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

## 10 Mathematics

#### RD Sharma 10 Mathematics ## Chapter 3: Pair of Linear Equations in Two Variables

Ex. 3.10Ex. 3.20OthersEx. 3.30Ex. 3.40Ex. 3.50Ex. 3.60Ex. 3.70Ex. 3.80Ex. 3.90

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.10, 3.20 solutions [Pages 12 - 31]

Ex. 3.10 | 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.

Ex. 3.10 | 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.

Ex. 3.10 | 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

Ex. 3.10 | 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.

Ex. 3.10 | 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.

Ex. 3.10 | 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.

Ex. 3.10 | 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.

Ex. 3.10 | 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

Ex. 3.10 | 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

Ex. 3.20 | 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

Ex. 3.20 | 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

Ex. 3.10 | 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.

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.20 solutions [Pages 0 - 31]

Solve the following systems of equations graphically:

x − 2y = 5

2x + 5y = 12

Solve the following systems of equations graphically:

x − 2y = 5

2x + 5y = 12

Solve the following systems of equations graphically:

x − 2y = 5

2x + 3y = 10

Solve the following systems of equations graphically:

x − 2y = 5

2x + 3y = 10

Solve the following systems of equations graphically:

3x + y + 1 = 0

2x − 3y + 8 = 0

Solve the following systems of equations graphically:

3x + y + 1 = 0

2x − 3y + 8 = 0

Solve the following systems of equations graphically:

2x + y − 3 = 0

2x − 3y − 7 = 0

Solve the following systems of equations graphically:

2x + y − 3 = 0

2x − 3y − 7 = 0

Solve the following systems of equations graphically:

x + y = 6

x − y = 2

Solve the following systems of equations graphically:

x + y = 6

x − y = 2

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

3x − 6y = 0

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

3x − 6y = 0

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

2x − 3y = 3

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

2x − 3y = 3

Solve the following systems of equations graphically:

2x + 3y = 4

x − y + 3 = 0

Solve the following systems of equations graphically:

2x + 3y = 4

x − y + 3 = 0

Solve the following systems of equations graphically:

2x − 3y + 13 = 0

3x − 2y + 12 = 0

Solve the following systems of equations graphically:

2x − 3y + 13 = 0

3x − 2y + 12 = 0

Solve the following systems of equations graphically:

2x + 3y + 5 = 0

3x − 2y − 12 = 0

Solve the following systems of equations graphically:

2x + 3y + 5 = 0

3x − 2y − 12 = 0

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

2x + 3y = 6

4x + 6y = 12

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

2x + 3y = 6

4x + 6y = 12

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

x − 2y = 5

3x − 6y = 15

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

x − 2y = 5

3x − 6y = 15

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

3x + y = 8

6x + 2y = 16

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

3x + y = 8

6x + 2y = 16

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

x − 2y + 11 = 0

3x − 6y + 33 = 0

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

x − 2y + 11 = 0

3x − 6y + 33 = 0

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

3x − 5y = 20

6x − 10y = −40

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

3x − 5y = 20

6x − 10y = −40

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

x − 2y = 6

3x − 6y = 0

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

x − 2y = 6

3x − 6y = 0

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

2y − x = 9

6y − 3x = 21

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

2y − x = 9

6y − 3x = 21

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

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

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

2y − = 8, 5y − x = 14 and y − 2x = 1

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

2y − = 8, 5y − x = 14 and y − 2x = 1

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

y = xy = 0 and 3x + 3y = 10

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

y = xy = 0 and 3x + 3y = 10

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

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

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

2x − 3y = 6, x + y = 1

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

2x − 3y = 6, x + y = 1

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

2y = 4x − 6, 2x = y + 3

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

2y = 4x − 6, 2x = y + 3

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

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

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

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

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

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

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

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

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

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

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

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

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

y = x, y = 2x and y + x = 6

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

y = x, y = 2x and y + x = 6

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

y = x, 3y = xx + y = 8

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

y = x, 3y = xx + y = 8

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

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

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

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

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

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

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

2x + 3y = 12,

x − y = 1

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

2x + 3y = 12,

x − y = 1

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.

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.

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.

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.

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

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

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

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

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

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

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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

3x − 4y = 7
5x + 2y = 3

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

3x − 4y = 7
5x + 2y = 3

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

4x − y = 4
3x + 2y = 14

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

4x − y = 4
3x + 2y = 14

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

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

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.30 solutions [Pages 0 - 46]

Solve the following systems of equations:

11x + 15y + 23 = 0

7x – 2y – 20 = 0

Solve the following systems of equations:

11x + 15y + 23 = 0

7x – 2y – 20 = 0

Solve the following systems of equations:

3x − 7y + 10 = 0
y − 2x − 3 = 0

Solve the following systems of equations:

3x − 7y + 10 = 0
y − 2x − 3 = 0

Solve the following systems of equations:

0.4x + 0.3y = 1.7
0.7x − 0.2y = 0.8

Solve the following systems of equations:

0.4x + 0.3y = 1.7
0.7x − 0.2y = 0.8

Solve the following systems of equations:

x/2 + y = 0.8

7/(x + y/2) = 10

Solve the following systems of equations:

x/2 + y = 0.8

7/(x + y/2) = 10

Solve the following systems of equations:

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

Solve the following systems of equations:

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

Solve the following systems of equations:

x/7 + y/3 = 5

x/2 - y/9 = 6

Solve the following systems of equations:

x/7 + y/3 = 5

x/2 - y/9 = 6

Solve the following systems of equations:

x/3 + y/4 =11

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

Solve the following systems of equations:

x/3 + y/4 =11

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

Solve the following systems of equations:

4u + 3y = 8

6u - 4y = -5

Solve the following systems of equations:

4u + 3y = 8

6u - 4y = -5

Solve the following systems of equations:

x + y/2 = 4

x/3 + 2y = 5

Solve the following systems of equations:

x + y/2 = 4

x/3 + 2y = 5

Solve the following systems of equations:

x + 2y = 3/2

2x + y = 3/2

Solve the following systems of equations:

x + 2y = 3/2

2x + y = 3/2

Solve the following systems of equations:

sqrt2x + sqrt3y = 0

sqrt3x - sqrt8y = 0

Solve the following systems of equations:

sqrt2x + sqrt3y = 0

sqrt3x - sqrt8y = 0

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

2x - 3/y = 9

3x + 7/y = 2,  y != 0

Solve the following systems of equations:

2x - 3/y = 9

3x + 7/y = 2,  y != 0

Solve the following systems of equations:

0.5x + 0.7y = 0.74
0.3x + 0.5y = 0.5

Solve the following systems of equations:

0.5x + 0.7y = 0.74
0.3x + 0.5y = 0.5

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

(x + y)/(xy) = 2

(x - y)/(xy) = 6

Solve the following systems of equations:

(x + y)/(xy) = 2

(x - y)/(xy) = 6

Solve the following systems of equations:

15/u + 2/v = 17

Solve the following systems of equations:

15/u + 2/v = 17

Solve the following systems of equations:

3/x - 1/y = -9

2/x + 3/y  = 5

Solve the following systems of equations:

3/x - 1/y = -9

2/x + 3/y  = 5

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

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

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

2/sqrtx + 3/sqrty = 2

4/sqrtx - 9/sqrty = -1

Solve the following systems of equations:

2/sqrtx + 3/sqrty = 2

4/sqrtx - 9/sqrty = -1

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

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

Solve the following systems of equations:

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

"xy"/(y- x) = 6

Solve the following systems of equations:

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

"xy"/(y- x) = 6

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

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

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

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

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

Solve the following systems of equations:

4/x + 3y = 14

3/x - 4y = 23

Solve the following systems of equations:

x + y = 5xy
3x + 2y = 13xy

Solve the following systems of equations:

4/x + 3y = 14

3/x - 4y = 23

Solve the following systems of equations:

x + y = 5xy
3x + 2y = 13xy

Solve the following systems of equations:

x+y = 2xy

(x - y)/(xy) = 6   x != 0, y != 0

Solve the following systems of equations:

x+y = 2xy

(x - y)/(xy) = 6   x != 0, y != 0

Solve the following systems of equations:

2(3u − ν) = 5uν
2(u + 3ν) = 5uν

Solve the following systems of equations:

2(3u − ν) = 5uν
2(u + 3ν) = 5uν

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

Solve the following systems of equations:

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

4/x + 15y = 21

3/x + 4y = 5

Solve the following systems of equations:

4/x + 15y = 21

3/x + 4y = 5

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

5/(x - 1) + 1/(y - 2)  = 2

Solve the following systems of equations:

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

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

Solve the following systems of equations:

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

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

Solve the following systems of equations:

5/(x - 1) + 1/(y - 2)  = 2

Solve the following systems of equations:

152x − 378y = −74
−378x + 152y = −604

Solve the following systems of equations:

152x − 378y = −74
−378x + 152y = −604

Solve the following systems of equations:

99x + 101y = 499
101x + 99y = 501

Solve the following systems of equations:

99x + 101y = 499
101x + 99y = 501

Solve the following systems of equations:

23x − 29y = 98
29x − 23y = 110

Solve the following systems of equations:

23x − 29y = 98
29x − 23y = 110

Solve the following systems of equations:

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

Solve the following systems of equations:

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

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.40 solutions [Pages 0 - 58]

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

x/a + y/b = a + b

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

x/a + y/b = a + b

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

x + 2y + 1 = 0
2x − 3y − 12 = 0

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

x + 2y + 1 = 0
2x − 3y − 12 = 0

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

3x + 2y + 25 = 0
2x + y + 10 = 0

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

3x + 2y + 25 = 0
2x + y + 10 = 0

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

2x + y = 35
3x + 4y = 65

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

2x + y = 35
3x + 4y = 65

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

2x − y = 6
x − y = 2

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

2x − y = 6
x − y = 2

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

(x + y)/(xy) = 2

(x - y)/(xy) = 6

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

(x + y)/(xy) = 2

(x - y)/(xy) = 6

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

ax + by = a − b
bx − ay = a + b

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

ax + by = a − b
bx − ay = a + b

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

x + ay = b
ax − by = c

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

x + ay = b
ax − by = c

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

ax + by = a2
bx + ay = b2

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

ax + by = a2
bx + ay = b2

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

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

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

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

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

x/a = y/b

ax + by = a^2 + b^2

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

x/a = y/b

ax + by = a^2 + b^2

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

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

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

ax + by = (a + b)/2

3x + 5y = 4

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

ax + by = (a + b)/2

3x + 5y = 4

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

2ax + 3by = a + 2b
3ax + 2by = 2a + b

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

2ax + 3by = a + 2b
3ax + 2by = 2a + b

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

5ax + 6by = 28
3ax + 4by = 18

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

5ax + 6by = 28
3ax + 4by = 18

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

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

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

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

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)

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)

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

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

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

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

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

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

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

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

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

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

6(bx - ay) = 3b - 2a

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

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

6(bx - ay) = 3b - 2a

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

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

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

mx – my = m2 + n2

x + y = 2m

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

mx – my = m2 + n2

x + y = 2m

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

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

ax - by = 2ab

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

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

ax - by = 2ab

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

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

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.50 solutions [Pages 0 - 75]

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

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

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

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

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

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

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

4x + ky + 8 = 0

2x + 2y + 2 = 0

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

4x - 5y = k

2x - 3y = 12

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

x + 2y = 3

5x + ky + 7 = 0

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

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

4x + 5y = 3

kx + 15y = 9

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

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

8x + 5y = 9

kx + 10y = 18

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)

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)

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)

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

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

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)

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)

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

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

kx - 5y = 2

6x + 2y = 7

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

x + 2y = 0

2x + ky = 5

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

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

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

2x + ky = 11
5x − 7y = 5

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

kx + 3y = 3

12x + ky = 6

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

4x + 6y - 11 = 0

2x + ky - 7 = 0

For what value of α, the system of equations

αx + 3y = α - 3

12x + αy =  α

will have no solution?

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.

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?

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

x + 2y + 7 = 0

2x + ky + 14 = 0

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

ax + by = c

lx + my = n

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

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

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)

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

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

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

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

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

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)

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

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.60 solutions [Pages 78 - 79]

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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?

Ex. 3.60 | 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?

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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.

Ex. 3.60 | 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

Ex. 3.60 | 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

Ex. 3.60 | 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?

Ex. 3.60 | 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?

Ex. 3.60 | 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.

Ex. 3.60 | 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.

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.70 solutions [Pages 85 - 86]

Ex. 3.70 | Q 1 | Page 85

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

Ex. 3.70 | Q 1 | Page 85

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

Ex. 3.70 | 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?

Ex. 3.70 | 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?

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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?

Ex. 3.70 | 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?

Ex. 3.70 | Q 6 | Page 86

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

Ex. 3.70 | Q 6 | Page 86

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

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | Q 13 | Page 86

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

Ex. 3.70 | Q 13 | Page 86

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

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

Ex. 3.70 | 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.

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.80 solutions [Pages 88 - 89]

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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?

Ex. 3.80 | 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?

Ex. 3.80 | 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

Ex. 3.80 | 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

Ex. 3.80 | 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

Ex. 3.80 | 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

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

Ex. 3.80 | 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.

#### Chapter 3: Pair of Linear Equations in Two Variables Exercise 3.90 solutions [Page 92]

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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?

Ex. 3.90 | 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?

Ex. 3.90 | 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?

Ex. 3.90 | 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?

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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.

Ex. 3.90 | 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

Ex. 3.10Ex. 3.20OthersEx. 3.30Ex. 3.40Ex. 3.50Ex. 3.60Ex. 3.70Ex. 3.80Ex. 3.90

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

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

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