Back to: RS Aggarwal Class 10 Solutions Linear equations in two variables

**RS Aggarwal Class 10 Solutions Chapter 3 Linear equations in two variables**

RS Aggarwal Class 10 Solutions

**Exercise 3A**

**Question 1:**

**Read More:**

- What is a Linear Equation
- Linear Equations In One Variable
- Linear Equations In Two Variables
- Graphical Method Of Solving Linear Equations In Two Variables

**Question 2:**

**Question 3:**

**Question 4:**

**Question 5:**

**Question 6:**

**Question 7:**

**Question 8:**

**Question 9:**

**Question 10:**

**Question 11:**

**Question 12:**

**Question 13:**

**Question 14:**

**Question 15:**

**Question 16:**

**Question 17:**

**Question 18:**

**Question 19:**

**Question 20:**

**Question 21:**

**Question 22:**

**Question 23:**

**Question 24:**

**Question 25:**

**Question 26:**

**Question 27:**

**Question 28:**

**Exercise 3B**

**Question 1:**

**Question 2:**

**Question 3:**

**Question 4:**

**Question 5:**

**Question 6:**

**Question 7:**

**Question 8:**

**Question 9:**

**Question 10:**

**Question 11:**

**Question 12:**

**Question 13:**

**Question 14:**

**Question 15:**

**Question 16:**

**Question 17:**

**Question 18:**

**Question 19:**

**Question 20:**

**Question 21:**

**Question 22:**

Taking L.C.M, we get

Multiplying (1) by 1 and (2) by

Subtracting (4) from (3), we get

Substituting x = ab in (3), we get

Therefore solution is x = ab, y = ab

**Question 23:**

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

6ax + 6by = 3a + 2b —(1)

6(bx – ay) = 3b – 2a

6bx – 6ay = 3b- 2a —(2)

6ax + 6by = 3a + 2b —(1)

6bx – 6ay = 3b – 2a —(2)

Multiplying (1) by a and (2) by b

Adding (3) and (4), we get

Substituting in (1), we get

Hence, the solution is

**Question 24:**

**Question 25:**

The given equations are

71x + 37y = 253 —(1)

37x + 71y = 287 —(2)

Adding (1) and (2)

108x + 108y = 540

108(x + y) = 540

—-(3)

Subtracting (2) from (1)

34x – 34y = 253 – 287 = -34

34(x – y) = -34

—(4)

Adding (3) and (4)

2x = 5 – 1= 4

Subtracting (4) from (3)

2y = 5 + 1 = 6

Hence solution is x = 2, y = 3

**Question 26:**

37x + 43y = 123 —-(1)

43x + 37y = 117 —-(2)

Adding (1) and (2)

80x + 80y = 240

80(x + y) = 240

x + y = —-(3)

Subtracting (1) from (2),

6x – 6y = -6

6(x – y) = -6

—-(4)

Adding (3) and (4)

2x = 3 – 1 = 2

Subtracting (4) from (3),

2y = 3 + 1 = 4

Hence solution is x = 1, y = 2

**Question 27:**

217x + 131y = 913 —(1)

131x + 217y = 827 —(2)

Adding (1) and (2), we get

348x + 348y = 1740

348(x + y) = 1740

x + y = 5 —-(3)

Subtracting (2) from (1), we get

86x – 86y = 86

86(x – y) = 86

x – y = 1 —(4)

Adding (3) and (4), we get

2x = 6

x = 3

putting x = 3 in (3), we get

3 + y = 5

y = 5 – 3 = 2

Hence solution is x = 3, y = 2

**Question 28:**

41x – 17y = 99 —(1)

17x – 41y = 75 —(2)

Adding (1) and (2), we get

58x – 58y = 174

58(x – y) = 174

x – y = 3 —(3)

subtracting (2) from (1), we get

24x + 24y = 24

24(x + y) = 24

x + y = 1 —(4)

Adding (3) and (4), we get

2x = 4 x = 2

Putting x = 2 in (3), we get

2 – y = 3

-y = 3 – 2 y = -1

Hence solution is x =2, y = -1

**Exercise 3C**

**Question 1:**

x + 2y + 1 = 0 —(1)

2x – 3y – 12 = 0 —(2)

By cross multiplication, we have

Hence, x = 3 and y = -2 is the solution

**Question 2:**

2x + 5y – 1 = 0 —(1)

2x + 3y – 3 = 0 —(2)

By cross multiplication we have

Hence the solution is x = 3, y = -1

**Question 3:**

3x – 2y + 3 = 0

4x + 3y – 47 = 0

By cross multiplication we have

Hence the solution is x = 5, y = 9

**Question 4:**

6x – 5y – 16 = 0

7x – 13y + 10 = 0

By cross multiplication we have

Hence the solution is x = 6, y = 4

**Question 5:**

3x + 2y + 25 = 0

2x + y + 10 = 0

By cross multiplication, we have

Hence the solution is x = 5, y = -20

**Question 6:**

2x + y – 35 = 0

3x + 4y – 65 = 0

By cross multiplication, we have

**Question 7:**

7x – 2y – 3 = 0

By cross multiplication, we have

Hence x = 1, y = 2 is the solution

**Question 8:**

**Question 9:**

ax + by – (a – b) = 0

bx – ay – (a + b) = 0

By cross multiplication, we have

**Question 10:**

2ax + 3by – (a + 2b) = 0

3ax + 2by – (2a + b) = 0

By cross multiplication, we have

**Question 11:**

By cross multiplication, we have

**Question 12:**

By cross multiplication, we have

**Question 13:**

By cross multiplication we have

**Question 14:**

Taking

u + v – 7 = 0

2u + 3v – 17 = 0

By cross multiplication, we have

Hence the solution is

**Question 15:**

Let

in the equation

5u – 2v + 1 = 0

15u + 7v – 10 = 0

**Question 16:**

**Exercise 3D**

**Question 1:**

**Question 2:**

**Question 3:**

3x – 5y – 7 = 0

6x – 10y – 3 = 0

Hence the given system of equations is inconsistent

**Question 4:**

2x – 3y – 5 = 0, 6x – 9y – 15 = 0

These equations are of the form

Hence the given system of equations has infinitely many solutions

**Question 5:**

kx + 2y – 5 = 0

3x – 4y – 10 = 0

These equations are of the form

This happens when

[latex]k\neq \frac { -3 }{ 2 }[/latex]

Thus, for all real value of k other that , the given system equations will have a unique solution

(ii) For no solution we must have

Hence, the given system of equations has no solution if [latex]k=\frac { -3 }{ 2 } [/latex]

**Question 6:**

x + 2y – 5 = 0

3x + ky + 15 = 0

These equations are of the form of

Thus for all real value of k other than 6, the given system of equation will have unique solution

(ii) For no solution we must have

Therefore k = 6

Hence the given system will have no solution when k = 6.

**Question 7:**

x + 2y – 3 = 0, 5x + ky + 7 = 0

These equations are of the form

(i) For a unique solution we must have

Thus, for all real value of k other than 10

The given system of equation will have a unique solution.

(ii) For no solution we must have

Hence the given system of equations has no solution if

For infinite number of solutions we must have

This is never possible since

There is no value of k for which system of equations has infinitely many solutions

**Question 8:**

8x + 5y – 9 = 0

kx + 10y – 15 = 0

These equations are of the form

Clearly, k = 16 also satisfies the condition

Hence, the given system will have no solution when k = 16.

**Question 9:**

kx + 3y – 3 = 0 —-(1)

12x + ky – 6 = 0 —(2)

These equations are of the form

Hence, the given system will have no solution when k = -6

**Question 10:**

3x + y – 1 = 0

(2k – 1)x + (k – 1)y – (2k + 1) = 0

These equations are of the form

Thus,

Hence the given equation has no solution when k = 2

**Question 11:**

(3k + 1)x + 3y – 2 = 0

these equations are of the form

Thus, k = -1 also satisfy the condition

Hence, the given system will have no solution when k = -1

**Question 12:**

The given equations are

3x – y – 5 = 0 —(1)

6x – 2y + k = 0—(2)

Equations (1) and (2) have no solution, if

**Question 13:**

kx + 2y – 5 = 0

3x + y – 1 = 0

These equations are of the form

Thus, for all real values of k other than 6, the given system of equations will have a unique solution

**Question 14:**

x – 2y – 3 = 0

3x + ky – 1 = 0

These equations are of the form of

Thus, for all real value of k other than -6, the given system of equations will have a unique solution

**Question 15:**

kx + 3y – (k – 3) = 0

12x + ky – k = 0

These equations are of the form

Thus, for all real value of k other than , the given system of equations will have a unique solution

**Question 16:**

4x – 5y – k = 0, 2x – 3y – 12 = 0

These equations are of the form

Thus, for all real value of k the given system of equations will have a unique solution

**Question 17:**

2x + 3y – 7 = 0

(k – 1)x + (k + 2)y – 3k = 0

These are of the form

This hold only when

Now the following cases arises

Case : I

Case: II

Case III

For k = 7, there are infinitely many solutions of the given system of equations

**Question 18:**

2x + (k – 2)y – k = 0

6x + (2k – 1)y – (2k + 5) = 0

These are of the form

For infinite number of solutions, we have

This hold only when

Case (1)

Case (2)

Case (3)

Thus, for k = 5 there are infinitely many solutions

**Question 19:**

kx + 3y – (2k +1) = 0

2(k + 1)x + 9y – (7k + 1) = 0

These are of the form

For infinitely many solutions, we must have

This hold only when

Now, the following cases arise

Case – (1)

Case (2)

Case (3)

Thus, k = 2, is the common value for which there are infinitely many solutions

**Question 20:**

5x + 2y – 2k = 0

2(k +1)x + ky – (3k + 4) = 0

These are of the form

For infinitely many solutions, we must have

These hold only when

Case I

Thus, k = 4 is a common value for which there are infinitely by many solutions.

**Question 21:**

x + (k + 1)y – 5 = 0

(k + 1)x + 9y – (8k – 1) = 0

These are of the form

For infinitely many solutions, we must have

**Question 22:**

(k – 1)x – y – 5 = 0

(k + 1)x + (1 – k)y – (3k + 1) = 0

These are of the form

For infinitely many solution, we must now

k = 3 is common value for which the number of solutions is infinitely many

**Question 23:**

(a – 1)x + 3y – 2 = 0

6x + (1 – 2b)y – 6 = 0

These equations are of the form

For infinite many solutions, we must have

Hence a = 3 and b = -4

**Question 24:**

(2a – 1)x + 3y – 5 = 0

3x + (b – 1)y – 2 = 0

These equations are of the form

These holds only when

**Question 25:**

2x – 3y – 7 = 0

(a + b)x + (a + b – 3)y – (4a + b) = 0

These equation are of the form

For infinite number of solution

Putting a = 5b in (2), we get

Putting b = -1 in (1), we get

Thus, a = -5, b = -1

**Question 27:**

The given equations are

2x + 3y = 7 —-(1)

a(x + y) – b(x – y) = 3a + b – 2 —(2)

Equation (2) is

ax + ay – bx + by = 3a + b – 2

(a – b)x + (a + b)y = 3a + b -2

Comparing with the equations

There are infinitely many solution

2a + 2b = 3a – 3b and 3(3a + b – 2) = 7(a + b)

-a = -5b and 9a + 3b – 6 = 7a + 7b

a = 5b and 9a – 7a + 3b – 7b = 6

or 2a – 4b = 6

or a – 2b = 3

thus equation in a, b are

a = 5b —(3)

a – 2b = 3 —(4)

putting a = 5b in (4)

5b – 2b = 3 or 3b = 3 Þ b = 1

Putting b = 1 in (3)

a = 5 and b = 1

**Question 28:**

We have 5x – 3y = 0 —(1)

2x + ky = 0 —(2)

Comparing the equation with

These equations have a non – zero solution if

**Exercise 3E**

**Question 1:**

Let the cost of 1 chair be Rs x and the cost of one table be Rs. y

The cost of 5 chairs and 4 tables = Rs(5x + 4y) = Rs. 2800

5x + 4y = 2800 —(1)

The cost of 4 chairs and 3 tables = Rs(4x + 3y) = Rs. 2170

4x + 3y = 2170 —(2)

Multiplying (1) by 3 and (2) by 4, we get

15x + 12y = 8400 —(3)

16x + 12y = 8680 —(4)

Subtracting (3) and (4), we get

x = 280

Putting value of x in (1), we get

5 × 280 + 4y = 2800

or 1400 + 4y = 2800

or 4y = 1400

[latex]y=\frac { 1400 }{ 4 } =350 [/latex]

Thus, cost of 1 chair = Rs. 280 and cost of 1 table = Rs. 350

**Question 2:**

Let the cost of a pen and a pencil be Rs x and Rs y respectively

Cost of 37 pens and 53 pencils = Rs(37x + 53y) = Rs 820

37x + 53y = 820 —(1)

Cost of 53 pens and 37 pencils = Rs(53x + 37y) = Rs 980

53x + 37y = 980 —(2)

Adding (1) and (2), we get

90x + 90y = 1800

x + y = 20 —(3)

y = 20 – x

Putting value of y in (1), we get

37x + 53(20 – x) = 820

37x + 1060 – 53x = 820

16x = 240

[latex]x=\frac { 240 }{ 16} =15 [/latex]

From (3), y = 20 – x = 20 – 15 = 5

x = 15, y = 5

Thus, cost of a pen = Rs 15 and cost of pencil = Rs 5

**Question 3:**

Let the number of 20 P and 25 P coins be x and y respectively

Total number of coins x + y = 50

i.e., x + y = 50 —(1)

Multiplying (1) by 5 and (2) by 1, we get

5x + 5y = 250 —(3)

4x + 5y = 230 —(4)

Subtracting (4) from (3), we get

x = 20

Putting x = 20 in (1),

y = 50 – x

= 50 – 20

= 30

Hence, number of 20 P coins = 20 and number of 25 P coins = 30

**Question 4:**

Let the two numbers be x and y respectively.

Given:

x + y = 137 —(1)

x – y = 43 —(2)

Adding (1) and (2), we get

2x = 180

[latex]y=\frac { 180 }{ 2} =90 [/latex]

Putting x = 90 in (1), we get

90 + y = 137

y = 137 – 90

= 47

Hence, the two numbers are 90 and 47.

**Question 5:**

Let the first and second number be x and y respectively.

According to the question:

2x + 3y = 92 —(1)

4x – 7y = 2 —(2)

Multiplying (1) by 7 and (2) by 3, we get

14 x+ 21y = 644 —(3)

12x – 21y = 6 —(4)

Adding (3) and (4), we get

[latex]26x=650\\ x=\frac { 650 }{ 26 } =25 [/latex][/latex]

Putting x = 25 in (1), we get

2 × 25 + 3y = 92

50 + 3y = 92

3y = 92 – 50

[latex]y=\frac { 42 }{ 3 } =14 [/latex]

y = 14

**Question 6:**

Let the first and second numbers be x and y respectively.

According to the question:

3x + y = 142 —(1)

4x – y = 138 —(2)

Adding (1) and (2), we get

[latex]7x=280\\x=\frac { 280 }{ 7 } =40 [/latex]

Putting x = 40 in (1), we get

3 × 40 + y = 142

y = 142 – 120

y = 22

Hence, the first and second numbers are 40 and 22.

**Question 7:**

Let the greater number be x and smaller be y respectively.

According to the question:

2x – 45 = y

2x – y = 45 —(1)

and

2y – x = 21

-x + 2y = 21 —(2)

Multiplying (1) by 2 and (2) by 1

4x – 2y = 90 —(3)

-x + 2y = 21 —(4)

Adding (3) and (4), we get

3x = 111

[latex]x=\frac { 111 }{ 3 } =37 [/latex]

Putting x = 37 in (1), we get

2 × 37 – y = 45

74 – y = 45

y = 29

Hence, the greater and the smaller numbers are 37 and 29.

**Question 8:**

Let the larger number be x and smaller be y respectively.

We know,

Dividend = Divisor × Quotient + Remainder

3x = y × 4 + 8

3x – 4y = 8 —(1)

And

5y = x × 3 + 5

-3x + 5y = 5 —(2)

Adding (1) and (2), we get

y = 13

putting y = 13 in (1)

Hence, the larger and smaller numbers are 20 and 13 respectively.

**Question 9:**

Let the required numbers be x and y respectively.

Then,

Therefore,

2x – y = -2 —(1)

11x – 5y = 24 —(2)

Multiplying (1) by 5 and (2) by 1

10x – 5y = -10 —(3)

11x – 5y = 24 —(4)

Subtracting (3) and (4) we get

x = 34

putting x = 34 in (1), we get

2 × 34 – y = -2

68 – y = -2

-y = -2 – 68

y = 70

Hence, the required numbers are 34 and 70.

**Question 10:**

Let the numbers be x and y respectively.

According to the question:

x – y = 14 —(1)

From (1), we get

x = 14 + y —(3)

putting x = 14 + y in (2), we get

Putting y = 9 in (1), we get

x – 9 = 14

x = 14 + 9 = 23

Hence the required numbers are 23 and 9

**Question 11:**

Let the ten’s digit be x and units digit be y respectively.

Then,

x + y = 12 —(1)

Let the ten’s digit of required number be x and its unit’s digit be y respectively

Required number = 10x + y

10x + y = 7(x + y)

10x + y = 7x + 7y

3x – 6y = 0 —(1)

Number found on reversing the digits = 10y + x

(10x + y) – 27 = 10y + x

10x – x + y – 10y = 27

9x – 9y = 27

(x – y) = 27

x – y = 3 —(2)

Multiplying (1) by 1 and (2) by 6

3x – 6y = 0 —(3)

6x – 6y = 18 —(4)

Subtracting (3) from (4), we get

[latex]3x=18\\ x=\frac { 18 }{ 3 } =6 [/latex]

Putting x = 6 in (1), we get

3 × 6 – 6y = 0

18 – 6y = 0

[latex]-6y=-18\\ y=\frac { -18 }{ -6 } =3 [/latex]

Number = 10x + y

= 10 × 6 + 3

= 60 + 3

= 63

Hence the number is 63.

**Question 12:**

Let the ten’s digit and unit’s digits of required number be x and y respectively.

Required number = 10x + y

Number obtained on reversing digits = 10y + x

According to the question:

10y + x × (10x + y) = 18

10y + x – 10x – y = 18

9y – 9x = 18

y – x = 2 —-(2)

Adding (1) and (2), we get

[latex]2y=14\\ y=\frac { 14 }{ 2 } =7 [/latex]

Putting y = 7 in (1), we get

x + 7 = 12

x = 5

Number = 10x + y

= 10 × 5 + 7

= 50 + 7

= 57

Hence, the number is 57.

**Question 13:**

Let the ten’s digit and unit’s digits of required number be x and y respectively.

Then,

x + y = 15 —(1)

Required number = 10x + y

Number obtained by interchanging the digits = 10y + x

10y + x × (10x + y) = 9

10y + x – 10x – y = 9

9y – 9x = 9

Add (1) and (2), we get

[latex]2y=16\\ y=\frac { 16 }{ 2 } =8 [/latex]

Putting y = 8 in (1), we get

x + 8 = 15

x = 15 – 8 = 7

Required number = 10x + y

= 10 × 7 + 8

= 70 + 8

= 78

Hence the required number is 78.

**Question 14:**

Let the ten’s and unit’s of required number be x and y respectively.

Then, required number =10x + y

According to the given question:

10x + y = 4(x + y) + 3

10x + y = 4x + 4y + 3

6x – 3y = 3

2x – y = 1 —(1)

And

10x + y + 18 = 10y + x

9x – 9y = -18

x – y = -2 —(2)

Subtracting (2) from (1), we get

x = 3

Putting x = 3 in (1), we get

2 × 3 – y = 1

y = 6 – 1 = 5

x = 3, y = 5

Required number = 10x + y

= 10 × 3 + 5

= 30 + 5

= 35

Hence, required number is 35.

**Question 15:**

Let the ten’s digit and unit’s digit of required number be x and y respectively.

We know,

Dividend = (divisor × quotient) + remainder

According to the given questiion:

10x + y = 6 × (x + y) + 0

10x – 6x + y – 6y = 0

4x – 5y = 0 —(1)

Number obtained by reversing the digits is 10y + x

10x + y – 9 = 10y + x

9x – 9y = 9

9(x – y) = 9

(x – y) = 1 —(2)

Multiplying (1) by 1 and (2) by 5, we get

4x – 5y = 0 —(3)

5x – 5y = 5 —(4)

Subtracting (3) from (4), we get

x = 5

Putting x = 5 in (1), we get

x =5 and y = 4

Hence, required number is 54.

**Question 16:**

Let the ten’s and unit’s digits of the required number be x and y respectively.

Then, xy = 35

Required number = 10x + y

Also,

(10x + y) + 18 = 10y + x

9x – 9y = -18

9(y – x) = 18 —(1)

y – x = 2

Now,

Adding (1) and (2),

2y = 12 + 2 = 14

y = 7

Putting y = 7 in (1),

7 – x = 2

x = 5

Hence, the required number = 5 × 10 + 7

= 57

**Question 17:**

Let the ten’s and units digit of the required number be x and y respectively.

Then, xy = 14

Required number = 10x + y

Number obtained on reversing the digits = 10y + x

Also,

(10x + y) + 45 = 10y + x

9(y – x) = 45

y – x = 5 —(1)

Now,

y + x = 9 —(2) (digits cannot be negative, hence -9 is not possible)

On adding (1) and (2), we get

2y = 14

y = 7

Putting y = 7 in (2), we get

7 + x = 9

x = (9 – 7) = 2

x = 2 and y = 7

Hence, the required number is = 2 × 10 + 7

= 27

**Question 18:**

Let the ten’s and unit’s digits of the required number be x and y respectively.

Then, xy = 18

Required number = 10x + y

Number obtained on reversing its digits = 10y + x

(10x + y) – 63 = (10y + x)

9x – 9y = 63

x – y = 7 —(1)

Now,

Adding (1) and (2), we get

Putting x = 9 in (1), we get

9 – y = 7

y = 9 – 7

y = 2

x = 9, y = 2

Hence, the required number = 9 × 10 + 2

= 92.

**Question 19:**

Let the ten’s digit be x and the unit digit be y respectively.

Then, required number = 10x + y

According to the given question:

10x + y = 4(x + y)

6x – 3y = 0

2x – y = 0 —(1)

And

10x + y = 2xy —(2)

Putting y = 2x from (1) in (2), we get

Putting x = 3 in (1), we get

2 × 3 – y = 0

y = 6

Hence, the required number = 3 × 10 + 6

= 36.

**Question 20:**

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

According to the question:

x + y = 8 —(1)

And

Multiplying (1) be 3 and (2) by 1

3x + 3y = 24 —(3)

4x – 3y = -3 —(4)

Add (3) and (4), we get

[latex]7x=21\\ x=\frac { 21 }{ 7 } =3 [/latex]

Putting x = 3 in (1), we get

3 + y= 8

y = 8 – 3

y = 5

x = 3, y = 5

Hence, the fraction is [latex s= 2]\frac { x }{ y } =\frac { 3 }{ 5 } [/latex]

**Question 21:**

Let numerator and denominator be x and y respectively.

Sum of numerator and denominator = x + y

3 less than 2 times y = 2y – 3

x + y =2y – 3

or x – y = -3 —(1)

When 1 is decreased from numerator and denominator, the fraction becomes:

2(x – 1) = y – 1

or 2x – 2 = y – 1

or 2x – y = 1 —(2)

Subtracting (1) from (2), we get

x = 1 + 3 = 4

Putting x = 4 in (1), we get

y = x + 3

= 4 + 3

= 7

the fraction is [latex s= 2]\frac { x }{ y } =\frac { 4 }{ 7 } [/latex]

**Question 22:**

Let the numerator and denominator be x and y respectively.

Then the fraction is [latex s= 2]\frac { x }{ y } [/latex]

Subtracting (1) from (2), we get

x = 15

Putting x = 15 in (1), we get

2 × 15 – y = 4

30 – y = 4

y = 26

x = 15 and y = 26

Hence the given fraction is [latex s= 2]\frac { 15 }{ 26 } [/latex]

**Question 23:**

Let the numerator and denominator be x and y respectively.

Then the fraction is [latex s= 2]\frac { x }{ y } [/latex].

According to the given question:

y = x + 11

y – x = 11 —(1)

and

-3y + 4x = -8 —(2)

Multiplying (1) by 4 and (2) by 1

4y – 4x = 44 —(3)

-3y + 4x = -8 —(4)

Adding (3) and (4), we get

y = 36

Putting y = 36 in (1), we get

y – x = 11

36 – x = 11

x = 25

x = 25, y = 36

Hence the fraction is [latex s= 2]\frac { 25 }{ 36 } [/latex]

**Question 24:**

Let the numerator and denominator be x and y respectively.

Then the fraction is [latex s= 2]\frac { x }{ y } [/latex].

Subtracting (1) from (2), we get

x = 3

Putting x = 3 in (1), we get

2 × 3 – 4

-y = -4 -6

y = 10

x = 3 and y = 10

Hence the fraction is [latex s= 2]\frac { 3 }{ 10 } [/latex]

**Question 25:**

Let the fraction be [latex s= 2]\frac { x }{ y } [/latex].

When 2 is added to both the numerator and the denominator, the fraction becomes:

3x – y = -4 —(1)

When 3 is added both to the numerator and the denominator, the fractions becomes:

5x – 2y = -9 —-(2)

Multiplying (1) by 2 and (2) by 1, we get

6x – 2y = -8 —(3)

5x – 2y = -9 —(4)

Subtracting (4) from (3), we get

x = 1

Putting x = 1 in (1),

3 × 1 – y = 4

y = 7

Required fraction is [latex s= 2]\frac { 1 }{ 7 } [/latex]

**Question 26:**

Let the two numbers be x and y respectively.

According to the given question:

x + y = 16 —(1)

And

—(2)

From (2),

xy = 48

We know,

Adding (1) and (3), we get

2x = 24

x = 12

Putting x = 12 in (1),

y = 16 – x

= 16 – 12

= 4

The required numbers are 12 and 4.

**Question 27:**

Let the present ages of the man and his son be x years and y years respectively.

Then,

Two years ago:

(x – 2) = 5(y – 2)

x – 2 = 5y – 10

x – 5y = -8 —(1)

Two years later:

(x + 2) = 3(y + 2) + 8

x + 2 = 3y + 6 + 8

x – 3y = 12 —(2)

Subtracting (2) from (1), we get

-2y = -20

y = 10

Putting y = 10 in (1), we get

x – 5 × 10 = -8

x – 50 = -8

x = 42

Hence the present ages of the man and the son are 42 years and 10 respectively.

**Question 28:**

Let the present ages of A and B be x and y respectively.

Five years ago:

(x – 5) = 3(y – 5)

x – 5 = 3y – 15

x – 3y = -10 —(1)

Ten years later:

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

x + 10 = 2y + 20

x – 2y = 10 —(2)

Subtracting (2) from (1), we get

y = 20

Putting y = 20 in (1), we get

x – 3y = -10

x – 3 × 20 = -10

x = -10 + 60 = 50

x = 50, y = 20

Hence, present ages of A and B are 50 years and 20 years respectively.

**Question 29:**

Let the present ages of woman and daughter be x and y respectively.

Then,

Their present ages:

x = 3y + 3

x – 3y = 3 —(1)

Three years later:

(x + 3) = 2(y + 3) + 10

x + 3 = 2y + 6 + 10

x – 2y = 13 —(2)

Subtracting (2) from (1), we get

y = 10

Putting y = 10 in (1), we get

x – 3 × 10 = 3

x = 33

x = 33, y = 10

Hence, present ages of woman and daughter are 33 and 10 years.

**Question 30:**

Let the present ages of the mother and her son be x and y respectively.

According to the given question:

x + 2y = 70 —(1)

and

2x + y = 95 —(2)

Multiplying (1) by 1 and (2) by 2, we get

x + 2y = 70 —(3)

4x + 2y = 190 —(4)

Subtracting (3) from (4), we get

[latex s= 2]3x=120\\ y=\frac { 120 }{ 3 } =40 [/latex]

Putting x = 40 in (1), we get

40 + 2y = 70

2y = 30

y = 15

x = 40, y = 15

Hence, the ages of the mother and the son are 40 years and 15 years respectively.

**Question 31:**

Let the present age of the man and the sum of the ages of the two sons be x and y respectively.

We are given x = 3y —(1)

After 5 years the age of man = x + 5

And age of each son is increased by 5 years

Age of two sons after 5 years = y + 5 + 5 = y + 10

Now,

x + 5 = 2(y + 10)

or x + 5 = 2y + 10

x – 2y = 15 —(2)

Putting x = 3y in (2)

3y – 2y = 15

y = 15

Putting y = 15 in (1),

x = 3 × 15 = 45

Age of the man = 45 years.

**Question 32:**

Let the present age of the man and his son be x and y respectively.

Ten years later:

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

x + 10 = 2y + 20

x – 2y = 10 —(1)

Ten years ago:

(x – 10) = 4(y – 10)

x – 10 = 4y – 40

x – 4y = – 30 —(2)

Subtracting (1) from (2), we get

-2y = -40

y = 20 years

Putting y = 20 in (1), we get

x – 2 × 20 = 10

x = 50

x = 50 years, y = 20 years

Hence, present ages of the man and his son are 50 years and 20 years respectively.

**Question 33:**

Let the monthly income of A and B be Rs. 5x and Rs. 4x respectively and let their expenditures be Rs. 7y and Rs. 5y respectively.

Then,

5x – 7y = 3000 —(1)

4x – 5y = 3000 —(2)

Multiplying (1) by 5 and (2) by 7 we get

25x – 35y = 15000 —(3)

28x – 35y = 21000 —(4)

Subtracting (3) from (4), we get

3x = 6000

x = 2000

Putting x = 2000 in (1), we get

5 × 2000 – 7y = 3000

-7y = 3000 – 10000

[latex s= 2]y=\frac { -7000 }{ -7 } =1000 [/latex]

x = 2000, y = 1000

Income of A = 5x = 5 × 2000 = Rs. 10000

Income of B = 4x = 4 × 2000 = Rs. 8000

**Question 34:**

Let Rs. x and Rs. y be the CP of a chair and table respectively

If profit is 25%, then SP of chair =

If profit is 10%, then SP of the table =

SP of a chair and table = Rs. 760

Further , If profit is 10%, then SP of a chair =[latex s= 2]Rs\frac { 110 }{ 100 } x [/latex]

If profit is 25%, then SP of a table =[latex s= 2]Rs\frac { 125 }{ 100 } y [/latex]

SP of a chair and table = Rs. 767.50

Adding (1) and (2),

Subtracting (2) from (1)

Adding (3) and (4),

Subtracting (4) from (3)

Hence, CP of a chair is Rs 300 and CP of table is Rs 350.

**Question 35:**

Let the CP of TV and fridge be Rs x and Rs y respectively.

Further,

2x – y = 30000 —(2)

Multiplying (2) by 2 and (1) by 1, we get

4x – 2y = 60000 —(3)

x + 2y = 65000 —(4)

Adding (3) and (4), we get

5x = 125000

x = 25000

Putting x = 25000 in (1), we get

25000 + 2y = 65000

2y = 40000

y = 20000

The cost of TV = Rs. 25000 and cost of fridge = Rs. 20000

**Question 36:**

Let the amounts invested at 12% and 10% be Rs x and Rs y respectively.

Then,

First case:

Second case:

Multiplying (1) by 6 and (2) by 5, we get

36x + 30y = 343500 —(3)

25x + 30y = 263750 —(4)

Subtracting (4) from (3), we get

[latex s= 2]11x=79750\\ x=\frac { 79750 }{ 11 } =7250 [/latex]

Putting x = 7250 in (1), we get

6 × 7250 + 5y = 57250

43500 + 5y = 57250

5y = 13750

y = 2750

x = 7250, y = 2750

Hence, amount invested at 12% = Rs 7250

And amount invested at 10% = Rs 2750

**Question 37:**

Let the number of student in class room A and B be x and y respectively.

When 10 students are transferred from A to B:

x – 10 = y + 10

x – y = 20 —(1)

When 20 students are transferred from B to A:

2(y – 20) = x + 20

2y – 40 = x + 20

-x + 2y = 60 —(2)

Adding (1) and (2), we get

y = 80

Putting y = 80 in (1), we get

x – 80 = 20

x = 100

Hence, number of students of A and B are 100 and 80 respectively.

**Question 38:**

Let P and Q be the cars starting from A and B respectively and let their speeds be x km/hr and y km/hr respectively.

Case- I

When the cars P and Q move in the same direction.

Distance covered by the car P in 7 hours = 7x km

Distance covered by the car Q in 7 hours = 7y km

Let the cars meet at point M.

AM = 7x km and BM = 7y km

AM – BM = AB

7x – 7y = 70

7(x – y) = 70

x – y = 10 —-(1)

Case II

When the cars P and Q move in opposite directions.

Distance covered by P in 1 hour = x km

Distance covered by Q in 1 hour = y km

In this case let the cars meet at a point N.

AN = x km and BN = y km

AN + BN = AB

x + y = 70 —(2)

Adding (1) and (2), we get

2x = 80

x = 40

Putting x = 40 in (1), we get

40 – y = 10

y = (40 – 10) = 30

x = 40, y = 30

Hence, the speeds of these cars are 40 km/ hr and 30 km/ hr respectively.

**Question 39:**

Let the original speed be x km/h and time taken be y hours

Then, length of journey = xy km

Case I:

Speed = (x + 5)km/h and time taken = (y – 3)hour

Distance covered = (x + 5)(y – 3)km

(x + 5) (y – 3) = xy

xy + 5y -3x -15 = xy

5y – 3x = 15 —(1)

Case II:

Speed (x – 4)km/hr and time taken = (y + 3)hours

Distance covered = (x – 4)(y + 3) km

(x – 4)(y + 3) = xy

xy -4y + 3x -12 = xy

3x – 4y = 12 —(2)

Multiplying (1) by 4 and (2) by 5, we get

20y × 12x = 60 —(3)

-20y + 15x = 60 —(4)

Adding (3) and (4), we get

3x = 120

or x = 40

Putting x = 40 in (1), we get

5y – 3 × 40 = 15

5y = 135

y = 27

Hence, length of the journey is (40 × 27) km = 1080 km

**Question 40:**

Let the speed of train and car be x km/hr and y km/hr respectively.

Multiplying (1) by 40 and (2) by 1, we get

5000u + 2400v = 80 —(3)

1300u + 2400v = 43 —(4)

subtracting (4) from (3), we get

[latex s= 2]3700u=37\\ u=\frac { 1 }{ 100 } [/latex]

Putting [latex]u=\frac { 1 }{ 100 } [/latex] in (1), we get

Hence, speeds of the train and the car are 100km/hr and 80 km/hr respectively.

**Question 41:**

Let the speed of the boat in still water be x km/hr and speed of the stream be y km/hr.

Then,

Speed upstream = (x – y)km/hr

Speed downstream = (x + y) km/hr

Time taken to cover 12 km upstream = [latex s= 2]\frac { 12 }{ x-y } hrs [/latex]

Time taken to cover 40 km downstream = [latex s= 2]\frac { 40 }{ x+y } hrs [/latex]

Total time taken = 8hrs

Again, time taken to cover 16 km upstream = [latex s= 2]\frac { 16 }{ x-y } [/latex]

Time taken to taken to cover 32 km downstream = [latex s= 2]\frac { 32 }{ x+y } [/latex]

Total time taken = 8hrs

Putting

12u + 40v = 8

3u + 10v = 2 —(1)

and

16u + 32v = 8

2u + 4v = 1 —(2)

Multiplying (1) by 4 and (2) by 10, we get

12u + 40v = 8 —(3)

20u + 40v = 10 —(4)

Subtracting (3) from (4), we get

[latex s= 2]8u=2\\ u=\frac { 1 }{ 4 } [/latex]

Putting [latex s= 2]u=\frac { 1 }{ 4 } [/latex] in (3), we get

On adding (5) and (6), we get

2x = 12

x = 6

Putting x = 6 in (6) we get

6 + y = 8

y = 8 – 6 = 2

x = 6, y = 2

Hence, the speed of the boat in still water = 6 km/hr and speed of the stream = 2km/hr

**Question 42:**

Let the fixed charges of taxi per day be Rs x and charges for travelling for 1km be Rs y.

For travelling 110 km, he pays

Rs x + Rs 110y = Rs 1130

x + 110y = 1130 —(1)

For travelling 200 km, he pays

Rs x + Rs 200y = Rs 1850

x + 200y = 1850 —(2)

Subtracting (1) from (2), we get

[latex s= 2]90y=1850-1130=720\\ y=\frac { 720 }{ 90 } =8 [/latex]

Putting y = 8 in (1),

x + 110 × 8 = 1130

x = 1130 – 880 = 250

Hence, fixed charges = Rs 250

And charges for travelling 1 km = Rs 8

**Question 43:**

Let the fixed hostel charges be Rs x and food charges per day be Rs y respectively.

For student A:

Student takes food for 25days and he has to pay: Rs 3500

Rs x + Rs 25y = Rs 3500

x + 25y = 3500 —(1)

For student B:

Student takes food for 28days and he has to pay: Rs 3800

Rs x + Rs 28y = Rs 3800

or x + 28y = 3800 —(2)

Subtracting (1) from (2), we get

3y = 3800 – 3500

3y= 300

y = 100

Putting y = 100 in (1),

x + 25 × 100 = 3500

or x = 3500 – 2500

or x = 1000

Thus, fixed charges for hostel = Rs 1000 and

Charges for food per day = Rs 100

**Question 44:**

Let the length = x meters and breadth = y meters

Then,

x = y + 3

x – y = 3 —-(1)

Also,

(x + 3)(y – 2) = xy

3y – 2x = 6 —-(2)

Multiplying (1) by 2 and (2) by 1

-2y + 2x = 6 —(3)

3y – 2x = 6 —(4)

Adding (3) and (4), we get

y = 12

Putting y = 12 in (1), we get

x – 12 = 3

x = 15

x = 15, y = 12

Hence length = 15 metres and breadth = 12 metres

**Question 45:**

Let the length of a rectangle be x meters and breadth be y meters.

Then, area = xy sq.m

Now,

xy – (x – 5)(y + 3) = 8

xy × [xy × 5y + 3x -15] = 8

xy × xy + 5y × 3x + 15 = 8

3x – 5y = 7 —(1)

And

(x + 3)(y + 2) – xy = 74

xy + 3y +2x + 6 × xy = 74

2x + 3y = 68 —(2)

Multiplying (1) by 3 and (2) by 5, we get

9x – 15y = 21 —(3)

10x + 15y = 340 —(4)

Adding (3) and (4), we get

Putting x = 19 in (3) we get

x = 19 meters, y = 10 meters

Hence, length = 19m and breadth = 10m

**Question 46:**

Let man’s 1 day’s work be [latex s= 2]\frac { 1 }{ x } [/latex] and 1 boy’s day’s work be [latex s= 2]\frac { 1 }{ y } [/latex]

Also let [latex s= 2]\frac { 1 }{ x } =u [/latex] and [latex s= 2]\frac { 1 }{ y } =v [/latex]

Multiplying (1) by 6 and (2) by 5 we get

Subtracting (3) from (4), we get

Putting [latex s= 2]u=\frac { 1 }{ 18 } [/latex] in (1), we get

x = 18, y = 36

The man will finish the work in 18 days and the boy will finish the work in 36 days when they work alone.

**Question 47:**

∠A +∠B + ∠C = 180◦

x + 3x + y = 180

4x + y = 180 —(1)

Also,

3y – 5x = 30

-5x + 3y = 30 —(2)

Multiplying (1) by 3 and (2) by 1, we get

12x + 3y = 540 —(3)

-5x + 3y = 30 —(4)

Subtracting (4) from (3), we get

17x = 510

x = 30

Putting x = 30 in (1), we get

4 × 30 + y = 180

y = 60

Hence ∠A = 30◦, ∠B = 3 × 30◦ = 90◦, ∠C = 60◦

Therefore, the triangle is right angled.

**Question 48:**

In a cyclic quadrilateral ABCD:

∠A = (x + y + 10)°,

∠B = (y + 20)°,

∠C = (x + y – 30)°,

∠D = (x + y)°

We have, ∠A + ∠C = 180° and ∠B + ∠D = 180°

Now,

∠A + ∠C = (x + y + 10)° + (x + y – 30)° = 180°

2x + 2y – 20° = 180°

x + y – 10° = 90°

x + y = 100 —(1)

Also,

∠B + ∠D = (y + 20)° +(x + y)° = 180°

x + 2y + 20° = 180°

x + 2y = 160° —(2)

Subtracting (1) from (2), we get

y = 160 – 100 = 60

Putting y = 60 in (1), we get

x = 100 – y

x = 100 – 60

x = 40

Therefore,

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