#### Chapters

Chapter 2: Polynomials

Chapter 3: Pair of Linear Equations in Two Variables

Chapter 4: Quadratic Equations

Chapter 5: Arithmetic Progression

Chapter 6: Co-Ordinate Geometry

Chapter 7: Triangles

Chapter 8: Circles

Chapter 9: Constructions

Chapter 10: Trigonometric Ratios

Chapter 11: Trigonometric Identities

Chapter 12: Trigonometry

Chapter 13: Areas Related to Circles

Chapter 14: Surface Areas and Volumes

Chapter 15: Statistics

Chapter 16: Probability

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

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.1, 3.2 [Pages 12 - 31]

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.

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

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

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.

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

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

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.

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.2 [Pages 0 - 31]

Solve the following systems of equations graphically:

x − 2y = 5

2*x* + 5*y* = 12

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:

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* − 2*y* = 6

3*x* − 6*y* = 0

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

2*x* − 3*y* = 3

Solve the following systems of equations graphically:

2x + 3y = 4

x − y + 3 = 0

Solve the following systems of equations graphically:

2*x* − 3*y* + 13 = 0

3*x* − 2*y* + 12 = 0

Solve the following systems of equations graphically:

2*x* + 3*y* + 5 = 0

3*x* − 2*y* − 12 = 0

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

2*x* + 3*y* = 6

4*x* + 6*y* = 12

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

*x* − 2*y* = 5

3*x* − 6*y* = 15

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

3*x* + *y* = 8

6*x* + 2*y* = 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 is inconsistent (i.e. has no solution) :

3*x** *− 5*y* = 20

6*x* − 10*y* = −40

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

*x* − 2*y* = 6

3*x* − 6*y* = 0

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

2*y* − *x* = 9

6*y* − 3*x* = 21

3*x* − 4*y* − 1 = 0

`2x - 8/3y + 5 = 0`

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

2*y* − *x *= 8, 5*y* − *x* = 14 and *y* − 2*x* = 1

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

*y* = *x*, *y* = 0 and 3*x* + 3*y* = 10

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

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

2*x* − 3*y** *= 6, *x* + *y* = 1

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

2*y* = 4*x* − 6, 2*x* = *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*.

2*x* − 5*y* + 4 = 0,

2*x* + *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*.

3*x* + 2*y* = 12,

5*x* − 2*y* = 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*.

2*x* + *y* − 11 = 0,

*x* − *y* − 1 = 0

*y*.

x + 2y − 7 = 0,

2x − y − 4 = 0

*y*.

3*x* + *y* − 5 = 0

2*x* − *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

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*, 3*y* = *x*, *x* + *y* = 8

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

2*x* + 3*y* = 12

*x* − *y* = 1

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

3*x* + 2*y* − 4 = 0, 2*x* − 3*y* − 7 = 0

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

3*x* + 2*y* − 11 = 0

2*x* − 3*y** *+ 10 = 0

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

2*x* + 3*y* = 12,

*x* − *y* = 1

Draw the graphs of *x* − *y* + 1 = 0 and 3*x* + 2*y* − 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 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.

2*x* + *y *= 6*x* − 2*y* = −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.

2*x* − *y *= 2

4*x* − *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

*x* in each system.

2x + 3y = 8

x − 2y = −3

Draw the graphs of the following equations:

2*x* − 3*y* + 6 = 0

2*x* + 3*y* − 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.

2*x* − 3*y* + 6 = 0

2*x* + 3*y* − 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

4*x* − 5*y* − 20 = 0

3*x* + 5*y* − 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 5*x* − *y* = 5 and 3*x* − *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.

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.

Solve the following system of equations graphically:

Shade the region between the lines and the y-axis

3*x* − 4*y* = 7

5*x* + 2*y* = 3

Solve the following system of equations graphically:

Shade the region between the lines and the y-axis

4*x* − *y* = 4

3*x* + 2*y* = 14

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

*x* + 3*y* = 6

2*x* − 3*y* = 12

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.3 [Pages 0 - 46]

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:

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:

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/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:

`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:

`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:

`2x - 3/y = 9`

`3x + 7/y = 2, y != 0`

Solve the following systems of equations:

0.5*x* + 0.7*y* = 0.74

0.3*x* + 0.5*y* = 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/(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:

`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:

`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:

`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:

`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:

`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:

`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:

`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:

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 = 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/(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:

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

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

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:

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

`5(1/x) - 4(1/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:

`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:

152*x* − 378*y* = −74

−378*x* + 152*y* = −604

Solve the following systems of equations:

99*x* + 101*y* = 499

101*x* + 99*y* = 501

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

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.4 [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 + 2y + 1 = 0

2x − 3y − 12 = 0

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

3*x* + 2*y* + 25 = 0

2*x* + *y* + 10 = 0

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

2*x* + *y* = 35

3*x* + 4*y* = 65

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

2*x* − *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

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

ax + by = a^{2}

bx + ay = b^{2}

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

`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 :

`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

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 :

`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

`(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 :

`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 :

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 :

mx – my = m^{2} + n^{2}

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 :

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

x + y - 2ab = 0

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.5 [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

x - 2y - 8 = 0

5x - 10y - 10 = 0

kx + 2y - 5 = 0

3x + y - 1 = 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.

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 :

2*x* + 3*y* − 5 = 0

6*x* + *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)

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

\[kx + 3y = 2k + 1\]

\[2\left( k + 1 \right)x + 9y = 7k + 1\]

*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)

*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 à·º, 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

(a - 1)x + 3y = 2

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

3x + 4y = 12

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

2x + 3y = 7

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

2x + 3y = 7

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

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

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.

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

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?

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.

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

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

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?

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

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.

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?

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

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.

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.

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.

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.

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

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.

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

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.

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.

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

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.

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.

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.

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?

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`

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 is`4/5`

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.

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.

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.

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.

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.

#### RD Sharma solutions for Class 10 Mathematics Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.9 [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.

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?

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?

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.

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.

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.

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.

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.

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

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

RD Sharma solutions for Class 10 Mathematics 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 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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

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