# NCERT Solutions for Class 9 Maths Chapter 4- Linear Equations In Two Variables

## NCERT Solutions Class 9 Maths Chapter 4 โ Free PDF Download

**NCERT Solutions for Class 9 Maths Chapter 4 **Linear Equations In Two Variables are considered to be very useful when you are preparing for the CBSE Class 9 Maths exams. Here, we bring to you detailed answers and solutions to the exercises of Solutions Of NCERT Class 9 Maths Chapter 4. Subject matter experts who created the solutions have collected these questions for you to revise from Chapter 4 of NCERT Textbook. We provide you the accurate solutions to all the questions that are covered in the NCERT books. These NCERT Solutions will rely on the syllabus and its guidelines. You will get enough practise solving these exercises and it will also help you to score high marks.

It helps to give you proper knowledge about the subject and the topic โLinear equationsโ. Does a linear equation in two variables have a solution? If yes, is it unique? What does the solution look like on the Cartesian plane? You shall also use the concepts you studied in Chapter 3 and the NCERT Solutions of Class 9 will also give you an idea about these concepts. These questions have been devised as per the **NCERT syllabus** and the guidelines.Chapter 4 Linear Equations In Two Variables

### Download PDF of NCERT Solutions for Class 9 Maths Chapter 4- Linear Equations In Two Variables

**List of Exercises in class 9 Maths Chapter 4**

Exercise 4.1 Solutions 2 Questions (1 Short Answer, 1 Main Question with 8 short answer questions under it)

Exercise 4.2 Solutions 4 Questions (2 MCQs, 1 Main Questions with 3 equations to solve as part of it, 1 Short Answer Questions)

Exercise 4.3 Solutions 8 Questions (4 Long Answer Questions, 2 Short Answer Questions, 1 MCQ, 1 Main Question with 5 Sub-Questions Under It)

Exercise 4.4 Solutions 2 Questions (1 main question with 2 sub-section under it, 1 Main question with 2 sub-section under it)

### Access Answers of Maths NCERT class 9 Chapter 4 โ Linear Equations In Two Variables

## Exercise 4.1 Page: 68

**1. The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this statement.**

**(Take the cost of a notebook to be โน x and that of a pen to be โน y)**

Solution:

Let the cost of a notebook to be = โน x

Let the cost of a pen to be = โน y

According to the question,

The cost of a notebook is twice the cost of a pen.

i.e., Cost of a notebook = 2รCost of a pen

x = 2รy

x = 2y

x-2y = 0

x-2y = 0 is the linear equation in two variables to represent the statement โThe cost of a notebook is twice the cost of a penโ.

**2. Express the following linear equations in the form ax + by + c = 0 and indicate the values of a, b and c in each case:**

**(ii) x โ(y/5)โ10 = 0**

Solution:

The equation x โ(y/5)-10 = 0 can be written as,

1x+(-1/5)y +(โ10) = 0

Now comparing x+(-1/5)y+(โ10) = 0 with ax+by+c = 0

We get,

a = 1

b = -(1/5)

c = -10

**(iii) โ2x+3y = 6**

Solution:

โ2x+3y = 6

Re-arranging the equation, we get,

โ2x+3yโ6 = 0

The equation โ2x+3yโ6 = 0 can be written as,

(โ2)x+3y+(โ 6) = 0

Now comparing (โ2)x+3y+(โ6) = 0 with ax+by+c = 0

We get, a = โ2

b = 3

c =6

**(iv) x = 3y**

Solution:

x = 3y

Re-arranging the equation, we get,

x-3y = 0

The equation x-3y=0 can be written as,

1x+(-3)y+(0)c = 0

Now comparing 1x+(-3)y+(0)c = 0 with ax+by+c = 0

We get, a = 1

b = -3

c =0

**(v) 2x = โ5y**

Solution:

2x = โ5y

Re-arranging the equation, we get,

2x+5y = 0

The equation 2x+5y = 0 can be written as,

2x+5y+0 = 0

Now comparing 2x+5y+0= 0 with ax+by+c = 0

We get, a = 2

b = 5

c = 0

**(vi) 3x+2 = 0**

Solution:

3x+2 = 0

The equation 3x+2 = 0 can be written as,

3x+0y+2 = 0

Now comparing 3x+0+2= 0 with ax+by+c = 0

We get, a = 3

b = 0

c = 2

**(vii) yโ2 = 0**

Solution:

yโ2 = 0

The equation yโ2 = 0 can be written as,

0x+1y+(โ2) = 0

Now comparing 0x+1y+(โ2) = 0with ax+by+c = 0

We get, a = 0

b = 1

c = โ2

**(viii) 5 = 2x**

Solution:

5 = 2x

Re-arranging the equation, we get,

2x = 5

i.e., 2xโ5 = 0

The equation 2xโ5 = 0 can be written as,

2x+0yโ5 = 0

Now comparing 2x+0yโ5 = 0 with ax+by+c = 0

We get, a = 2

b = 0

c = -5

## Exercise 4.2 Page: 70

**1. Which one of the following options is true, and why?**

**y = 3x+5 has**

**A unique solution****Only two solutions****Infinitely many solutions**

Solution:

Let us substitute different values for x in the linear equation y = 3x+5,

x | 0 | 1 | 2 | โฆ. | 100 |

y, where y=3x+5 | 5 | 8 | 11 | โฆ. | 305 |

From the table, it is clear that x can have infinite values, and for all the infinite values of x, there are infinite values of y as well.

Hence, (iii) infinitely many solutions is the only option true.

**2. Write four solutions for each of the following equations:**

**(i) 2x+y = 7**

Solution:

To find the four solutions of 2x+y =7 we substitute different values for x and y

Let x = 0

Then,

2x+y = 7

(20)+y = 7

y = 7

(0,7)

Let x = 1

Then,

2x+y = 7

(2ร1)+y = 7

2+y = 7

y = 7-2

y = 5

(1,5)

Let y = 1

Then,

2x+y = 7

(2x)+1 = 7

2x = 7-1

2x = 6

x = 6/2

x = 3

(3,1)

Let x = 2

Then,

2x+y = 7

(2ร2)+y = 7

4+y = 7

y =7-4

y = 3

(2,3)

The solutions are (0, 7), (1,5), (3,1), (2,3)

**(ii) ฯx+y = 9**

Solution:

To find the four solutions of ฯx+y = 9 we substitute different values for x and y

Let x = 0

Then,

ฯx+y = 9

(ฯ0)+y = 9

y = 9

(0,9)

Let x = 1

Then,

ฯx +y = 9

(ฯร1)+y = 9

ฯ+y = 9

y = 9-

(1, 9-)

Let y = 0

Then,

ฯx+y = 9

ฯx+0 = 9

ฯx = 9

x = 9/

(9/,0)

Let x = -1

Then,

ฯx + y = 9

(ร-1) + y = 9

-+y = 9

y = 9+ฯ

(-1,9+)

The solutions are (0,9), (1,9-), (9/,0), (-1,9+)

**(iii) x = 4y**

Solution:

To find the four solutions of x = 4y we substitute different values for x and y

Let x = 0

Then,

x = 4y

0 = 4y

4y= 0

y = 0/4

y = 0

(0,0)

Let x = 1

Then,

x = 4y

1 = 4y

4y = 1

y = 1/4

(1,1/4)

Let y = 4

Then,

x = 4y

x= 4ร4

x = 16

(16,4)

Let y =

Then,

x = 4y

x = 4ร1

x = 4

(4,1)

The solutions are (0,0), (1,1/4), (16,4), (4,1)

**3. Check which of the following are solutions of the equation xโ2y = 4 and which are not:**

**(i) (0, 2)**

**(ii) (2, 0)**

**(iii) (4, 0)**

**(iv) (โ2, 4โ2)**

**(v) (1, 1)**

Solutions:

**(i) (0, 2)**

(x,y) = (0,2)

Here, x=0 and y=2

Substituting the values of x and y in the equation xโ2y = 4, we get,

xโ2y = 4

โน 0 โ (2ร2) = 4

But, -4 4

(0, 2) is **not** a solution of the equation xโ2y = 4

**(ii) (2, 0)**

(x,y) = (2, 0)

Here, x = 2 and y = 0

Substituting the values of x and y in the equation x -2y = 4, we get,

x -2y = 4

โน 2-(2ร0) = 4

โน 2 -0 = 4

But, 2 โ 4

(2, 0) is **not** a solution of the equation x-2y = 4

**(iii) (4, 0)**

Solution:

(x,y) = (4, 0)

Here, x= 4 and y=0

Substituting the values of x and y in the equation x -2y = 4, we get,

xโ2y = 4

โน 4 โ 2ร0 = 4

โน 4-0 = 4

โน 4 = 4

(4, 0) is a solution of the equation xโ2y = 4

**(iv) (โ2,4โ2)**

Solution:

(x,y) = (โ2,4โ2)

Here, x = โ2 and y = 4โ2

Substituting the values of x and y in the equation xโ2y = 4, we get,

x โ2y = 4

โน โ2-(2ร4โ2) = 4

โ2-8โ2 = 4

But, -7โ2 โ 4

(โ2,4โ2) is **not** a solution of the equation xโ2y = 4

**(v) (1, 1)**

Solution:

(x,y) = (1, 1)

Here, x= 1 and y= 1

Substituting the values of x and y in the equation xโ2y = 4, we get,

x โ2y = 4

โน 1 -(2ร1) = 4

โน 1-2 = 4

But, -1 โ 4

(1, 1) is **not** a solution of the equation xโ2y = 4

**4. Find the value of k, if x = 2, y = 1 is a solution of the equation 2x+3y = k.**

Solution:

The given equation is

2x+3y = k

According to the question, x = 2 and y = 1.

Now, Substituting the values of x and y in the equation2x+3y = k,

We get,

(2ร2)+(3ร1) = k

โน 4+3 = k

โน 7 = k

k = 7

The value of k, if x = 2, y = 1 is a solution of the equation 2x+3y = k, is 7.

## Exercise 4.3 Page: 74

**1. Draw the graph of each of the following linear equations in two variables:**

**(i) x+y = 4**

Solution:

To draw a graph of linear equations in two variables, let us find out the points to plot.

To find out the points, we have to find the values which x and y can have, satisfying the equation.

Here,

x+y = 4

Substituting the values for x,

When x = 0,

x+y = 4

0+y = 4

y = 4

When x = 4,

x+y = 4

4+y = 4

y = 4โ4

y = 0

x | y |

0 | 4 |

4 | 0 |

The points to be plotted are (0, 4) and (4,0)

**(ii) xโy = 2**

Solution:

To draw a graph of linear equations in two variables, let us find out the points to plot.

To find out the points, we have to find the values which x and y can have, satisfying the equation.

Here,

xโy = 2

Substituting the values for x,

When x = 0,

xโy = 2

0 โ y = 2

y = โ 2

When x = 2,

xโy = 2

2โy = 2

โ y = 2โ2

โy = 0

y = 0

x | y |

0 | โ 2 |

2 | 0 |

The points to be plotted are (0, โ 2) and (2, 0)

**(iii) y=3x**

Solution:

To draw a graph of linear equations in two variables, let us find out the points to plot.

To find out the points, we have to find the values which x and y can have, satisfying the equation.

Here,

y = 3x

Substituting the values for x,

When x = 0,

y = 3x

y = 3ร0

y = 0

When x = 1,

y = 3x

y = 3ร1

y = 3

x | y |

0 | 0 |

1 | 3 |

The points to be plotted are (0, 0) and (1, 3)

**(iv) 3 = 2x+y**

Solution:

To draw a graph of linear equations in two variables, let us find out the points to plot.

To find out the points, we have to find the values which x and y can have, satisfying the equation.

Here,

3 = 2x+y

Substituting the values for x,

When x = 0,

3 = 2x+y

3 = 2ร0+y

3 = 0+y

y = 3

When x = 1,

3= 2x+y

3 = 2ร1+y

3 = 2+y

y = 3โ2

y = 1

x | y |

0 | 3 |

1 | 1 |

The points to be plotted are (0, 3) and (1, 1)

**2. Give the equations of two lines passing through (2, 14). How many more such lines are there, and why?**

Solution:

We know that infinite number of lines passes through a point.

Equation of 2 lines passing through (2,14) should be in such a way that it satisfies the point.

Let the equation be, 7x = y

7xโy = 0

When x = 2 and y = 14

(7ร2)-14 = 0

14โ14 = 0

0 = 0

L.H.S = R.H.S

Let another equation be, 4x = y-6

4x-y+6 = 0

When x = 2 and y = 14

(4ร2โ14+6 = 0

8โ14+6 = 0

0 = 0

L.H.S = R.H.S

Since both the equations satisfies the point (2,14), than say that the equations of two lines passing through (2, 14) are 7x = y and 4x = y-6

We know that, infinite number of line passes through one specific point. Since there is only one point (2,14) here, there can be infinite lines that passes through the point.

**3. If the point (3, 4) lies on the graph of the equation 3y = ax+7, find the value of a.**

Solution:

The given equation is

3y = ax+7

According to the question, x = 3 and y = 4

Now, Substituting the values of x and y in the equation 3y = ax+7,

We get,

(3ร4) = (aร3)+7

โน 12 = 3a+7

โน 3a = 12โ7

โน 3a = 5

โน a = 5/3

The value of a, if the point (3,4) lies on the graph of the equation 3y = ax+7 is 5/3.

**4. The taxi fare in a city is as follows: For the first kilometer, the fare is **โน**8 and for the subsequent distance it is **โน**5 per km. Taking the distance covered as x km and total fare as **โน** y, write a linear equation for this information, and draw its graph.**

Solution:

Given,

Total distance covered = x

Total fare = y

Fare for the first kilometer = 8 per km

Fare after the first 1km = 5 per km

If x is the total distance, then the distance after one km = (x-1)km

i.e., Fare after the first km = 5(x-1)

According to the question,

The total fare = Fare of first km+ fare after the first km

y = 8+5(x-1)

y = 8+5(x-1)

y = 8+5x โ 5

y = 5x+3

Solving the equation,

When x = 0,

y = 5x+3

y = 5ร0+3

y = 3

When y = 0,

y = 5x+3

o = 5x+3

5x = -3

x = -3/5

x | y |

0 | 3 |

-3/5 | 0 |

The points to be plotted are (0, 3) and (-3/5, 0)**5. From the choices given below, choose the equation whose graphs are given in Fig. 4.6 and Fig. 4.7.**

For Fig. 4. 6

(i) y = x

(ii) x+y = 0

(iii) y = 2x

(iv) 2+3y = 7x

Solution:

The points given in the figure 4.6 are (0,0), (-1,1), (1,-1)

Substituting the values for x and y from these points in the equations, we get,

(i) y = x

(0,0) โน 0 = 0

(-1, 1) โน -1 โ 1 โโโโโโโโโ equation not satisfied

(1, -1) โน 1โ -1 โโโโโโโโโ equation not satisfied

(ii) x+y = 0

(0,0) โน 0+0 = 0

(-1, 1) โน -1+1 = 0

(1, -1) โน 1+(-1) =0

(iii) y = 2x

(0,0) โน 0 = 2ร0

0 = 0

(-1, 1) โน 1 = 2ร(-1)

1โ -2 โโโโโโโโโ equation not satisfied

(1, -1) โน -1 = 2ร1

-1 โ 2 โโโโโโโโโ equation not satisfied

(iv) 2+3y = 7x

(0,0) โน 2+(30) = 7ร0

2 โ 0 โโโโโโโโโ equation not satisfied

(-1, 1) โน 2+(3ร1) = 7ร-1

5 โ -7 โโโโโโโโโ equation not satisfied

(1, -1) โน 2+(3ร-1) = 7ร1

-1 โ 7 โโโโโโโโโ equation not satisfied

Since, only equation x+y = 0 satisfies all the points, the equation whose graphs are given in Fig. 4.6 is

x+y = 0

For Fig. 4. 7

(i) y = x+2

(ii) y = xโ2

(iii) y = โx+2

(iv) x+2y = 6

Solution:

The points given in the figure 4.7 are (0,2), (2,0), (-1,3)

Substituting the values for x and y from these points in the equations, we get,

(i) y = x+2

(0,2) โน2 = 0+2

2 = 2

(2, 0) โน 0= 2+2

0 โ 4 โโโโโโโโโ equation not satisfied

(-1, 3) โน 3 = -1+2

3 โ 1 โโโโโโโโโ equation not satisfied

(ii) y = xโ2

(0,2) โน 2 = 0โ2

2 โ -2 โโโโโโโโโ equation not satisfied

(2, 0) โน 0 = 2โ2

0= 0

(-1, 3) โน 3= โ1โ2

3 โ โ3 โโโโโโโโโ equation not satisfied

(iii) y = โx+2

(0,2) โน 2 = -0+2

2 = 2

(2, 0) โน 0 = -2+2

0 = 0

(-1, 3) โน 3= -(-1)+2

3 = 3

(iv) x+2y = 6

(0,2) โน 0+(2ร2) = 6

4 โ 6 โโโโโโโโโ equation not satisfied

(2, 0) โน 2+(2ร0) = 6

2 โ 6 โโโโโโโโโ equation not satisfied

(-1, 3) โน -1+(2ร3) = 6

5 โ 6 โโโโโโโโโ equation not satisfied

Since, only equation y = โx+2satisfies all the points, the equation whose graphs are given in Fig. 4.7 is

y = โx+2

**6. If the work done by a body on application of a constant force is directly proportional to the distance travelled by the body, express this in the form of an equation in two variables and draw the graph of the same by taking the constant force as 5 units. Also read from the graph the work done when the distance travelled by the body is**

**(i) 2 units**

**(ii) 0 unit**

Solution:

Let the distance traveled by the body be x and the force applied on the body be y.

It is given that,

The work done by a body is directly proportional to the distance travelled by the body.

According to the question,

y โ x

y = 5x (5 is a constant of proportionality)

Solving the equation,

(i) when x = 2 units,

then y = 5ร2 = 10 units

(2, 10)

(ii) when x = 0 units,

then y = 5ร0 = 0 units.

(0, 0)

The points to be plotted are (2, 10) and (0, 0)

**7. Yamini and Fatima, two students of Class IX of a school, together contributed โน 100 towards the Prime Ministerโs Relief Fund to help the earthquake victims. Write a linear equation which satisfies this data. (You may take their contributions as โน x and โน y.) Draw the graph of the same.**

Solution:

Let Yaminiโs donation be โนx and Fatimaโs donation be โนy

According to the question;

x+y = 100

We know that,

when x = 0 , y = 100

when x = 50, y = 50

when x = 100, y = 0

The points to be plotted are (0,100), (50,50), (100,0)

**8. In countries like USA and Canada, temperature is measured in Fahrenheit, whereas in countries like India, it is measured in Celsius. Here is a linear equation that converts Fahrenheit to Celsius:**

**(i) Draw the graph of the linear equation above using Celsius for x-axis and Fahrenheit for y-axis.**

**(ii) If the temperature is 30ยฐC, what is the temperature in Fahrenheit?**

**(iii) If the temperature is 95ยฐF, what is the temperature in Celsius?**

**(iv) If the temperature is 0ยฐC, what is the temperature in Fahrenheit and if the temperature is 0ยฐF, what is the temperature in Celsius?**

**(v) Is there a temperature which is numerically the same in both Fahrenheit and Celsius? If yes, find it.**

Solution:

(i) According to the question,

F = (9/5)C + 32

Solving the equation,

We get,

When C = 0, F = 32

When C = -10 , F = 14

The points to be plotted are (0, 32), (-10, 14)

(ii) When C = 30,

F = (9/5)C +32

F = (9ร30)/5+32

= (9ร6)+32

= 54+32

= 86^{o}F

(iii) When F = 95,

95 = (9/5)C +32

(9/5)C = 95-32

(9/5)C =63

C = (63ร5)/9

=35^{o}C

(iv) When C = 0,

F = (9/5)C +32

F = (9ร0)/5 +32

=0+32

=32^{o}F

When F = 0,

0 = (9/5)C+32

(9/5)C = 0-32

(9/5)C = -32

C = (-32ร5)/9

=-17.7777

=-17.8^{o}C

(v) When F = C,

C = (9/5)C+32

C โ (9/5)C = 32

(5-9)C/5 =32

(-4/5)C = 32

(-4/5)C = (-32ร5)/4

= โ 40^{o}C

Hence, -40^{o} is the temperature which is numerically the same in both Fahrenheit and Celsius.

## Exercise 4.4 Page: 77

**1. Give the geometric representations of y = 3 as an equation**

**(i) in one variable**

**(ii) in two variables**

Solution:

- In one variable, y = 3

(ii) In two variable, 0x+y = 3

When x = 0, y = 3

When x = 1, y = 3

**2. Give the geometric representations of 2x+9 = 0 as an equation**

**(i) in one variable**

**(ii) in two variables**

Solution:

(i) In one variable,

2x+9 = 0

2x = -9

x = -9/2

x = -4.5

(ii) In two variables,

2x+9 = 0

2x+0y+9 = 0

When y = 0, x = -4.5

When y = 1, x = -4.5

### Summary of NCERT Solutions for Class 9 Maths Chapter 4 Linear Equations In Two Variables

โLinear Equations In Two Variablesโ is chapter 4 of the **NCERT Textbook**, and it falls under the unit 2 Algebra. For the final exam, from the unit Algebra you usually get 7 questions that is 1 multiple choice question of 1 mark, 2 short answer with reasoning for a total of 4 marks, 3 short answer questions of total 9 marks and 1 long answer question of 6 marks. Thus, the total weightage for the unit is 20 marks. Thus, the total weightage for the unit is 20 marks. Topics covered under this chapter is listed below.

Chapter 4 โ Linear Equations In Two Variables |

4.1 โ Introduction |

4.2 โ Linear Equations |

4.3 โ Solution of a Linear Equation |

4.4 โ Graph of a Linear Equation In Two Variables |

4.5 โ Equations of Lines Parallel to the x-axis and y-axis |

### NCERT Solutions for Class 9 Maths Chapter 4 โ Linear Equations In Two Variables

In this chapter, the knowledge of linear equations in one variable is recalled and extended to that of two variables. Any equation which can be put in the form ax + by + c = 0, where a, b and c are real numbers, and a and b are not both zero, is called a linear equation in two variables. The NCERT solutions for Class 9 Maths offer chapter wise explanation of the exercises provided in the textbook and students can easily understand this Linear Equation concept of Algebra with the help of easy examples provided.

**Key Benefits of NCERT Solutions for Class 9 Maths Chapter 4-Linear Equations In Two Variables**

In this chapter, you have studied about Linear Equations In Two Variables. And here we will see how the **NCERT Solutions** for Class 9 Maths Chapter 4 can benefit the students:

- It is created on the basis of the NCERT syllabus
- It features all questions under each exercise section of the textbook
- Students get a clear idea about the concept and topic
- Solving the questions of the solutions will help them to self evaluate their performance
- Students can prepare for the exams on the basis of their knowledge gap

## Frequently Asked Questions on NCERT Solutions for Class 9 Maths Chapter 4

### Give me a summary of exercises present in NCERT Solutions for Class 9 Maths Chapter 4.

There are five exercises 5 exercises present in NCERT Solutions for Class 9 Maths Chapter 4. viz,

4.1 โ Introduction

4.2 โ Linear Equations

4.3 โ Solution of a Linear Equation

4.4 โ Graph of a Linear Equation In Two Variables

4.5 โ Equations of Lines Parallel to the x-axis and y-axis

### Is it necessary to learn all the questions present in NCERT Solutions for Class 9 Maths Chapter 4?

Yes, if you want to score good in your finals then have to practice all the questions and formulae related to it. The solutions present in BYJUโS website are very accurate and clear. Students can start practicing NCERT Solutions for Class 9 Maths Chapter 4 to score higher marks. These solutions can be helpful not only for the exam preparation but also in solving homework and assignments.

### Is NCERT Solutions for Class 9 Maths Chapter 4 difficult to understand?

No not at all, if you practice regularly NCERT Solutions for Class 9 Maths Chapter 4 is not much difficult to understand. The main aim of these solutions is to provide a fundamental aspect of Maths, which in turn, helps the students to understand every concept clearly.