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

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 3x + 4y − 12 = 0 and the path of another train B is given by the equation 6x + 8y − 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 `bb(a_1/a_2,b_1/b_2)` and `bb(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 representation of the pair so formed is :

1. Intersecting lines
2. Parallel lines
3. 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.

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:

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 − 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:

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 + 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:

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:

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) :

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) :

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 − x = 8, 5y − x = 14 and y − 2x = 1

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

y = x, y = 0 and 3x + 3y = 10

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 :

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.

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.

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

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, 3y = x, x + 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:

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

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

2x + 3y = 12,

x − y = 1

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

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 + y = 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 − y = 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.

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.

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.

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.

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

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

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

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:

0.5x + 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/(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:

`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:

`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:

152x − 378y = −74
−378x + 152y = −604

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:

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

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 :

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 

`(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 = 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 :

`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 = 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 :

`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 

`x/a + y/b = a + b`

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

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.

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

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

Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:

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

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

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.

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.

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

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

Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:

The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

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

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

Form the pair of linear equation in the following problem, and find its solution (if they exist) by the elimination method:

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

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`

Form the pair of linear equation in the following problem, and find its solutions (if they exist) by the elimination method:

Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

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.

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.