Math is tough but solving them is impossible said no one ever!! Make use of these NCERT Solutions For Class 10 Maths for finding solutions and also note that NCERT Solutions for Class 10 Maths PDF are available online for extra reference. This chapter includes NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2. Practice more so you score more!!

## NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Exercise 3.2

Question 1.

From the pair of linear equations in the following problems and find their solutions graphically.

(i) 10 students of Class X took part in 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.

(ii) 5 pencils and 7 pens together cost ₹ 50, whereas 7 pencils and 5 pens together cost ₹ 45. Find the cost of one pencil and one pen.

Solution:

(i) Let the number of boys participating in mathematics quiz is x and the number of girls participating in mathematics quiz is y.

Therefore the equations form in this situations are

(ii) Let the cost of 1 pencil is ₹ x

and the cost of 1 pen is ₹ y

Therefore the linear equations on the given situations are

5x + 7y = 50 … (i)

and 7x + 5y = 46 … (ii)

Again from equation (ii)

7x + 5y = 46

∴ x = \(\frac{46-5 y}{7}\)

Multiply equation. (1) by 7 and (ii) by 5 the get

35x + 49y = 350 … (i)

35x + 25y = 230 …(ii)

Substracting equation (ii) from equation (i) we get

24y = 120

y = 5

Putting the value of y in equation (j)

We get

5x + 35 = 50

5x = 15

x = 5

Cost of one pencil = ₹ 3

Cost of one pen = ₹ 5

Question 2.

On comparing the ratios \(\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}\) and \(\frac{c_{1}}{c_{2}}\) find out whether the lines representing the following pairs of linear equations intersect at a point, parallel or coincident.

(i) 5x – 4y + 8 = 0

7x + 6y-9 = 0

(ii) 9x + 3y + 12 = 0

18x + 6y + 24 = 0

(iii) 6x – 3y + 10 = 0

2x – y + 9 = 0

Solution:

(i) We have,

5x – 4y + 8 = 0

and 7x + 6y – 9 = 0

Here, a_{1} = 5, b_{1} = – 4, c_{1} = 8

and a_{2} = 7, b_{2} = 6 and c_{2} = – 9

Therefore, \(\frac{5}{7} \neq \frac{-4}{6}\)

So, \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

Therefore the following pair of linear equations intersect each other at point.

(ii) We have,

9x + 3y + 12 = 0

18x + 6y + 24 = 0

Here, a_{1} = 9, b_{1} = 3, c_{1} = 12

and a_{2} = 18, b_{2} = 6 and c_{2} = 24

Therefore, \(\frac{9}{18}=\frac{3}{6}=\frac{12}{24}=\frac{1}{2}\)

So, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

Therefore the lines representing the given pair of linear equations are coincident.

(iii) We have given,

6x – 3y + 10 = 0

2x – y + 9 = 0

Here, a_{1} = 6, b_{1} = – 3, c_{1} = 10

and a_{2} = 2, b_{2} = – 1 and c_{2} = 9

Therefore,

\(\frac{6}{2}=\frac{-3}{-1} \neq \frac{10}{9}\)

So, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}\)

Therefore the lines representing the given pair of linear equations are coincident.

Question 3.

On comparing the ratios \(\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}\) and \(\frac{c_{1}}{c_{2}}\), find out whether the following pair of linear equations are consistent or inconsistent.

(i) 3% + 2y = 5

2x – 3y = 7

(ii) 2x – 3y = 8

4x – 6y = 9

(iii) \(\frac { 3 }{ 2 }\)x + \(\frac { 5 }{ 3 }\)y = 7

9x – 10y = 14

(iv) 5x – 3y = 11

– 10x + 6y = – 22

(v) \(\frac { 4 }{ 3 }\)x + 2y = 8

2x + 3y = 12

Solution:

(i) We have given,

3x + 2y = 5

2x – 3y = 7

We can also write

3x + 2y – 5 = 0

2x – 3y – 7 = 0

Here, a_{1} = 3, b_{1} = 2, c_{1} = – 5

and a_{2} = 2, b_{2} = – 3 and c_{2} = – 7

So, \(\frac { 3 }{ 2 }\) ≠ \(\frac { 2 }{ – 3 }\)

Therefore, \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

So the given pair of linear equations are consistent.

(ii) We have given,

2x – 3y = 8

4x – 4y = 9

We can also write

2x – 3y – 8 = 0

4x – 6y – 3 = 0

Here, a_{1} = 2, b_{1} = – 3, c_{1} = – 8

and a_{2} = 4, b_{2} = – 6 and c_{2} = – 3

So, \(\frac{2}{4}=\frac{-3}{-6} \neq \frac{-8}{-9}\)

Therefore,

So, the given pair of linear equations are inconsistent.

(iii) We have given,

\(\frac { 3 }{ 2 }\)x + \(\frac { 5 }{ 3 }\)y = 7

9x – 10y = 14

We can also write,

\(\frac { 3 }{ 2 }\)x + \(\frac { 5 }{ 3 }\)y – 7 = 0

9x – 10y – 14 = 0

Here, a_{1} = \(\frac { 3 }{ 2 }\), b_{1} = \(\frac { 5 }{ 3 }\), c_{1} = – 7

and a_{2} = 9, b_{2} = – 10 and c_{2} = – 14

So, the given pair of linear equations consistent.

(iv) We have given,

5x – 3y = 11

– 10x + 6y = – 22

We can also write,

5x – 3y – 11 = 0

– 10x + 6y + 22 = 0

So, \(\frac { 5 }{ -10 }\) = \(\frac { – 3 }{ 6 }\) = \(\frac { -11 }{ 22 }\)

Here, a_{1} = 5, b_{1} = – 3, c_{1} = – 11

and a_{2} = – 10, b_{2} = 6 and c_{2} = 22

Therefore, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

So, the given pair of linear equations consistent.

(v) We have given

\(\frac { 4 }{ 3 }\)x + 2y = 8

2x + 3y = 12

We can also write,

\(\frac { 4 }{ 3 }\)x + 2y – 8 = 0

2x + 3y – 12 = 0

Here, a_{1} = \(\frac { 4 }{ 3 }\), b_{1} = 2, c_{1} = 8

and a_{2} = 2, b_{2} = 3 and c_{2} = – 12

So, the given pair of linear equations consistent.

Question 4.

Which of the following pairs of linear equations are consistent inconsistent ? If consistent, obtain the solution graphically:

(i) x + y = 5

2x + 2y = 10

(ii) x – y = 8

3x – 3y = 16

(iii) 2x + y – 6 = 0

4x – 2y – 4 = 0

(iv) 2x – 2y – 2 = 0

4x – 4y – 5 = 0

Solution:

We have given

x + y = 5

2x + 2y = 10

We can also write,

x + y – 5 = 0

2x + 2y – 10 = 0

Here, a_{1} = 1, b_{1} = 1, c_{1} = – 5

and a_{2} = 2, b_{2} = 2 and c_{2} = – 10

So, \(\frac { 1 }{ 2 }\) = \(\frac { 1 }{ 2 }\) = \(\frac { – 5 }{ – 10 }\)

Therefore, \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

So, the given pair of linear equations are consistent from equation (i) we have,

So the given pair of linear equations are inconsistent.

(iii) We have given

2x + y -6 = 0 … (i)

4x – 2y – 4 = 0 … (ii)

Here, a_{1} = 2, b_{1} = 1, c_{1} = – 6

and a_{2} = 4, b_{2} = – 2 and c_{2} = – 4

Therefore, \(\frac{2}{4} \neq \frac{1}{-2}\)

So, \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\)

Therefore, the given linear equations are consistent From equation (i)

Therefore, the given linear equations are inconsistent.

Question 5.

Half the perimeter of rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Solution:

Let the length of the rectangular garden by y

and the breadth of the rectangular garden by y

According to question

x = y + 4

x – y = 4 … (i)

Perimeter of rectangular garden = 72

2(x + y)= 72

2x + 2y = 72 … (ii)

Multiplying equation (i) and (ii) and adding both equations, we get

2x = 40

x = 20 m

Putting this value in equation (i) we get

20 – y = 4

y = 16 m

Hence the length of garden is 20 m and the width of garden is 16 m.

Question 6.

Given the linear equation 2x + 3y – 8 = 0 write another linear equaton in two variables such that the geometrical representation of the pair so formed is (i) intersecting lines (ii) parallel lines, (iii) coincident lines.

Solution:

Given equation is 2x + 3y – 8 = 0 i.e., a_{1} = 2, b_{1} = 3, c_{1} = 8

(i)

one such equation can be 5x + 4y + 1 = 0 (Try forming other equations. How many such equation can be ?)

One such equation can be 2x + 3y + 5 = 0 (Try forming other equations. How many such equation can be ?]

(iii) Try yourself.

Question 7.

Draw the graphs of the equation x- y +1 = of the vertices of the triangle formed by these line

Solution:

Verticles of the triangle are (-1, 0) (4, 0) and (2, 3).

We hope you found these NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables Ex 3.2 helped you with finding the solutions easily.

Make sure to practice the remaining parts of the Class 10 Maths NCERT Solutions English Medium Chapter 3.

- Pair of Linear Equations in Two Variables Class 10 Ex 3.1
- Pair of Linear Equations in Two Variables Class 10 Ex 3.2
- Pair of Linear Equations in Two Variables Class 10 Ex 3.3
- Pair of Linear Equations in Two Variables Class 10 Ex 3.4
- Pair of Linear Equations in Two Variables Class 10 Ex 3.5
- Pair of Linear Equations in Two Variables Class 10 Ex 3.6
- Pair of Linear Equations in Two Variables Class 10 Ex 3.7