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A linear equation is a polynomial with degree 1. A pair of linear equations in two variables consists of two equations, each representing a straight line. These equations play a fundamental role in algebra and are widely used in various mathematical and real-world applications. They can be solved using different methods like substitution, elimination, and graphical representation. Linear equations have many real-life applications in fields like physics, economics, engineering, and computer science.
This article on NCERT Class 10 Maths Chapter 3 Solutions of Pair of Linear Equations in Two Variables provides clear and step-by-step solutions for exercise problems in NCERT Class 10 Maths Book. These solutions of Pair of Linear Equations in Two Variables Class 10 are designed by Subject Matter Experts according to the latest CBSE syllabus, ensuring that students grasp the concepts effectively. NCERT solutions for other subjects and classes can be downloaded in NCERT solutions.
Also read,
Linear equations are polynomials with degree one. Eg:
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
The graph of a pair of linear equations in two variables is represented by two lines.
(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
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:
To eliminate
Now, substituting
Thus, the solution is
NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables (Intext Questions and Exercise)
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
Q1 Form the pair of linear equations in the following problems and find their solutions graphically.
Answer:
Let the number of boys is x and the number of girls is y.
Now, according to the question,
Total number of students in the class = 10, i.e.
And the number of girls is 4 more than the number of boys,i.e.
Different points (x, y) for equation (1)
X | 5 | 6 | 4 |
Y | 5 | 4 | 6 |
Different points (x,y) satisfying (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 x be the price of 1 pencil and y be the price of 1 pen,
Now, According to the question
And
Now, the points (x,y), that satisfies the equation (1) are
X | 3 | -4 | 10 |
Y | 5 | 10 | 0 |
And, the points(x,y) that satisfies the equation (2) are
X | 3 | 8 | -2 |
Y | 5 | -2 | 12 |
The Graph,
As we can see from the Graph, both line intersects at point (3,5) that is, x = 3 and y = 5 which means cost of 1 pencil is 3 and the cost of 1 pen is 5.
Answer:
Given Equations,
Comparing these equations with
As we can see
It means that both lines intersect at exactly one point.
Q2 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means that both lines are coincident.
Q2 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means that both lines are parallel to each other.
Q3 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have exactly one solution and thus pair of linear equations is consistent.
Q3 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have no solution and thus pair of linear equations is inconsistent.
Q3 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have exactly one solution and thus pair of linear equations is consistent.
Q3 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have an infinite number of solutions and thus pair of linear equations is consistent.
Q3 On comparing the ratios
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have an infinite number of solutions and thus pair of linear equations is consistent.
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have an infinite number of solutions and thus pair of linear equations is consistent.
The points (x,y) which satisfies in both equations are
X | 1 | 3 | 5 |
Y | 4 | 2 | 0 |
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have no solution and thus pair of linear equations is inconsistent.
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have exactly one solution and thus pair of linear equations is consistent.
Now The points(x, y) satisfying the equation are,
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 intersects at point (2,2) and hence the solution of both equations is x = 2 and y = 2.
Answer:
Given Equations,
Comparing these equations with
As we can see
It means the given equations have no solution and thus pair of linear equations is inconsistent.
Answer:
Let
Now, According to the question, the length is 4 m more than its width.i.e.
Also given Half Parameter of the rectangle
Now, as we have two equations, on adding both equations, we get,
Putting this in equation (1),
Hence Length and width of the rectangle are 20m and 16 respectively.
Answer:
Given the equation,
As we know that the condition for the intersection of lines
So Any line with this condition can be
Here,
As
the line satisfies the given condition.
Answer:
Given the equation,
As we know that the condition for the lines
So Any line with this condition can be
Here,
As
Answer:
Given the equation,
As we know that the condition for the coincidence of the lines
So any line with this condition can be
Here,
As
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.
Pair of Linear Equations in Two Variables Class 10 Solutions Exercise: 3.2
Q1 Solve the following pair of linear equations by the substitution method. (i)
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of
Hence, Solution of the given equations is
Q1 Solve the following pair of linear equations by the substitution method (ii)
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of t in (3)
Hence, Solution of the given equations is s = 9 and t = 6.
Q1 Solve the following pair of linear equations by the substitution method. (iii)
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
This is always true, and hence this pair of the equation has infinite solutions.
As we have
One of many possible solutions is
Q1 Solve the following pair of linear equations by the substitution method. (iv)
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of
Hence, Solution of the given equations is,
Q1 Solve the following pair of linear equations by the substitution method. (v)
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of x in (3)
Hence, Solution of the given equations is,
x = 0 , y = 0 .
Q1 Solve the following pair of linear equations by the substitution method. (vi)
Answer:
Given
From (1) we have,
Putting this in (2) we get,
putting this value in (3) we get,
Hence
Q2 Solve
Answer:
Given two equations,
Now, from (1), we have
Substituting this in (2), we get
Substituting this value of
Hence, Solution of the given equations is,
Now,
As it satisfies
Hence Value of m is -1.
(i) 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 let the bigger number is y.
Now, According to the question,
And
Now, the substituting value of
Substituting this in (2)
Hence the two numbers are 13 and 39.
Answer:
Let the larger angle be x and 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
Hence the two supplementary angles are
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
Now, Substituting this value of x in (3)
Hence, The cost of one bat is 500 Rs and the cost of one ball 50 Rs.
Answer:
Let the fixed charge is x and the per km charge is y.
Now According to the question
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 denominator of the fraction is 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
Substituting this value of y in (3),
Hence, Present age of Jacob is 40 years and the present age of Jacob's son is 10 years.
Pair of Linear Equations in Two Variables Class 10 Solutions Exercise: 3.3
Q1 Solve the following pair of linear equations by the elimination method and the substitution method :
(i)
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
Hence,
Q1 Solve the following pair of linear equations by the elimination method and the substitution method :
(ii)
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
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
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 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 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 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 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 per day charge is 3 Rs.
If interested, students can also check exercises here:
Chapter No. | Chapter Name |
Chapter 1 | |
Chapter 2 | |
Chapter 3 | Pair of Linear Equations in Two Variables |
Chapter 4 | |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 | |
Chapter 14 | |
Chapter 15 |
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
As NCERT Class 10 Maths Chapter 3 solutions are solved by the subject matter experts, the answers to all the questions are reliable. NCERT Class 10 Maths Chapter 3 Solutions give the step-by-step explanations to all the questions which makes it easy for the students to understand. Using the NCERT Class 10 Maths chapter 3 Solutions, students will be able to confirm the right answers once they are done solving the questions themselves.
In this article, you have gone through the NCERT solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables and have a good knowledge of answering structurally. It's time to practice various kinds of problems based on a pair of linear equations in two variables.
After the completion of the NCERT syllabus, you can check the past 5-year papers of board exams. Class 10 Maths Chapter 3 Test Paper with Solution will increase your dealing ability with a variety of questions.
NCERT syllabus coverage and previous year papers are enough tools to get a good score in the board examination. After covering NCERT and the previous year papers of this chapter, you can jump to the next chapters.
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.
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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.
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After the KCET exam, you will need to participate in online counseling.
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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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