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Solving a pair of linear equations is like finding where two friends agree to meet—on the same path, at the same point. A pair of linear equations in two variables consists of two linear equations in each corresponding x and y expression, and when we say "solving" the pair means we are looking for the point where they intersect on the Cartesian plane. NCERT Solutions for Class 10 Maths Chapter 3 will provide both clear and in-depth solutions that accurately answer all the exercise questions in the NCERT textbook. In this chapter, students will work with multiple methods of solving linear equations, such as graphical methods, algebraic methods, substitution methods, and elimination methods.
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A pair of linear equations shows that every problem has a point where paths cross—if we solve it correctly. These equations play a key role in algebra and find widespread use in various mathematical and real-world applications. Our academic team here at Careers360 comprises experienced experts with years of teaching experience who have developed these NCERT Solutions for Class 10 content based on the modified CBSE curriculum. For a detailed syllabus, study materials, and downloadable PDFs, check out this link: NCERT.
Below are the NCERT class 10 math chapter 3 solutions for exercise questions.
Class 10 Maths Chapter 3 Solutions Pair of Linear Equations in Two Variables Exercise: 3.1 Total Questions: 7 Page number: 28-29 |
Q1: Form the pair of linear equations in the following problems and find their solutions graphically.
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.
And, given that the number of girls is 4 more than the number of boys it means;
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,
As we can see 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: Form the pair of linear equations in the following problems and find their solutions graphically.
Answer:
Let the price of 1 pencil be x, and y be the price of 1 pen.
Now, according to the question
And
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,
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.
Answer:
Given Equations,
Comparing these equations with
It is observed that;
It means that both lines intersect at exactly one point and have a unique solution.
Answer:
Given Equations,
Comparing these equations with
It is observed that;
It means that both lines are coincident and have infinitely many solutions.
Q2 (iii): On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
It is observed that;
It means that both lines are parallel and thus have no solution.
Answer:
Given Equations,
Or,
Comparing these equations with
It is observed that;
It means that the given equations have a unique solution and thus the pair of linear equations is consistent.
Answer:
Given Equations,
Or,
Comparing these equations with
It is observed that;
It means the given equations have no solution, and thus the pair of linear equations is inconsistent.
Answer:
Given Equations,
Or,
Comparing these equations with
It is observed that;
It means the given equations have exactly one solution, and thus the pair of linear equations is consistent.
Answer:
Given Equations,
Or,
Comparing these equations with
It is observed that;
It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.
Answer:
Given Equations,
Or,
Comparing these equations with
It is observed that;
It means the given equations have an infinite number of solutions, and thus a pair of linear equations is consistent.
Answer:
Given Equations,
Comparing these equations with
It is observed that;
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 |
Answer:
Given Equations,
Comparing these equations with
It is observed that:
It means the given equations have no solution, and thus the pair of linear equations is inconsistent.
Answer:
Given Equations,
Comparing these equations with
It is observed that;
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
X |
0 |
2 |
3 |
Y |
6 |
2 |
0 |
And The points(x,y) satisfying the equation
X |
0 |
1 |
2 |
Y |
-2 |
0 |
2 |
GRAPH:
As we can see, both lines intersect at point (2,2) and hence the solution of both equations is x = 2 and y = 2.
Answer:
Given Equations,
Comparing these equations with
It is observed that;
It means the given equations have no solution, and thus the pair of linear equations is inconsistent.
Answer:
Let
Now, according to the question, the length is 4 m more than its width, so we can write it as
Or,
Also given Half Parameter of the rectangle = 36 it means
Now, as we have two equations, add both equations, and we get,
We get the value of
Now, putting this in equation (1), we get;
Hence, the Length and width of the rectangle are 20m and 16m, respectively.
Answer:
Given the equation,
We know that the condition for the intersection of lines for the equations in the form
So any line with this condition can be
Proof,
Hence,
Therefore, the pair of lines has a unique solution, thus forming intersecting lines.
Answer:
Given the equation,
As we know that the condition for the parallel lines for the equations in the form
So any line with this condition can be
Proof,
Hence,
Therefore, the pair of lines has no solutions; thus lines are parallel.
Answer:
Given the equation,
As we know that the condition for the coincidence of the lines for the equations in the form
So any line with this condition can be
Proof,
Hence,
Therefore, the pair of lines has infinitely many solutions; thus lines are coincident.
Answer:
Given two equations,
And
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:
As we can see from the graph that both lines intersect at the point (2,3), And the vertices of the Triangle are ( -1,0), (2,3) and (4,0). The area of the triangle is shaded with a green color.
Class 10 Maths Chapter 3 Solutions Pair of Linear Equations in Two Variables Exercise: 3.2 Total Questions: 3 Page number: 33-34 |
Q1(i): Solve the following pair of linear equations by the substitution method.
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of x in (3)
Hence, the solution of the given equations is x = 9 and y = 5.
Q1(ii): Solve the following pair of linear equations by the substitution method
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of t in (3)
Hence, the solution of the given equations is s = 9 and t = 6.
Q1(iii): Solve the following pair of linear equations by the substitution method.
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
This is always true, and hence this pair of equations has infinite solutions.
As we have
One of many possible solutions is x = 1, and y = 0.
Q1(iv): Solve the following pair of linear equations by the substitution method.
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of x in (3)
Hence, the solution of the given equations is
x = 2 and y = 3.
Q1(v): Solve the following pair of linear equations by the substitution method.
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of y in (3)
Hence, the solution of the given equations is,
x = 0, and y = 0 .
Q1(vi): Solve the following pair of linear equations by the substitution method.
Answer:
Given,
From (1) we have,
Putting this in (2), we get,
Putting this value in (3), we get,
Hence, x = 2 and y = 3.
Q2: Solve 2x + 3y = 11 and 2x − 4y = −24 and hence find the value of ‘m’ for which y = mx + 3.
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of x in (3)
Hence, the solution of the given equations is,
x = −2, and y = 5.
Now,
As it satisfies
Hence, the value of m is -1.
The difference between the two numbers is 26, and one number is three times the other. Find them.
Answer:
Let two numbers be x and y, and the bigger number is y.
Now, according to the question,
And
Now, substituting the value of y from (2) in (1), we get,
Substituting this in (2)
Hence, the two numbers are 13 and 39.
The larger of the two supplementary angles exceeds the smaller by 18 degrees. Find them.
Answer:
Let the larger angle be x and the smaller angle be y
Now, as we know, the sum of supplementary angles is 180. so,
Also given in the question,
Now, from (2) we have,
Substituting this value in (1)
Now, substituting this value of x in (3), we get
Hence, the two supplementary angles are
Q3: Form the pair of linear equations for the following problems and find their solution by the substitution method.
(iii) The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, she buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.
Answer:
Let the cost of 1 bat is x and the cost of 1 ball is y.
Now, according to the question,
Now, from (1) we have
Substituting this value of y in (2)
Now, Substituting this value of x in (3)
Hence, the cost of one bat is 500 Rs and the cost of one ball is 50 Rs.
Answer:
Let the fixed charge is x and the per km charge is y.
Now, according to the question
And,
Now, from (1) we have,
Substituting this value of x in (2), we have
Now, substituting this value in (3)
Hence, the fixed charge is 5 Rs and the per km charge is 10 Rs.
Now, Fair for 25 km :
Hence, fair for 25km is 255 Rs.
Answer:
Let the numerator of the fraction be x, and the denominator of the fraction be y
Now, according to the question,
Also,
Now, from (1) we have
Substituting this value of y in (2)
Substituting this value of x in (3)
Hence, the required fraction is
Answer:
Let x be the age of Jacob and y be the age of Jacob's son.
Now, according to the question
Also,
Now,
From (1) we have,
Substituting this value of x in (2)
Substituting this value of y in (3),
Hence, the present age of Jacob is 40 years, and the present age of Jacob's son is 10 years.
Class 10 Maths Chapter 3 Solutions Pair of Linear Equations in Two Variables Exercise: 3.3 Total Questions: 2 Page number: 36-37 |
Q1(i): Solve the following pair of linear equations by the elimination method and the substitution method :
Answer:
Elimination Method:
Given, equations
Now, multiplying (1) by 3 we get
Now, adding (2) and (3), we get
Substituting this value in (1), we get
Hence,
Substitution method :
Given, equations
Now, from (1) we have,
Substituting this value in (2)
Substituting this value of x in (3)
Hence,
Q1(ii): Solve the following pair of linear equations by the elimination method and the substitution method :
Answer:
Elimination Method:
Given, equations
Now, multiplying (2) by 2 we get
Now, adding (1) and (3), we get
Putting this value in (2) we get
Hence,
Substitution method :
Given, equations
Now, from (2) we have,
Substituting this value in (1)
Substituting this value of x in (3)
Hence,
Answer:
Elimination Method:
Given, equations
Now, multiplying (1) by 3 we get
Now, subtracting (3) from (2), we get
Putting this value in (1), we get
Hence,
Substitution method :
Given, equations
Now, from (2) we have,
Substituting this value in (1)
Substituting this value of x in (3)
Hence,
Answer:
Elimination Method:
Given, equations
Now, multiplying (2) by 2, we get
Now, adding (1) and (3), we get
Putting this value in (2), we get
Hence,
Substitution method :
Given, equations
Now, from (2) we have,
Substituting this value in (1)
Substituting this value of x in (3)
Hence,
Answer:
Let the numerator of the fraction be x, and the denominator is y,
Now, according to the question,
Also,
Now, subtracting (1) from (2), we get
Putting this value in (1)
Hence,
And the fraction is:
Answer:
Let the age of Nuri be x and the age of Sonu be y.
Now, according to the question
Also,
Now, subtracting (1) from (2), we get
Putting this value in (2)
Hence, the age of Nuri is 50 and the age of Nuri is 20.
Answer:
Let the unit digit of the number be x and the 10's digit be y.
Now, according to the question,
Also
Now adding (1) and (2), we get,
Now putting this value in (1)
Hence, the number is 18.
Answer:
Let the number of Rs 50 notes be x and the number of Rs 100 notes be y.
Now, according to the question,
And
Now, subtracting (1) from (2), we get
Putting this value in (1).
Hence, Meena received 10, 50 Rs notes and 15, 100 Rs notes.
Answer:
Let the fixed charge be x, and per day charge is y.
Now, according to the question,
And
Now, Subtracting (2) from (1). We get,
Putting this in (1)
Hence, the fixed charge is 15 Rs and the per-day charge is 3 Rs.
Also, read,
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Linear equations are polynomials with degree one. Eg: 3x=7,2x+6y=10
S. No. |
Types of Linear Equation |
General form |
Description |
Solutions |
1. |
Linear Equation in one Variable |
ax + b = 0 |
Where a ≠ 0 and a & b are real numbers |
One Solution |
2. |
Linear Equation in Two Variables |
ax + by + c = 0 |
Where a ≠ 0 & b ≠ 0 and a, b & c are real numbers |
Infinite Solutions possible |
3. |
Linear Equation in Three Variables |
ax + by + cz + d = 0 |
Where a ≠ 0, b ≠ 0, c ≠ 0 and a, b, c, d are real numbers |
Infinite Solutions possible |
The simultaneous system of linear equations in two variables is in format,
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
The simultaneous system of linear equations can be solved using two methods,
1. Graphical Method
2. Algebraic Method
Two lines represent the graph of a pair of linear equations in two variables.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions, each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
The pair of linear equations can be solved using the algebraic method in two other methods, namely,
1. Substitution Method
2. Elimination Method
Substitution Method
In the substitution method, one of the linear equations is converted to an equation based on any one of the variables. Eg. The equation x−y=1 can be converted into x=y+1. Now, this value of x is substituted in the second linear equation, which makes the equation as a linear equation of one variable, which is much easier to solve.
Elimination Method
In the elimination method, the given system of equations is manipulated to eliminate one of the variables by adding or subtracting the equations.
For example,
Consider the system of equations: 2x+3y=8 4x−3y=10
To eliminate y, we add both equations:
(2x+3y)+(4x−3y)=8+10
6x=18
x=3
Now, substituting
Thus, the solution is
The topics discussed in the NCERT Solutions for class 10, chapter 3, Pair of linear equations in two variables are:
Access all NCERT Class 10 Maths solutions from one place using the links below.
Also, read,
Students can use the following links to check the solutions to Maths and science-related questions.
After completing the NCERT textbooks, students should practice exemplar exercises for a better understanding of the chapters and clarity. The following links will help students find exemplar exercises.
Students can use the following links to check the latest NCERT syllabus and read some reference books.
In the substitution method, one of the linear equations is converted to an equation based on any one of the variables. Eg. The equation
The elimination method is one of the algebraic methods used to solve linear equations.
In the elimination method, the given system of equations is manipulated to eliminate one of the variables by adding or subtracting the equations.
For example,
Consider the system of equations:
To eliminate
Now, substituting
Thus, the solution is
The graph of a pair of linear equations in two variables is represented by two lines. Graph the linear equations and find the intersection points. The solution of the linear equations depends on the intersection points.
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations.
(ii) If the lines coincide, then there are infinitely many solutions — each point on the line being a solution.
(iii) If the lines are parallel, then the pair of equations has no solution.
Yes, two linear equations have infinitely many solutions when they represent the same line. For the linear equations
(i) If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.
(ii) If the lines coincide, then there are infinitely many solutions — each point on the line being a solution. In this case, the pair of equations is dependent (consistent).
(iii) If the lines are parallel, then the pair of equations has no solution. In this case, the pair of equations is inconsistent.
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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