Complete Class 10 Maths Chapter 3 Solutions – Expert Guide

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Class 10 Maths Chapter 3 solutions provide step-by-step explanations for mastering Linear Equations in Two Variables. These solutions help students build problem-solving skills with detailed examples and expert tips. Follow this guide to enhance your understanding and ace your CBSE board exams with confidence. Perfect for NCERT and CBSE syllabus preparation.

Understanding Linear Equations in Two Variables is crucial for Class 10 students. This section provides clear and simple solutions to commonly asked questions, ensuring easy comprehension. Explore these sample questions with step-by-step answers, designed to help you grasp concepts effortlessly and boost your exam confidence.

Class 10 Maths Chapter 3 – Sample Questions & Answers

Find the solution for the equation 2x + 3y = 12 when x = 2.
Substituting x = 2 in the equation, we get 2(2) + 3y = 12. This simplifies to 4 + 3y = 12. Subtracting 4 from both sides, we get 3y = 8. Dividing by 3, y = 8/3.

Find the value of k if (3, k) is a solution to the equation 5x - 2y = 7.
Substituting x = 3, we get 5(3) - 2k = 7, which simplifies to 15 - 2k = 7. Subtracting 15 from both sides, -2k = -8. Dividing by -2, k = 4.

Find the point where the line x + 2y = 6 intersects the y-axis.
On the y-axis, x = 0. Substituting x = 0 in the equation, we get 0 + 2y = 6. Dividing both sides by 2, we get y = 3. The point of intersection is (0, 3).

If 2x - 5y = 10 and y = 2, find the value of x.
Substituting y = 2, we get 2x - 5(2) = 10, which simplifies to 2x - 10 = 10. Adding 10 to both sides, 2x = 20. Dividing by 2, x = 10.

Find two solutions for the equation x - y = 4.
If x = 6, then 6 - y = 4. Subtracting 6, we get -y = -2, so y = 2. One solution is (6,2).
If x = 5, then 5 - y = 4. Subtracting 5, -y = -1, so y = 1. Another solution is (5,1).

Check if the point (2, 1) satisfies the equation 3x + 4y = 10.
Substituting x = 2, y = 1, we get 3(2) + 4(1) = 6 + 4 = 10. Since both sides are equal, the point (2,1) satisfies the equation.

Write the equation of a line passing through (0, -3) with slope 2.
Using the equation y = mx + c, where m = 2 and c = -3, the equation is y = 2x - 3.

Find the x-intercept of the equation 4x - y = 8.
At the x-intercept, y = 0. Substituting, we get 4x - 0 = 8, which simplifies to 4x = 8. Dividing by 4, x = 2. The x-intercept is (2,0).

Find the equation of the line passing through (3,2) and parallel to 2x + 3y = 6.
The given equation has a slope of -2/3. A parallel line has the same slope, so using y = mx + c, we get 2 = (-2/3)(3) + c. This simplifies to 2 = -2 + c, so c = 4. The equation is y = (-2/3)x + 4.

Determine whether the lines 3x + 4y = 12 and 6x + 8y = 24 are parallel.
Dividing the second equation by 2, we get 3x + 4y = 12, which is the same as the first equation. Since both represent the same line, they are coincident, not parallel.

Find the midpoint of the line segment joining (2, 3) and (6, 7).
The midpoint formula is ((x1 + x2)/2, (y1 + y2)/2). Substituting values, we get ((2+6)/2, (3+7)/2) = (8/2, 10/2) = (4,5).

A line passes through (1, 2) and has a slope of -3. Find its equation.
Using y = mx + c, substituting values, 2 = -3(1) + c, we get 2 = -3 + c. Adding 3 to both sides, c = 5. The equation is y = -3x + 5.

Solve for x and y: 2x + 3y = 12 and x - y = 2.
From x - y = 2, we solve for x: x = y + 2. Substituting in 2x + 3y = 12, we get 2(y+2) + 3y = 12, which simplifies to 2y + 4 + 3y = 12. Simplifying further, 5y = 8, so y = 8/5. Substituting y = 8/5 in x = y + 2, we get x = 8/5 + 10/5 = 18/5.

Find the distance between the points (3, 4) and (7, 1).
Using the distance formula, we get sqrt((7-3)² + (1-4)²) = sqrt(4² + (-3)²) = sqrt(16 + 9) = sqrt(25) = 5.

Find the equation of the perpendicular bisector of the segment joining (4,2) and (8,6).
The midpoint is ((4+8)/2, (2+6)/2) = (6,4). The slope of the segment is (6-2)/(8-4) = 4/4 = 1. The perpendicular slope is -1. Using y = mx + c, substituting 4 = (-1)(6) + c, we get 4 = -6 + c, so c = 10. The equation is y = -x + 10.

Determine whether the points (1,2), (3,6), and (5,10) are collinear.
Find the slopes between (1,2) and (3,6) and (3,6) and (5,10). The first slope is (6-2)/(3-1) = 4/2 = 2. The second slope is (10-6)/(5-3) = 4/2 = 2. Since both slopes are equal, the points are collinear.

Find the area of the triangle with vertices (2, 3), (4, 7), and (6, 5).
Using the formula 1/2 | x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2) |, we get 1/2 | 2(7-5) + 4(5-3) + 6(3-7) | = 1/2 | 2(2) + 4(2) + 6(-4) | = 1/2 | 4 + 8 - 24 | = 1/2 | -12 | = 6.

These Class 10 Chapter 3 questions and answers simplify concepts and provide practical understanding. Practice these to strengthen your skills in solving linear equations in two variables with confidence.

Best Indian Books for Class 10 Maths Chapter 3 Solutions

1. NCERT Mathematics for Class 10 – NCERT Publications
   Covers all syllabus-based Linear Equations in Two Variables problems with step-by-step NCERT solutions. Includes exercise problems, additional practice, and conceptual explanations.

2. RS Aggarwal Class 10 Mathematics – RS Aggarwal (Bharti Bhawan)
   Features a variety of problem types, from basic to advanced level questions, covering linear equations, graph-based solutions, and word problems.

3. RD Sharma Mathematics for Class 10 – RD Sharma (Dhanpat Rai Publications)
   Offers in-depth theory and conceptual clarity, along with multiple solved examples and advanced-level practice exercises for deeper understanding.

4. All in One Mathematics Class 10 – Arihant Experts (Arihant Publications)
   Provides detailed explanations with extra solved examples, NCERT-based questions, and additional practice problems covering real-life applications.

5. CBSE Mathematics for Class 10 – RS Aggarwal & V Aggarwal (Arya Publications)
   Includes NCERT exercise solutions, extra problems for board exam preparation, and multiple-choice questions to strengthen conceptual understanding.

6. Mathematics for Class 10 – ML Aggarwal (Avichal Publishing Company)
   Structured with stepwise solutions, short-trick methods, and well-explained graphical solutions for Linear Equations in Two Variables.

7. U-Like CBSE Mathematics Question Bank Class 10 – U-Like Publications
   Covers previous years’ questions, sample papers, and NCERT-based exercises, ideal for board exam preparation.

8. Xam Idea Mathematics Class 10 – VK Global Publications
   Includes NCERT-based problems, HOTS (Higher Order Thinking Skills) questions, and practice worksheets for better problem-solving techniques.

9. Together with Mathematics for Class 10 – Rachna Sagar Publications
   Features practice questions, past board questions, and concept revision exercises, helpful for quick grasping of linear equations.

10. CBSE Sample Papers & Solutions Class 10 Mathematics – Oswaal Books
    Provides solved sample papers, detailed stepwise solutions, and extra practice questions aligned with board exam patterns.

11. Foundation Mathematics Class 10 – SK Goyal (Arihant Publications)
    Covers conceptual depth, multiple problem-solving techniques, and graph-based solutions for linear equations.

12. Evergreen Self-Study Mathematics Class 10 – Evergreen Publications
    Offers comprehensive stepwise solutions, NCERT-based problem-solving, and exam-oriented practice.

13. Exam Idea Mathematics Class 10 – VK Global Publications
    Features detailed solutions, extra exercises, and previous year’s board questions for thorough revision.

14. Pradeep’s Fundamental Mathematics Class 10 – Pradeep Publications
    Known for its concept-based approach, simple explanations, and diverse problem-solving techniques.

15. Super 10 Sample Papers for Class 10 Mathematics – Disha Experts (Disha Publications)
    Includes chapter-wise sample papers, revision notes, and problem-solving exercises for board exams.

16. Mathematics ExamMate Class 10 – Full Marks Publications
    Offers chapter summaries, short tricks, and practice problems, helpful for quick revision and self-assessment.

17. Success Package Class 10 Mathematics – Arihant Publications
    Provides NCERT-based explanations, sample questions, and self-practice exercises.

18. Self Tutor Mathematics Class 10 – S Chand Publications
    Features detailed topic explanations, worked-out examples, and multiple-choice questions for CBSE exams.

19. Goyal Brothers Mathematics Class 10 – Goyal Brothers Prakashan
    Includes practice worksheets, solved examples, and objective-type questions for quick learning.

20. Mathematics NCERT Exemplar Solutions Class 10 – NCERT
    Contains higher-level practice problems, tricky questions, and board-level solutions, making it ideal for in-depth preparation.

Understanding Linear Equations in Two Variables is a crucial step in building strong mathematical foundations. This chapter introduces the concept of equations with two variables and their graphical representations, helping students develop logical thinking and problem-solving skills.

Concept of Linear Equations in Two Variables
A linear equation in two variables is of the form ax + by + c = 0, where a, b, and c are real numbers, and x and y are variables. The solutions to these equations form straight lines on a graph, helping visualize how equations interact with each other.

Graphical Representation and Interpretation
One of the key aspects of this chapter is plotting these equations on a coordinate plane. By finding two or more solutions and marking them on a graph, students can understand the relationship between the variables. This skill is particularly useful in real-life scenarios, such as calculating budgets, business profits, and even physics-based motion problems.

Methods to Solve Linear Equations in Two Variables
There are several ways to solve these equations:

• Substitution Method – Expressing one variable in terms of another and substituting it into the second equation.
• Elimination Method – Adding or subtracting the given equations to eliminate one variable, making it easier to solve for the other.
• Graphical Method – Plotting both equations and identifying their point of intersection, which represents the solution.

Each method provides a different approach to solving equations, allowing students to develop versatile problem-solving strategies.

Real-World Applications of Linear Equations
This topic extends beyond textbooks and finds applications in various fields. In economics, these equations help analyze supply and demand. In physics, they assist in calculating speed, distance, and time relationships. Even in daily life, concepts like splitting expenses among friends or calculating ingredient proportions in recipes use the principles of linear equations.

Common Mistakes and How to Avoid Them
Students often make calculation errors while simplifying equations. Missing negative signs, incorrect arithmetic operations, or misplacing numbers while plotting graphs can lead to incorrect solutions. Regular practice, double-checking steps, and using different methods for cross-verification can help minimize these errors.

Why Practicing Different Types of Questions is Important
NCERT exercises, reference books, and sample papers provide a variety of problems, including word problems, graphical interpretation, and algebraic solutions. Working through diverse question types enhances conceptual clarity and prepares students for board exams.

Practicing previous years' CBSE questions, HOTS problems, and application-based questions further sharpens analytical skills. Students should attempt self-assessment tests, timed practice exercises, and peer discussions to strengthen their grasp on linear equations.

By understanding the logic behind equations, exploring different solving techniques, and applying these concepts to real-life problems, students can build confidence and accuracy in mathematics.

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