CBSE Class 10 Maths Chapter 3: Linear Equations in Two Variables NCERT Solutions
This resource provides detailed NCERT Solutions for Class 10 Mathematics, Chapter 3, focusing on Linear Equations in Two Variables. It covers how to represent real-world problems algebraically by forming linear equations and then solve them using graphical methods. The solutions explain the process of setting up equations based on given conditions, finding pairs of values that satisfy these equations, and plotting them on a graph to find the point of intersection, which represents the solution. This chapter is crucial for understanding systems of equations and their applications. These solutions are designed to help students grasp the concepts thoroughly and prepare effectively for their board examinations by offering clear, step-by-step explanations.
Quick info
| Board | CBSE |
|---|---|
| Class | Class 10 |
| Subject | Mathematics |
| Session | 2026 |
| Language | English |
| Type | NCERT Solutions |
| Chapter | Chapter 3 |
Chapter summary
Chapter 3 of the NCERT Class 10 Mathematics textbook deals with Linear Equations in Two Variables. This solution set provides step-by-step guidance to represent word problems algebraically and graphically. It focuses on forming pairs of linear equations from given scenarios and solving them by plotting the lines on a graph to find their intersection point. The exercises cover setting up equations and interpreting graphical solutions.
Learning outcomes
- Understand the concept of linear equations in two variables.
- Represent real-world situations algebraically.
- Formulate pairs of linear equations from given word problems.
- Solve systems of linear equations graphically.
- Interpret the graphical representation of linear equations.
- Identify the solution of a system of linear equations from a graph.
Topics covered
Paper topics
- Linear equations in two variables
- Algebraic representation of linear equations
- Graphical representation of linear equations
- Forming equations from word problems
- Solving systems of linear equations
- Intersection of lines
- Age problems
Important topics
- Forming algebraic equations from word problems
- Graphical method of solving linear equations
- Interpreting graphical solutions
- Representing real-life situations with equations
PDF preview
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Questions and Solutions
Question 1
Let the present age of Aftab be represented by x years and the present age of his daughter be represented by y years.
According to the first condition (Seven years ago):
Aftab's age seven years ago was x - 7.
His daughter's age seven years ago was y - 7.
The problem states that Aftab was seven times as old as his daughter then:
x - 7 = 7(y - 7)
Expanding this equation:
x - 7 = 7y - 49
Rearranging the terms to form a linear equation:
x - 7y = -49 + 7
x - 7y = -42 \quad \dots (1)
According to the second condition (Three years from now):
Aftab's age three years from now will be x + 3.
His daughter's age three years from now will be y + 3.
The problem states that Aftab will be three times as old as his daughter then:
x + 3 = 3(y + 3)
Expanding this equation:
x + 3 = 3y + 9
Rearranging the terms to form a linear equation:
x - 3y = 9 - 3
x - 3y = 6 \quad \dots (2)
Algebraic Representation:
The situation can be represented algebraically by the following pair of linear equations:
x - 7y = -42
x - 3y = 6
Graphical Representation:
To represent these equations graphically, we need to find at least two points for each line.
For equation (1): x - 7y = -42 or x = 7y - 42
If y = 5, then x = 7(5) - 42 = 35 - 42 = -7. So, one point is (-7, 5).
If y = 6, then x = 7(6) - 42 = 42 - 42 = 0. So, another point is (0, 6).
If y = 7, then x = 7(7) - 42 = 49 - 42 = 7. So, another point is (7, 7).
For equation (2): x - 3y = 6 or x = 3y + 6
If y = 0, then x = 3(0) + 6 = 6. So, one point is (6, 0).
If y = -1, then x = 3(-1) + 6 = -3 + 6 = 3. So, another point is (3, -1).
If y = -2, then x = 3(-2) + 6 = -6 + 6 = 0. So, another point is (0, -2).
Now, we plot these points on a graph and draw the lines. The line representing x - 7y = -42 passes through (-7, 5), (0, 6), and (7, 7). The line representing x - 3y = 6 passes through (6, 0), (3, -1), and (0, -2).
The graphical representation shows two intersecting lines. The point of intersection is where the solution lies. (Note: The actual graph plotting is not possible in this text format but would be shown on paper).
Common mistakes
- Errors in forming the correct algebraic equations from word problems.
- Mistakes in calculating coordinates for plotting the graph.
- Incorrectly interpreting the intersection point of the lines on the graph.
- Calculation errors while simplifying equations.
Revision tips
- Practice converting word problems into algebraic equations accurately.
- Ensure all calculations for finding coordinate points are double-checked.
- Understand how to plot lines correctly on a graph.
- Focus on interpreting the graphical solution in the context of the original problem.
Practice MCQs
Q1. What is the algebraic representation of the statement 'Seven years ago, I was seven times as old as you were then'?
Explanation: Let x be the current age of Aftab and y be the current age of his daughter. Seven years ago, their ages were (x-7) and (y-7) respectively. The problem states Aftab's age was seven times his daughter's age, leading to the equation x - 7 = 7(y - 7).
Q2. If the algebraic representation of a problem is x - 3y = 6, which of the following points lies on this line?
Explanation: Substitute the coordinates into the equation: For (0, -2), 0 - 3(-2) = 6. For (6, 0), 6 - 3(0) = 6. For (3, -1), 3 - 3(-1) = 3 + 3 = 6. All points satisfy the equation.
Q3. What does the intersection point of the two lines represent in a graphical solution of linear equations?
Explanation: The point where the lines representing two linear equations intersect is the only point that satisfies both equations simultaneously, thus representing the unique solution to the system.
Q4. In the graphical method, if two lines are parallel, what does it indicate about the system of equations?
Explanation: Parallel lines never intersect, meaning there is no common point that satisfies both equations. Therefore, the system has no solution.
Frequently asked questions
What is the main focus of Chapter 3, Linear Equations in Two Variables for Class 10 Maths?
Chapter 3 focuses on understanding and solving problems involving two unknown quantities using linear equations. It covers both algebraic methods for setting up equations and graphical methods for finding their solutions.
How are real-world problems represented algebraically in this chapter?
Real-world problems are translated into linear equations by assigning variables to unknown quantities and using the given information to establish relationships between them, forming one or more equations.
What is the graphical method for solving linear equations?
The graphical method involves plotting the lines represented by each linear equation on a graph. The point(s) where the lines intersect (or if they are parallel or coincident) provide the solution(s) to the system of equations.
How do these NCERT Solutions help students prepare for exams?
These solutions offer clear, step-by-step explanations for each problem, helping students understand the concepts and methods. They provide a reliable way to check answers and learn problem-solving techniques for exam revision.
What does the intersection point of two lines on a graph signify in this context?
The intersection point of two lines on a graph represents the unique solution to the system of linear equations, as it is the only coordinate pair that satisfies both equations simultaneously.
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