NCERT Solutions Class 10 Maths Chapter 3 Linear Equations in Two Variables

NCERT Solutions Class 10 Maths Chapter 3 enables students to attempt textbook questions on their own so that they can easily revise the key concepts. It is important to attempt these exercises in order to test whether you have understood the concept or not, and many questions for exams are directly from these exercises themselves.

All questions in NCERT Class 10 Maths chapter 3 solutions have been answered step by step to understand the methods for solving pairs of linear equations. This chapter is also one of the key building blocks of the understanding of systems of equations, which are utilized in fields like physics, economics, and engineering. This chapter shows various ways in which pairs of linear equations can be represented both algebraically and graphically.

For the last minute exam preparation, combining the NCERT Solutions Class 10 Maths along with the CBSE Class 10 Notes is a great way for students to revise the key concepts and formulas. These notes will assist students to quickly review their understanding of concepts which they need to fully grasp the materials and ultimately increase their exam performance. Steady revision of the NCERT Solutions and the revision notes will help students prepare for their board exams. Students can download ncert solutions for class 10 maths chapter 3 pdf from below:

1.0NCERT Solutions Class 10 Maths Chapter 3 Linear Equations in Two Variables: All Exercises

Solving NCERT Textbook exercises can help students grasp basic understanding of the key concepts as it provides a wide range of problems covering all the topics. 

2.0What Will Students Learn in Chapter 3: Linear Equations in Two Variables?

• Understanding general forms of equations and what they represent graphically in the form of a straight line.
• Learn how to solve linear equations through graphs and finding the point of intersection.
• Solve equations by various Algebraic Techniques.
• Determine whether the system of equations has a unique solution, no solution (parallel lines), or infinite solutions (identical lines).
• Applying these concepts to real-life problems involving relationships between two variables.

3.0Class 10 Maths Chapter 3 Linear Equations in Two Variables : Key Topics

Chapter 3 focuses on Pair of Linear Equations in Two Variables includes fundamental concepts, methods of solving linear equations, and applications. Below is a breakdown of the key topics covered in this chapter. Students should focus on these areas to understand, solve, and apply linear equations effectively.

1. Methods of solution of a linear equation in two variable
2. Plotting linear equations in two variables on the graph
3. Consistency and nature of the graphs
4. Equations reducible to a pair of linear equations in two variables
5. Systems of a pair of linear equations in two variables

4.0Summary of Consistency and Nature of the Graphs

5.0Brief Overview of Class 10 Chapter 3 Pair of Linear Equation in Two Variables

Introduction

1. What is a linear equation in two variables?

• Definition of a linear equation: An equation that represents a straight line when plotted on the coordinate plane.
• General form: ax + by = c, where a, b, and c are constants, and x and y are variables.

2. Examples of linear equations in two variables:

• 2x + 3y = 5
• x – y = 3

Graph of a Linear Equation

1. Understanding the graph of a linear equation

• A linear equation in two variables represents a straight line.
• Plotting a linear equation on a graph involves finding at least two points that satisfy the equation and then drawing a straight line through them.

2. Steps to graph a linear equation: 

• Find two solutions (coordinates) by substituting different values for x and solving for y, or vice versa.
• Plot these points on the graph.
• Draw a straight line through the points.

Solutions of a Linear Equation

1. What does a solution represent?

• The solution to a linear equation is a pair of values (x, y) that satisfy the equation.

2. How to find solutions:

• Choose a value for x (or y) and substitute it into the equation to solve for the other variable.
• Example: For the equation 3x – 2y = 6, if x = 2, then substitute into the equation to find y.

Consistent, Inconsistent, and Dependent Equations

• Consistent system: A system of equations that has at least one solution (intersecting lines).
• Inconsistent system: A system of equations that has no solution (parallel lines).
• Dependent system: A system of equations that has infinitely many solutions (coincident lines).

Algebraic Methods to Solve a Pair of Linear Equations

1. Substitution method: Solve one equation for one variable and substitute it into the other equation.
2. Elimination method: Add or subtract the equations to eliminate one variable, then solve for the other variable.
3. Graphical method: Plot both equations on a graph and find the point of intersection.

Word Problems Involving Linear Equations

1. Setting up the equation:

• Translate a real-world problem into a linear equation in two variables.
• Solve the equation to find the required quantities.

6.0Sample NCERT Solutions for Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables

1. Aftab tells his daughter, "Seven years ago, I was seven times as old as you were then. Also, three years from now, I shall be three times as old as you will be." Represent this situation algebraically and graphically.

Solution:

Let the present age of Aftab's daughter be x years.

Let the present age of Aftab be y years (y > x).

Seven years ago:

Aftab's age: y - 7

Daughter's age: x - 7

According to the problem: y - 7 = 7(x - 7)

Simplifying: y - 7 = 7x - 49

Equation 1: 7x - y - 42 = 0

Three years later:

 Aftab's age: y + 3

 Daughter's age: x + 3

 According to the problem: y + 3 = 3(x + 3)

 Simplifying: y + 3 = 3x + 9

 Equation 2: 3x - y + 6 = 0

 Therefore, the algebraic representation is:

  7x - y - 42 = 0

  3x - y + 6 = 0

When plotted on a graph, the intersection point of the two lines is (12, 42).

Thus, the present age of Aftab's daughter is 12 years, and the present age of Aftab is 42 years.

2. The coach of a cricket team buys 3 bats and 6 balls for ₹ 3900. Later, she buys another bat and 3 more balls of the same kind for ₹ 1300. Represent this situation algebraically and geometrically.

Solution:

Algebraic Representation:

Let the cost of 1 bat be ₹ x.

Let the cost of 1 ball be ₹ y.

According to the problem:

3x + 6y = 3900

x + 3y = 1300

3. On comparing the ratios a₁/a₂, b₁/b₂, and c₁/c₂, find out whether the lines representing the following pairs of linear equations intersect at a point, are 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

Solutions:

(i) 5x - 4y + 8 = 0  ...(1)

7x + 6y - 9 = 0  ...(2)

 a₁/a₂ = 5/7

 b₁/b₂ = -4/6 = -2/3

Since a₁/a₂ ≠ b₁/b₂, the lines represented by equations (1) and (2) intersect at a point.

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

18x + 6y + 24 = 0

a₁/a₂ = 9/18 = 1/2

b₁/b₂ = 3/6 = 1/2

c₁/c₂ = 12/24 = 1/2

Since a₁/a₂ = b₁/b₂ = c₁/c₂, the lines represented by the equations are coincident.

(iii) 6x - 3y + 10 = 0

2x - y + 9 = 0

a₁/a₂ = 6/2 = 3/1

b₁/b₂ = -3/-1 = 3/1

c₁/c₂ = 10/9

Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines represented by the equations are parallel.

3. On comparing the ratios a₁/a₂, b₁/b₂, and c₁/c₂, find out whether the following pairs of linear equations are consistent or inconsistent.

(i) 3x + 2y = 5 ; 2x - 3y = 7

(ii) 2x - 3y = 8 ; 4x - 6y = 9

(iii) (3/2)x + (5/3)y = 7 ; 9x - 10y = 14

(iv) 5x - 3y = 11 ; -10x + 6y = -22

(v) (4/3)x + 2y = 8 ; 2x + 3y = 12

Solutions:

(i) 3x + 2y = 5 ; 2x - 3y = 7

Rewrite the equations in the form ax + by + c = 0:

3x + 2y - 5 = 0

2x - 3y - 7 = 0

Calculate the ratios:

a₁/a₂ = 3/2

b₁/b₂ = 2/(-3) = -2/3

Since a₁/a₂ ≠ b₁/b₂, the equations have a unique solution.

Therefore, the equations are consistent.

(ii) 2x - 3y = 8 ; 4x - 6y = 9

Calculate the ratios:

a₁/a₂ = 2/4 = 1/2

b₁/b₂ = -3/(-6) = 1/2

c₁/c₂ = 8/9

Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the equations have no solution.

Therefore, the equations are inconsistent.

(iii) (3/2)x + (5/3)y = 7 ; 9x - 10y = 14

Calculate the ratios:

a₁/a₂ = (3/2)/9 = 1/6

b₁/b₂ = (5/3)/(-10) = -1/6

Since a₁/a₂ ≠ b₁/b₂, the equations have a unique solution.

Therefore, the equations are consistent.

(iv) 5x - 3y = 11 ; -10x + 6y = -22

Calculate the ratios:

a₁/a₂ = 5/(-10) = -1/2

b₁/b₂ = -3/6 = -1/2

c₁/c₂ = 11/(-22) = -1/2

Since a₁/a₂ = b₁/b₂ = c₁/c₂, the equations have infinitely many solutions.

Therefore, the equations are consistent.

(v) (4/3)x + 2y = 8 ; 2x + 3y = 12

Calculate the ratios:

a₁/a₂ = (4/3)/2 = 2/3

b₁/b₂ = 2/3

c₁/c₂ = 8/12 = 2/3

Since a₁/a₂ = b₁/b₂ = c₁/c₂, the equations have infinitely many solutions.

Therefore, the equations are consistent.

7.0Benefits of Class 10 Maths Chapter 3 Linear Equations in Two Variables 

• Analytical Skills: Enhances logical reasoning and problem-solving abilities.
• Foundation for Advanced Math: Prepares for higher topics like algebra and calculus.
• Real-World Applications: Models situations like budgeting, cost, and profit analysis.
• System of Equations: Introduces students to solving multiple-variable problems.
• Graphical Interpretation: Improves understanding of graphs and geometry.
• Competitive Exams: Builds skills needed for quantitative reasoning in exams.
• Logical Thinking: Strengthens critical thinking and attention to detail.
• Interdisciplinary Use: Relevant in fields like physics, economics, and data science.
• Career Pathways: Opens doors to careers in finance, engineering, and data analysis.
• Confidence in Math: Boosts students' confidence and readiness for more complex topics.