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NCERT Solutions for Exercise 3.2 Class 10 Maths Chapter 3 - Pair of Linear Equations in two variables

NCERT Solutions for Exercise 3.2 Class 10 Maths Chapter 3 - Pair of Linear Equations in two variables

Updated on Apr 30, 2025 05:37 PM IST | #CBSE Class 10th

The exercise explains how to solve a pair of linear equations through graphical methods. The lesson establishes how two linear equations display as straight lines on a Cartesian coordinate system, where their point of intersection reveals the solution. The visualisation enables learners to determine the solution possibilities based on the line behaviour of lines intersecting or being parallel, or coinciding. The process reveals how algebra functions together with geometric principles to solve problems that exist in the real world.

This Story also Contains
  1. NCERT Solutions Class 10 Maths Chapter 3: Exercise 3.1
  2. Access Solution of Pair of Linear Equations in Two Variables Class 10 Chapter 3 Exercise: 3.1
  3. Topics covered in Chapter 3 Pair of Linear Equations in Two Variables: Exercise 3.1
  4. NCERT Solutions of Class 10 Subject Wise
  5. NCERT Exemplar Solutions of Class 10 Subject Wise
NCERT Solutions for Exercise 3.2 Class 10 Maths Chapter 3 - Pair of Linear Equations in two variables
NCERT Solutions for Exercise 3.2 Class 10 Maths Chapter 3 - Pair of Linear Equations in two variables

Understandings of two-variable linear equation behaviour in graphical representations are taught in this essential section of the NCERT Solutions for Class 10 Maths. Students who perform exercise gain deeper comprehension about linear equation graphical solutions and system consistency, and how it determines unique versus infinite solutions. The clear presentation of the solutions given in the NCERT Books enables students' understanding of the connection between graphical interpretations and algebraic processes before moving on to complex multisystem topics in higher education.

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NCERT Solutions Class 10 Maths Chapter 3: Exercise 3.1

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Access Solution of Pair of Linear Equations in Two Variables Class 10 Chapter 3 Exercise: 3.1

Q1 (i) Form the pair of linear equations in the following problems and find their solutions graphically. 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.

Answer:

Let the number of boys be x and the number of girls be y.

Now, according to the question,

Total number of students in the class = 10, i.e.

x+y=10.....(1)

And, given that the number of girls is 4 more than the number of boys it means; x=y+4

xy=4..........(2)

Different points (x, y) satisfying equation (1)

X
5
6
4
Y
5
4
6

Different points (x,y) satisfying equation (2)

X
5
6
7
y
1
2
3


Graph,

1635919752095


From the graph, both lines intersect at the point (7,3). That is x = 7 and y = 3, which means the number of boys in the class is 7 and the number of girls in the class is 3.

Q1 (ii) 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.

Answer:

Let the price of 1 pencil be x, and y be the price of 1 pen.

Now, according to the question

5x+7y=50......(1)

And

7x+5y=46......(2)

Now, the points (x,y) that satisfy the equation (1) are

X
3
-4
10
Y
5
10
0

And, the points (x,y) that satisfy the equation (2) are

X
3
8
-2
Y
5
-2
12

The Graph,

Graph


From the graph, both lines intersect at point (3,5), that is, x = 3 and y = 5, which means the cost of 1 pencil is 3 and the cost of 1 pen is 5.

Q2 (i) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (i) 5x4y+8=0;7x+6y9=0

Answer:

Given Equations,

5x4y+8=0and7x+6y9=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=57,b1b2=46andc1c2=89

It is observed that;

a1a2b1b2

It means that both lines intersect at exactly one point and have a unique solution.

Q2 (ii) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (ii) 9x+3y+12=0;18x+6y+24=0

Answer:

Given Equations,

9x+3y+12=0 and 18x+6y+24=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=918=12,b1b2=36=12andc1c2=1224=12

It is observed that;

a1a2=b1b2=c1c2

It means that both lines are coincident and have infinitely many solutions.

Q2 (iii) On comparing the ratios a1a2, b1b2and c1c2 , find out whether the lines representing the following pairs of linear equations intersect at a point, are parallel or coincident: (iii) 6x3y+10=0;2xy+9=0

Answer:

Given Equations,

6x3y+10=0 and2xy+9=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=62=3,b1b2=31=3andc1c2=109

It is observed that;

a1a2=b1b2c1c2

It means that both lines are parallel and thus have no solution.

Q3 (i) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (i) 3x+2y=5;2x3y=7

Answer:

Given Equations,

3x+2y=5 and 2x3y=7

Or, 3x+2y5=0 and 2x3y7=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=32,b1b2=23andc1c2=57

It is observed that;

a1a2b1b2

It means that the given equations have a unique solution and thus the pair of linear equations is consistent.

Q3 (iI) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (ii) 2x3y=8;4x6y=9

Answer:

Given Equations,

2x3y=8 and 4x6y=9

Or, 2x3y8=0 and 4x6y9=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=24=12,b1b2=36=12andc1c2=89

It is observed that;

a1a2=b1b2c1c2

It means the given equations have no solution, and thus the pair of linear equations is inconsistent.

Q3 (iii) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (iii) 32x+53y=7;9x10y=14

Answer:

Given Equations,

32x+53y=7 and 9x10y=14

Or, 32x+53y7=0 and 9x10y14=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=3/29=318=16,b1b2=5/310=530=16andc1c2=714=12

It is observed that;

a1a2b1b2

It means the given equations have exactly one solution, and thus the pair of linear equations is consistent.

Q3 (iv) On comparing the ratios a1a2, b1b2and c1c2, find out whether the lines representing the following pairs of linear equations are consistent, or inconsistent: (iv) 5x3y=11;10x+6y=22

Answer:

Given Equations,

5x3y=11 and 10x+6y=22

Or, 5x3y11=0 and 10x+6y+22=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=510=12,b1b2=36=12andc1c2=1122=12

It is observed that;

a1a2=b1b2=c1c2

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

Q3 (v) On comparing the ratios a1a2, b1b2and c1c2, find out whether the following pair of linear equations are consistent, or inconsistent (v) 43x+2y=8;2x+3y=12

Answer:

Given Equations,

43x+2y=8 and 2x+3y=12

Or, 43x+2y8=0 and 2x+3y12=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=4/32=46=23,b1b2=23andc1c2=812=23

It is observed that;

a1a2=b1b2=c1c2

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

Q4 (i) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: x+y=5;2x+2y=10

Answer:

Given Equations,

x+y=5 and 2x+2y=10

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=12,b1b2=12andc1c2=510=12

It is observed that;

a1a2=b1b2=c1c2

It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.

The points (x,y) which satisfy both equations are

X
1
3
5
Y
4
2
0

Graph

Q4 (ii) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: xy=8;3x3y=16

Answer:

Given Equations,

xy=8 and 3x3y=16

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=13,b1b2=13=13andc1c2=816=12

It is observed that:

a1a2=b1b2c1c2

It means the given equations have no solution, and thus the pair of linear equations is inconsistent.

Q4 (iii) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: 2x+y6=0;4x2y4=0

Answer:

Given Equations,

2x+y6=0 and 4x2y4=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=24=12,b1b2=12=12andc1c2=64=32

It is observed that;

a1a2b1b2

It means the given equations have exactly one solution, and thus the pair of linear equations is consistent.

The points(x, y) satisfying the equation 2x+y6=0 are,

X
0
2
3
Y
6
2
0


And The points(x,y) satisfying the equation 4x2y4=0 are,

X
0
1
2
Y
-2
0
2


GRAPH:

1635919975632


As we can see, both lines intersect at point (2,2) and hence the solution of both equations is x = 2 and y = 2.

Q4 (iv) Which of the following pairs of linear equations are consistent/inconsistent? If consistent, obtain the solution graphically: 2x2y2=0;4x4y5=0

Answer:

Given Equations,

2x2y2=0,4x4y5=0

Comparing these equations with a1x+b1y+c1=0anda2x+b2y+c2=0 , we get

a1a2=24=12,b1b2=24=12andc1c2=25=25

It is observed that;

a1a2=b1b2c1c2

It means the given equations have no solution, and thus the pair of linear equations is inconsistent.

Q5 Half the perimeter of a rectangular garden, whose length is 4 m more than its width, is 36 m. Find the dimensions of the garden.

Answer:

Let l be the length of the rectangular garden and b be the width.

Now, according to the question, the length is 4 m more than its width, so we can write it as l=b+4

Or, lb=4....(1)

Also given Half Parameter of the rectangle = 36 it means l+b=36....(2)

Now, as we have two equations, add both equations, and we get,

l+b+lb=4+36

2l=40

l=20

We get the value of l, which is 20m

Now, putting this in equation (1), we get;

20b=4

b=204

b=16

Hence, the Length and width of the rectangle are 20m and 16m, respectively.

Q6 (i) Given the linear equation 2x+3y8=0, write another linear equation in two variables such that the geometrical representation of the pair so formed is: intersecting lines

Answer:

Given the equation,

2x+3y8=0

We know that the condition for the intersection of lines for the equations in the form a1x+b1y+c1=0 and a2x+b2y+c2=0 is,

a1a2b1b2

So any line with this condition can be 4x+3y16=0

Proof,

a1a2=24=12

b1b2=33=1

Hence, 121 it means a1a2b1b2

Therefore, the pair of lines has a unique solution, thus forming intersecting lines.

Q6 (ii) Given the linear equation 2x+3y8=0 , write another linear equation in two variables such that the geometrical representation of the pair so formed is parallel lines

Answer:

Given the equation,

2x+3y8=0

As we know that the condition for the parallel lines for the equations in the form a1x+b1y+c1=0 and a2x+b2y+c2=0 is,

a1a2=b1b2c1c2

So any line with this condition can be 4x+6y8=0

Proof,

a1a2=24=12

b1b2=36=12

c1c2=88=1

Hence, 12=121 it means a1a2=b1b2c1c2

Therefore, the pair of lines has no solutions; thus lines are parallel.

Q6 (iii) Given the linear equation 2x+3y8=0 , write another linear equation in two variables such that the geometrical representation of the pair so formed is: coincident lines

Answer:

Given the equation,

2x+3y8=0

As we know that the condition for the coincidence of the lines for the equations in the form a1x+b1y+c1=0 and a2x+b2y+c2=0 is,

a1a2=b1b2=c1c2

So any line with this condition can be 4x+6y16=0

Proof,

a1a2=24=12

b1b2=36=12

c1c2=816=12

Hence, 12=12=12 it means a1a2=b1b2=c1c2

Therefore, the pair of lines has infinitely many solutions; thus lines are coincident.

Q7 Draw the graphs of the equations xy+1=0and 3x+2y12=0 . Determine the coordinates of the vertices of the triangle formed by these lines and the x-axis, and shade the triangular region.

Answer:

Given two equations,

xy+1=0.........(1)

And

3x+2y12=0.........(2)

The points (x,y) satisfying (1) are

X
0
3
6
Y
1
4
7

And The points(x,y) satisfying (2) are,

X
0
2
4
Y
6
3
0


GRAPH:

1635920037829


From the graph, we can see that both lines intersect at the point (2,3), and therefore the vertices of the Triangle are ( -1,0), (2,3) and (4,0). The area of the triangle is shaded with a green colour.


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Topics covered in Chapter 3 Pair of Linear Equations in Two Variables: Exercise 3.1

1. Graphical Representation of Linear Equations: The process includes plotting linear equations on Cartesian graphs by using tables of values to draw their resulting straight lines.

2. Finding Solutions through Intersection Points: The intersection point between two lines indicates all solutions of the system while offering accurate insights into the equations.

3. Understanding Consistency of Systems: Knowing about System Consistency involves examining the graph to identify the number of solutions, along with determining whether the system follows a consistent or inconsistent or dependent pattern.

4. Types of Solutions: Unique solutions when lines intersect, No solutions when lines remain parallel and Infinite solutions occur with coinciding lines.

5. Verification of Solutions: The final verification involves testing whether graphically obtained solutions match both equations through substitution and transforming these results into calculations for accuracy verification.

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NCERT Solutions of Class 10 Subject Wise

Students must check the NCERT solutions for class 10 of Mathematics and Science Subjects.

JEE Main Important Mathematics Formulas

As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters

NCERT Exemplar Solutions of Class 10 Subject Wise

Students must check the NCERT Exemplar solutions for class 10 of Mathematics and Science Subjects.

Frequently Asked Questions (FAQs)

1. Check whether the equation xy - 9 = 3 a linear equation in two variables ?

The concepts related to linear equation is discussed in ex 3.2 class 10. Practice the problems discussed in this exercise to command the concepts. For this question, because of the term xy is of degree 2,  xy - 9 = 3 is not a linear equation in two variables.

2. In a graph, how many quadrants are there?

To understand the concepts of quadrants go through the problems discussed in class 10 maths ex 3.2. In a graph, there are four quadrants. a point can be represent in in a plain using the (x, y) coordinates.

3. When two lines intersect on a plane, how many choices are there, and what are they?

To get in depth understanding of related concepts practice problems enumerated in the class 10 ex 3.2.  as per these concepts, when two lines are in a plane, there are three alternative solutions. They really are. 

  • Two lines may intersect at times.
  • Two lines may not intersect at times, and they may be parallel to each other.
  • Two lines may be coincident at times.
4. In the NCERT solutions for Class 10 Maths chapter 3 exercise 3.2 How many questions and what types of questions are covered ?

In Class 10th Maths chapter 3 exercise 3.2, there are seven questions based on the notion of graphical representation of a system of equations.

5. How many solutions are there for linear equations in two variables if the equations are inconsistent and independent?

If the equations are consistent and dependent, there are no solutions to linear equations in two variables.

6. What is the criterion for two-variable linear equations that are both consistent and dependent?

You can go through the 10th class maths exercise 3.2 answers to get deeper understanding of the concepts related to equations are consistent and dependent. The requirement for linear equations in two variables is:  

a1/a2 = b1/b2 = c1/c2

7. According to NCERT solutions for Class 10 Maths chapter 3 exercise 3.2 , What is the graphical method of solution of a pair of linear equations?

The basic strategy to represent the linear equations on the graph and determine the point of intersection is the graphical method of solution of a pair of linear equations.

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Questions related to CBSE Class 10th

Have a question related to CBSE Class 10th ?

Hello

Since you are a domicile of Karnataka and have studied under the Karnataka State Board for 11th and 12th , you are eligible for Karnataka State Quota for admission to various colleges in the state.

1. KCET (Karnataka Common Entrance Test): You must appear for the KCET exam, which is required for admission to undergraduate professional courses like engineering, medical, and other streams. Your exam score and rank will determine your eligibility for counseling.

2. Minority Income under 5 Lakh : If you are from a minority community and your family's income is below 5 lakh, you may be eligible for fee concessions or other benefits depending on the specific institution. Some colleges offer reservations or other advantages for students in this category.

3. Counseling and Seat Allocation:

After the KCET exam, you will need to participate in online counseling.

You need to select your preferred colleges and courses.

Seat allocation will be based on your rank , the availability of seats in your chosen colleges and your preferences.

4. Required Documents :

Domicile Certificate (proof that you are a resident of Karnataka).

Income Certificate (for minority category benefits).

Marksheets (11th and 12th from the Karnataka State Board).

KCET Admit Card and Scorecard.

This process will allow you to secure a seat based on your KCET performance and your category .

check link for more details

https://medicine.careers360.com/neet-college-predictor

Hope this helps you .

Hello Aspirant,  Hope your doing great,  your question was incomplete and regarding  what exam your asking.

Yes, scoring above 80% in ICSE Class 10 exams typically meets the requirements to get into the Commerce stream in Class 11th under the CBSE board . Admission criteria can vary between schools, so it is advisable to check the specific requirements of the intended CBSE school. Generally, a good academic record with a score above 80% in ICSE 10th result is considered strong for such transitions.

hello Zaid,

Yes, you can apply for 12th grade as a private candidate .You will need to follow the registration process and fulfill the eligibility criteria set by CBSE for private candidates.If you haven't given the 11th grade exam ,you would be able to appear for the 12th exam directly without having passed 11th grade. you will need to give certain tests in the school you are getting addmission to prove your eligibilty.

best of luck!

According to cbse norms candidates who have completed class 10th, class 11th, have a gap year or have failed class 12th can appear for admission in 12th class.for admission in cbse board you need to clear your 11th class first and you must have studied from CBSE board or any other recognized and equivalent board/school.

You are not eligible for cbse board but you can still do 12th from nios which allow candidates to take admission in 12th class as a private student without completing 11th.

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A block of mass 0.50 kg is moving with a speed of 2.00 ms-1 on a smooth surface. It strikes another mass of 1.00 kg and then they move together as a single body. The energy loss during the collision is

Option 1)

0.34\; J

Option 2)

0.16\; J

Option 3)

1.00\; J

Option 4)

0.67\; J

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times.  Assume that the potential energy lost each time he lowers the mass is dissipated.  How much fat will he use up considering the work done only when the weight is lifted up ?  Fat supplies 3.8×107 J of energy per kg which is converted to mechanical energy with a 20% efficiency rate.  Take g = 9.8 ms−2 :

Option 1)

2.45×10−3 kg

Option 2)

 6.45×10−3 kg

Option 3)

 9.89×10−3 kg

Option 4)

12.89×10−3 kg

 

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1)

2,000 \; J - 5,000\; J

Option 2)

200 \, \, J - 500 \, \, J

Option 3)

2\times 10^{5}J-3\times 10^{5}J

Option 4)

20,000 \, \, J - 50,000 \, \, J

A particle is projected at 600   to the horizontal with a kinetic energy K. The kinetic energy at the highest point

Option 1)

K/2\,

Option 2)

\; K\;

Option 3)

zero\;

Option 4)

K/4

In the reaction,

2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}

Option 1)

11.2\, L\, H_{2(g)}  at STP  is produced for every mole HCL_{(aq)}  consumed

Option 2)

6L\, HCl_{(aq)}  is consumed for ever 3L\, H_{2(g)}      produced

Option 3)

33.6 L\, H_{2(g)} is produced regardless of temperature and pressure for every mole Al that reacts

Option 4)

67.2\, L\, H_{2(g)} at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, Mg_{3}(PO_{4})_{2} will contain 0.25 mole of oxygen atoms?

Option 1)

0.02

Option 2)

3.125 × 10-2

Option 3)

1.25 × 10-2

Option 4)

2.5 × 10-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1)

decrease twice

Option 2)

increase two fold

Option 3)

remain unchanged

Option 4)

be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1)

Molality

Option 2)

Weight fraction of solute

Option 3)

Fraction of solute present in water

Option 4)

Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol-1) is

Option 1)

twice that in 60 g carbon

Option 2)

6.023 × 1022

Option 3)

half that in 8 g He

Option 4)

558.5 × 6.023 × 1023

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t2) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m2 , the number of rotations made by the pulley before its direction of motion if reversed, is

Option 1)

less than 3

Option 2)

more than 3 but less than 6

Option 3)

more than 6 but less than 9

Option 4)

more than 9

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