y- x) = 6

Exercise 3.3 | Q 29 | Page 45

Solve the following systems of equations:

22/(x + y) + 15/(x - y) = 5

55/(x + y) + 45/(x - y) = 14

Exercise 3.3 | Q 30 | Page 45

Solve the following systems of equations:

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

15/(x + y) + 7/(x - y) = 10

Exercise 3.3 | Q 31 | Page 45

Solve the following systems of equations:

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

9/(x + y) - 4/(x - y) = 1

Exercise 3.3 | Q 32 | Page 45

Solve the following systems of equations:

1/(2(x + 2y)) + 5/(3(3x - 2y)) = (-3)/2

5/(4(x + 2y)) - 3'/(5(3x - 2y)) = 61/60

Exercise 3.3 | Q 33 | Page 46

Solve the following systems of equations:

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

10/(x + 1) + 2/(y - 1) = 5/2 where x != -1 and y != 1

Exercise 3.3 | Q 34 | Page 46

Solve the following systems of equations:

x + y = 5xy
3x + 2y = 13xy

Exercise 3.3 | Q 34 | Page 46

Solve the following systems of equations:

4/x + 3y = 14

3/x - 4y = 23

Exercise 3.3 | Q 35 | Page 46

Solve the following systems of equations:

x+y = 2xy

(x - y)/(xy) = 6   x != 0, y != 0

Exercise 3.3 | Q 36 | Page 46

Solve the following systems of equations:

2(3u − ν) = 5uν
2(u + 3ν) = 5uν

Exercise 3.3 | Q 37 | Page 46

Solve the following systems of equations:

2/(3x + 2y) + 3/(3x - 2y) = 17/5

5/(3x + 2y) + 1/(3x - 2y) = 2

Exercise 3.3 | Q 38 | Page 46

Solve the following systems of equations:

x − y + z = 4
x + y + z = 2
2x + y − 3z = 0

Exercise 3.3 | Q 39 | Page 46

Solve the following systems of equations:

44/(x + y) + 30/(x - y) = 10

55/(x + y) + 40/(x - y) = 13

Exercise 3.3 | Q 40 | Page 46

Solve the following systems of equations:

10/(x + y) + 2/(x - y) = 4

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

Exercise 3.3 | Q 40 | Page 46

Solve the following systems of equations:

4/x + 15y = 21

3/x + 4y = 5

Exercise 3.3 | Q 41 | Page 46

Solve the following systems of equations:

2(1/x) + 3(1/y) = 13

5(1/x) - 4(1/y) = -2

Exercise 3.3 | Q 41 | Page 46

Solve the following systems of equations:

1/(3x + y) + 1/(3x - y) = 3/4

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

Exercise 3.3 | Q 42 | Page 46

Solve the following systems of equations:

5/(x - 1) + 1/(y - 2)  = 2

Exercise 3.3 | Q 42 | Page 46

Solve the following systems of equations:

(7x - 2y)/"xy" = 5

(8x + 7y)/"xy" = 15

Exercise 3.3 | Q 43 | Page 46

Solve the following systems of equations:

152x − 378y = −74
−378x + 152y = −604

Exercise 3.3 | Q 44 | Page 46

Solve the following systems of equations:

99x + 101y = 499
101x + 99y = 501

Exercise 3.3 | Q 45 | Page 46

Solve the following systems of equations:

23x − 29y = 98
29x − 23y = 110

Exercise 3.3 | Q 46 | Page 46

Solve the following systems of equations:

x − y + z = 4
x − 2y − 2z = 9
2x + y + 3z = 1

Exercise 3.4 [Pages 57 - 58]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.4 [Pages 57 - 58]

Exercise 3.4 | Q 1 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

x + 2y + 1 = 0
2x − 3y − 12 = 0

Exercise 3.4 | Q 2 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

3x + 2y + 25 = 0
2x + y + 10 = 0

Exercise 3.4 | Q 3 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

2x + y = 35
3x + 4y = 65

Exercise 3.4 | Q 4 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

2x − y = 6
x − y = 2

Exercise 3.4 | Q 5 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

(x + y)/(xy) = 2

(x - y)/(xy) = 6

Exercise 3.4 | Q 6 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

ax + by = a − b
bx − ay = a + b

Exercise 3.4 | Q 7 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

x + ay = b
ax − by = c

Exercise 3.4 | Q 8 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

ax + by = a2
bx + ay = b2

Exercise 3.4 | Q 9 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

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

15/(x + y) + 7/(x - y) = 10

where x != 0 and y != 0

Exercise 3.4 | Q 10 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

2/x + 3/y = 13

5/4 - 4/y = -2

where x != 0 and y != 0

Exercise 3.4 | Q 11 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

x/a = y/b

ax + by = a^2 + b^2

Exercise 3.4 | Q 12 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication

x/a + y/b = 2

ax - by = a^2 - b^2

Exercise 3.4 | Q 14 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

ax + by = (a + b)/2

3x + 5y = 4

Exercise 3.4 | Q 15 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

2ax + 3by = a + 2b
3ax + 2by = 2a + b

Exercise 3.4 | Q 16 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication

5ax + 6by = 28
3ax + 4by = 18

Exercise 3.4 | Q 17 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

(a + 2b)x + (2a − b)y = 2
(a − 2b)x + (2a + b)y = 3

Exercise 3.4 | Q 18 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

x(a - b + (ab)/(a -  b)) = y(a + b - (ab)/(a + b))

x + y = 2a^2

Exercise 3.4 | Q 19 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication

bx + cy  = a + b

ax (1/(a - b) - 1/(a + b)) + cy(1/(b -a) - 1/(b + a)) = (2a)/(a + b)

Exercise 3.4 | Q 20 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication

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

(a + b)(a + y) =  4ab

Exercise 3.4 | Q 21 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication

a^2x + b^2y = c^2

b^2x + a^2y = d^2

Exercise 3.4 | Q 22 | Page 57

Solve each of the following systems of equations by the method of cross-multiplication :

57/(x + y) + 6/(x - y) = 5

38/(x + y) + 21/(x - y) = 9

Exercise 3.4 | Q 23 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

2(ax – by) + a + 4b = 0

2(bx + ay) + b – 4a = 0

Exercise 3.4 | Q 24 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

6(ax + by) = 3a + 2b

6(bx - ay) = 3b - 2a

Exercise 3.4 | Q 25 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

a^2/x - b^2/y = 0

(a^2b)/x + (b^2a)/y = a + b, x , y != 0

Exercise 3.4 | Q 26 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

mx – my = m2 + n2

x + y = 2m

Exercise 3.4 | Q 27 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

(ax)/b - (by)/a = a + b

ax - by = 2ab

Exercise 3.4 | Q 28 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication :

b/a x + a/b y - (a^2 + b^2) = 0

x + y - 2ab = 0

Exercise 3.4 | Q 29 | Page 58

Solve each of the following systems of equations by the method of cross-multiplication

x/a + y/b = a + b

Exercise 3.5 [Pages 73 - 75]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.5 [Pages 73 - 75]

Exercise 3.5 | Q 1 | Page 73

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

x − 3y = 3
3x − 9y = 2

Exercise 3.5 | Q 2 | Page 73

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

2x + y - 5 = 0

4x + 2y - 10 = 0

Exercise 3.5 | Q 3 | Page 73

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

3x - 5y = 20

6x - 10y = 40

Exercise 3.5 | Q 4 | Page 73

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

x - 2y - 8 = 0

5x - 10y - 10 = 0

Exercise 3.5 | Q 5 | Page 73

Find the value of k for which the system
kx + 2y = 5
3x + y = 1
has (i) a unique solution, and (ii) no solution.

Exercise 3.5 | Q 5 | Page 73

In the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

kx + 2y - 5 = 0

3x + y - 1 = 0

Exercise 3.5 | Q 6 | Page 73

Find the value of k for which the following system of equations has a unique solution:

4x + ky + 8 = 0

2x + 2y + 2 = 0

Exercise 3.5 | Q 7 | Page 73

Find the value of k for which the following system of equations has a unique solution:

4x - 5y = k

2x - 3y = 12

Exercise 3.5 | Q 8 | Page 73

Find the value of k for which the following system of equations has a unique solution:

x + 2y = 3

5x + ky + 7 = 0

Exercise 3.5 | Q 9 | Page 73

Find the value of k for which each of the following systems of equations has infinitely many solutions :

2x + 3y − 5 = 0
6x + ky − 15 = 0

Exercise 3.5 | Q 10 | Page 73

Find the value of k for which each of the following systems of equations has infinitely many solutions :

4x + 5y = 3

kx + 15y = 9

Exercise 3.5 | Q 11 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions

kx - 2y + 6 = 0

4x + 3y + 9 = 0

Exercise 3.5 | Q 12 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

8x + 5y = 9

kx + 10y = 18

Exercise 3.5 | Q 13 | Page 73

Find the value of k for which each of the following system of equations have infinitely many solutions :

2x - 3y = 7

(k + 2)x - (2k + 1)y - 3(2k -1)

Exercise 3.5 | Q 14 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

2x + 3y = 2

(k + 2)x + (2k + 1)y - 2(k - 1)

Exercise 3.5 | Q 15 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

x + (k + 1)y =4

(k + 1)x + 9y - (5k + 2)

Exercise 3.5 | Q 16 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

$kx + 3y = 2k + 1$
$2\left( k + 1 \right)x + 9y = 7k + 1$

Exercise 3.5 | Q 17 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

2x + (k - 2)y = k

6x + (2k - 1)y - (2k + 5)

Exercise 3.5 | Q 18 | Page 73

Find the value of k for which each of the following system of equations have infinitely many solutions :

2x + 3y = 7

(k + 1)x + (2k - 1)y - (4k + 1)

Exercise 3.5 | Q 19 | Page 73

Find the value of k for which each of the following system of equations has infinitely many solutions :

2x +3y = k

(k - 1)x + (k + 2)y = 3k

Exercise 3.5 | Q 20 | Page 73

Find the value of k for which each of the following system of equations have no solution

kx - 5y = 2

6x + 2y = 7

Exercise 3.5 | Q 21 | Page 73

Find the value of k for which each of the following system of equations have no solution

x + 2y = 0

2x + ky = 5

Exercise 3.5 | Q 22 | Page 73

Find the value of k for which each of the following system of equations have no solution :

3x - 4y + 7 = 0

kx + 3y - 5 = 0

Exercise 3.5 | Q 23 | Page 73

Find the value of k for which each of the following system of equations have no solution :

2x - ky + 3 = 0

3x + 2y - 1 = 0

Exercise 3.5 | Q 24 | Page 73

Find the value of k for which each of the following system of equations have no solution :

2x + ky = 11
5x − 7y = 5

Exercise 3.5 | Q 25 | Page 73

Find the value of k for which the following system of equations has a unique solution:

kx + 3y = 3

12x + ky = 6

Exercise 3.5 | Q 26 | Page 74

For what value of ෺, the following system of equations will be inconsistent?

4x + 6y - 11 = 0

2x + ky - 7 = 0

Exercise 3.5 | Q 27 | Page 74

For what value of α, the system of equations

αx + 3y = α - 3

12x + αy =  α

will have no solution?

Exercise 3.5 | Q 29 | Page 74

Prove that there is a value of c (≠ 0) for which the system

6x + 3y = c - 3

12x + cy = c

has infinitely many solutions. Find this value.

Exercise 3.5 | Q 30 | Page 74

Find the values of k for which the system
2x + ky = 1
3x – 5y = 7
will have (i) a unique solution, and (ii) no solution. Is there a value of k for which the
system has infinitely many solutions?

Exercise 3.5 | Q 31 | Page 74

For what value of k, the following system of equations will represent the coincident lines?

x + 2y + 7 = 0

2x + ky + 14 = 0

Exercise 3.5 | Q 32 | Page 74

Obtain the condition for the following system of linear equations to have a unique solution

ax + by = c

lx + my = n

Exercise 3.5 | Q 33 | Page 74

Determine the values of a and b so that the following system of linear equations have infinitely many solutions:

(2a - 1)x + 3y - 5 = 0

3x + (b - 1)y - 2 = 0

Exercise 3.5 | Q 34 | Page 57

Find the values of a and b for which the following system of linear equations has infinite the number of solutions:

2x - 3y = 7

(a + b)x - (a + b - 3)y = 4a + b

Exercise 3.5 | Q 35 | Page 57

Find the values of p and q for which the following system of linear equations has infinite a number of solutions:

2x - 3y = 9

(p + q)x + (2p - q)y = 3(p + q + 1)

Exercise 3.5 | Q 36

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

(a - b)x + (a + b)y = 3a + b - 2

Exercise 3.5 | Q 36.1 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

(2a - 1)x - 3y = 5

3x + (b - 2)y = 3

Exercise 3.5 | Q 36.2 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x - (2a + 5)y = 5

(2b + 1)x - 9y = 15

Exercise 3.5 | Q 36.3 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

(a - 1)x + 3y = 2

6x + (1 + 2b)y = 6

Exercise 3.5 | Q 36.4 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

3x + 4y = 12

(a + b)x + 2(a - b)y = 5a - 1

Exercise 3.5 | Q 36.5 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

(a - 1)x + (a + 1)y = (3a - 1)

Exercise 3.5 | Q 36.6 | Page 75

Find the values of a and b for which the following system of equations has infinitely many solutions:

2x + 3y = 7

(a - 1)x + (a + 2)y = 3a

Exercise 3.6 [Pages 78 - 79]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.6 [Pages 78 - 79]

Exercise 3.6 | Q 1 | Page 78

5 pens and 6 pencils together cost Rs 9 and 3 pens and 2 pencils cost Rs 5. Find the cost of
1 pen and 1 pencil.

Exercise 3.6 | Q 2 | Page 78

7 audio cassettes and 3 video cassettes cost Rs 1110, while 5 audio cassettes and 4 video
cassettes cost Rs 1350. Find the cost of an audio cassette and a video cassette.

Exercise 3.6 | Q 3 | Page 78

Reena has pens and pencils which together are 40 in number. If she has 5 more pencils and
5 less pens, then the number of pencils would become 4 times the number of pens. Find the
original number of pens and pencils.

Exercise 3.6 | Q 4 | Page 78

4 tables and 3 chairs, together, cost Rs 2,250 and 3 tables and 4 chairs cost Rs 1950. Find the cost of 2 chairs and 1 table.

Exercise 3.6 | Q 5 | Page 79

3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost R 324.
Find the total cost of 1 bag and 10 pens.

Exercise 3.6 | Q 6 | Page 79

5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77. Find the total cost of 1 book and 2 pens.

Exercise 3.6 | Q 7 | Page 79

A and B each have a certain number of mangoes. A says to B, “if you give 30 of your mangoes, I will have twice as many as left with you.” B replies, “if you give me 10, I will have thrice as many as left with you.” How many mangoes does each have?

Exercise 3.6 | Q 8 | Page 79

On selling a T.V. at 5%gain and a fridge at 10% gain, a shopkeeper gains Rs 2000. But if he sells the T.V. at 10% gain and the fridge at 5% loss. He gains Rs 1500 on the transaction. Find the actual prices of T.V. and fridge.

Exercise 3.6 | Q 9 | Page 79

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

Exercise 3.6 | Q 10 | Page 79

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.

Exercise 3.6 | Q 12 | Page 79

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

Exercise 3.6 | Q 13 | Page 79

A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes, I will have twice as many as left with you." B replies, "if you give me 10, I will have thrice as many as left with you." How many mangoes does each have?

Exercise 3.6 | Q 15 | Page 79

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs 2000. But if he sells the T.V. at 10% gain the fridge at 5% loss. He gains Rs 1500 on the transaction. Find the actual prices of T.V. and fridge.

Exercise 3.7 [Pages 85 - 86]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.7 [Pages 85 - 86]

Exercise 3.7 | Q 1 | Page 85

The sum of two numbers is 8. If their sum is four times their difference, find the numbers.

Exercise 3.7 | Q 2 | Page 85

The sum of digits of a two digit number is 13. If the number is subtracted from the one obtained by interchanging the digits, the result is 45. What is the number?

Exercise 3.7 | Q 3 | Page 86

A number consist of two digits whose sum is five. When the digits are reversed, the number becomes greater by nine. Find the number.

Exercise 3.7 | Q 4 | Page 86

The sum of digits of a two number is 15. The number obtained by reversing the order of digits of the given number exceeds the given number by 9. Find the given number.

Exercise 3.7 | Q 5 | Page 86

The sum of a two-digit number and the number formed by reversing the order of digit is 66. If the two digits differ by 2, find the number. How many such numbers are there?

Exercise 3.7 | Q 6 | Page 86

The sum of two numbers is 1000 and the difference between their squares is 256000. Find the numbers.

Exercise 3.7 | Q 7 | Page 86

The sum of a two digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by 3, find the number.

Exercise 3.7 | Q 8 | Page 86

A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits are reversed. Find the number.

Exercise 3.7 | Q 9 | Page 86

A two-digit number is 3 more than 4 times the sum of its digits. If 8 is added to the number, the digits are reversed. Find the number.

Exercise 3.7 | Q 10 | Page 86

A two-digit number is 4 more than 6 times the sum of its digits. If 18 is subtracted from the number, the digits are reversed. Find the number.

Exercise 3.7 | Q 11 | Page 86

A two-digit number is 4 times the sum of its digits and twice the product of the digits. Find the number.

Exercise 3.7 | Q 12 | Page 86

A two-digit number is such that the product of its digits is 20. If 9 is added to the number, the digits interchange their places. Find the number.

Exercise 3.7 | Q 13 | Page 86

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

Exercise 3.7 | Q 14 | Page 86

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.

Exercise 3.7 | Q 15 | Page 86

Seven times a two-digit number is equal to four times the number obtained by reversing the digits. If the difference between the digits is 3. Find the number.

Exercise 3.8 [Pages 88 - 89]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.8 [Pages 88 - 89]

Exercise 3.8 | Q 1 | Page 88

The numerator of a fraction is 4 less than the denominator. If the numerator is decreased by 2 and denominator is increased by 1, then the denominator is eight times the numerator. Find the fraction.

Exercise 3.8 | Q 2 | Page 88

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

Exercise 3.8 | Q 3 | Page 89

A fraction becomes 1/3 if 1 is subtracted from both its numerator and denominator. It 1 is added to both the numerator and denominator, it becomes 1/2. Find the fraction.

Exercise 3.8 | Q 4 | Page 89

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

Exercise 3.8 | Q 5 | Page 89

Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is x/y

If the numerator is multiplied by 2 and the denominator is reduced by 5, the fraction becomes 6/5. Thus, we have

(2x)/(y-5)=6/5

⇒ 10x=6(y-5)

⇒ 10x=6y-30

⇒ 10x-6y+30 =0

⇒ 2(5x-3y+15)=0

⇒ 5x - 3y+15=0

If the denominator is doubled and the numerator is increased by 8, the fraction becomes 2/5. Thus, we have

(x+8)/(2y)=2/5

⇒ 5(x+8)=4y

⇒ 5x+40=4y

⇒ 5x-4y+40=0

So, we have two equations

5x-3y+15=0

5x-4y+40=0

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

x/((-3)xx40-(-4)xx15)=-y/(5xx40-5xx15)=1/(5xx(-4)-5xx(-3))

⇒ x/(-120+60)=(-y)/(200-75)=1/(-20+15)

⇒x/(-60)=-y/125=1/-5

⇒ x= 60/5,y=125/5

⇒ x=12,y=25
Hence, the fraction is 12/25

Exercise 3.8 | Q 6 | Page 89

Let the numerator and denominator of the fraction be x and y respectively. Then the fraction is x/y

If 3 is added to the denominator and 2 is subtracted from the numerator, the fraction becomes 1/4. Thus, we have

(x-2)/(y+3)=1/4

⇒ 4(x-2)=y+3

⇒ 4x-8=y+3

⇒ 4x-y-11=0

If 6 is added to the numerator and the denominator is multiplied by 3, the fraction becomes 2/3. Thus, we have

(x+6)/(3y)=2/3

⇒ 3(x+6)=6y

⇒ 3x +18 =6y

⇒ 3x-6y+18=0

⇒ 3(x-2y+6)=0

⇒ x-3y+6=0

Here x and y are unknowns. We have to solve the above equations for x and y.

By using cross-multiplication, we have

x/((-1)xx6-(-2)xx(-11))=(-y)/(4xx6-1xx(-11))=1/(4xx(-2)-1xx(-1))

⇒ x/(-6-22)=-y/(24+11)=1/(-8+1)

⇒ x/-28=-y/35=1/-7

⇒ x= 28/7,y=35/7

⇒ x= 4,y=5

Hence, the fraction is4/5

Exercise 3.8 | Q 7 | Page 89

The sum of a numerator and denominator of a fraction is 18. If the denominator is increased by 2, the fraction reduces to 1/3. Find the fraction.

Exercise 3.8 | Q 8 | Page 89

If 2 is added to the numerator of a fraction, it reduces to 1/2 and if 1 is subtracted from the denominator, it reduces to 1/3. Find the fraction.

Exercise 3.8 | Q 9 | Page 89

The sum of the numerator and 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.

Exercise 3.8 | Q 10 | Page 89

The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.

Exercise 3.8 | Q 11 | Page 89

The sum of the numerator and denominator of a fraction is 12. If the denominator is increased by 3, the fraction becomes 1/2. Find the fraction.

Exercise 3.9 [Page 92]

### RD Sharma solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.9 [Page 92]

Exercise 3.9 | Q 1 | Page 92

A father is three times as old as his son. After twelve years, his age will be twice as that of his son then. Find the their present ages.

Exercise 3.9 | Q 2 | Page 92

Ten years later, A will be twice as old as B and five years ago, A was three times as old as B. What are the present ages of A and B?

Exercise 3.9 | Q 3 | Page 92

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?

Exercise 3.9 | Q 4 | Page 92

Six years hence a man's age will be three times the age of his son and three years ago he was nine times as old as his son. Find their present ages.

Exercise 3.9 | Q 5 | Page 92

Ten years ago, a father was twelve times as old as his son and ten years hence, he will be twice as old as his son will be then. Find their present ages.

Exercise 3.9 | Q 6 | Page 92

The present age of a father is three years more than three times the age of the son. Three years hence father's age will be 10 years more than twice the age of the son. Determine their present ages.

Exercise 3.9 | Q 7 | Page 92

Father's age is three times the sum of age of his two children. After 5 years his age will be twice the sum of ages of two children. Find the age of father.

Exercise 3.9 | Q 8 | Page 92

Father's age is three times the sum of age of his two children. After 5 years his age will be twice the sum of ages of two children. Find the age of father.

Exercise 3.9 | Q 9 | Page 92

Two years ago, a father was five times as old as his son. Two year later, his age will be 8 more than three times the age of the son. Find the present ages of father and son.

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

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Exercise 3.6Exercise 3.7Exercise 3.8Exercise 3.9

## RD Sharma solutions for Class 10 Maths chapter 3 - Pair of Linear Equations in Two Variables

RD Sharma solutions for Class 10 Maths chapter 3 (Pair of Linear Equations in Two Variables) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 10 Maths solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 10 Maths chapter 3 Pair of Linear Equations in Two Variables are Elimination Method, Substitution Method, Cross - Multiplication Method, Equations Reducible to a Pair of Linear Equations in Two Variables, Simple Situational Problems, Graphical Method of Solution of a Pair of Linear Equations, Determinant of Order Two, Linear Equation in Two Variables, Simultaneous Linear Equations, Cramer’s Rule, Pair of Linear Equations in Two Variables, Relation Between Co-efficient, Inconsistency of Pair of Linear Equations, Algebraic Conditions for Number of Solutions, Simple Situational Problems, Pair of Linear Equations in Two Variables, Graphical Method of Solution of a Pair of Linear Equations, Substitution Method, Elimination Method, Cross - Multiplication Method, Equations Reducible to a Pair of Linear Equations in Two Variables, Consistency of Pair of Linear Equations, Linear Equation in Two Variables.

Using RD Sharma Class 10 solutions Pair of Linear Equations in Two Variables exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 10 prefer RD Sharma Textbook Solutions to score more in exam.

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