Class 10 Mathematics Chapter 3 Pair Of Linear Equation In Two Variable

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Class 10 Mathematics -> Chapter 3: Pair of Linear Equations in Two Variables

I. Chapter Summary:

In this chapter, students will learn about pair of linear equations in two variables and methods to solve them. The chapter explores the different ways of representing these equations graphically, algebraically, and solving them using substitution, elimination, and graphical methods. By the end of the chapter, students will be able to understand the relationship between two variables, solve the equations, and interpret the solutions graphically.

II. Key Concepts Covered:

  1. Linear Equation in Two Variables:
    • An equation of the form ax + by = c, where x and y are variables and a, b, c are constants.
    • Example: $3x + 4y = 10$
  2. Pair of Linear Equations in Two Variables:
    • Two linear equations involving two variables x and y.
    • Example:
      1. $2x + 3y = 5$
      2. $x – y = 1$
  3. Methods of Solving Pair of Linear Equations:
    • Substitution Method:
      • Solve one equation for one variable and substitute it into the second equation.
    • Elimination Method:
      • Multiply or divide the equations to eliminate one variable and solve for the other.
    • Graphical Method:
      • Plot both equations on the graph and find the point of intersection.
  4. Types of Solutions:
    • Unique Solution:
      • The lines intersect at exactly one point.
    • No Solution:
      • The lines are parallel and never intersect.
    • Infinite Solutions:
      • The lines coincide, i.e., they are the same line.

III. Important Questions:

(A) Multiple Choice Questions (MCQs) (1 Mark):

  1. Which of the following represents a linear equation in two variables?
    • (A) $2x + 3y = 7$
    • (B) $x^2 + y^2 = 1$
    • (C) $2x – 4y = x + y$
    • (D) None of the above
    • Answer: (A)
  2. The pair of linear equations in two variables has:
    • (A) One solution
    • (B) No solution
    • (C) Infinite solutions
    • (D) All of the above
    • Answer: (D)
  3. What is the slope of the line representing the equation $3x + 4y = 8$?
    • (A) $-frac{3}{4}$​
    • (B) $3frac{3}{4}$​
    • (C) $4frac{3}{4}$​
    • (D) $-frac{4}{3}$​
    • Answer: (A)
  4. Which method is used to eliminate one variable from two equations?
    • (A) Substitution
    • (B) Elimination
    • (C) Graphical
    • (D) None of the above
    • Answer: (B)

(B) Short Answer Questions (2/3 Marks):

  1. Solve the pair of equations using the substitution method:
    $3x + 4y = 10, quad x – y = 1$.
  2. Using the elimination method, solve:
    $2x + 3y = 5, quad 4x – y = 3$.
  3. A pair of linear equations in two variables is given as:
    $x + 2y = 8, quad 2x – y = 7$.
    Solve the pair by any method.
  4. What will be the graphical representation of the system of equations:
    $2x + 3y = 6, quad x – y = 2$?

(C) Long Answer Questions (5 Marks):

  1. Solve the following system of linear equations by the elimination method:
    $2x + 3y = 5, quad 4x – 2y = 3$.
  2. A pair of linear equations is given:
    $5x + 6y = 12, quad 3x – 2y = 4$.
    Solve the system by substitution and check if the solution is correct by substituting into the original equations.
  3. Solve the following system of equations graphically:
    $3x – y = 7, quad 2x + y = 4$
  4. Find the values of x and y for the pair of equations:
    $7x + 3y = 11, quad 2x + 5y = 10$

(D) HOTS (Higher Order Thinking Skills) Questions:

  1. Explain why a system of linear equations in two variables has a unique solution, infinite solutions, or no solution. Provide examples for each case.
  2. If a pair of linear equations in two variables has no solution, how can their graphical representation help in identifying this condition? Provide an example.

IV. Key Formulas/Concepts:

  1. Substitution Method:
    • Solve one equation for one variable and substitute it in the other equation.
  2. Elimination Method:
    • Multiply or divide the equations to align the coefficients of one variable and eliminate it.
  3. Graphical Method:
    • Plot the two equations on the same graph and find the point of intersection, which gives the solution.
  4. Key Formula for Graph:
    • Slope of the line for equation $ax + by = c, quad m = -frac{a}{b}$​.

V. Deleted Portions (CBSE 2025-2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025-2026):

Unit/Chapter Estimated Marks Type of Questions Typically Asked
Pair of Linear Equations in Two Variables 6 1 Mark (MCQs), 2/3 Marks (Short Answers), 5 Marks (Long Answers)

VII. Previous Year Questions (PYQs):

  1. 2019:
    • (1 Mark) Find the value of x if $2x + 3y = 7, quad y = 1$.
    • (2 Marks) Solve the following pair of equations: $x + y = 5, quad 3x – 2y = 7$.
    • (5 Marks) Solve the pair of equations graphically:
      $x + y = 6, quad x – y = 2$.
  2. 2020:
    • (3 Marks) Solve the following pair of equations by substitution method:
      $3x + 4y = 12, quad 2x – y = 1$.

VIII. Real-World Application Examples to Connect with Topics:

  1. Economics:
    • Linear equations can be used to calculate profit-loss models and determine optimal pricing strategies in economics.
  2. Engineering:
    • The elimination and substitution methods are widely used in engineering for solving systems of equations in various fields like electrical circuits.
  3. Business Planning:
    • In business management, linear equations help in financial planning, budgeting, and forecasting sales.

IX. Student Tips & Strategies for Success:

  1. Time Management:
    • Allocate sufficient time to practice solving equations using all three methods (substitution, elimination, and graphical).
  2. Exam Preparation:
    • Practice previous year questions and ensure understanding of all solution methods.
  3. Stress Management:
    • Break your study time into smaller intervals and take short breaks. Stay positive and focus on your progress.

X. Career Guidance & Exploration (Class-Specific):

  • For Classes 9-10:
    • This chapter lays the foundation for future studies in Engineering (especially Electrical Engineering, Civil Engineering), Economics, Physics, and Computer Science.
  • For Classes 11-12:
    • Understanding linear equations helps in preparing for JEE (Joint Entrance Examination) and NEET exams, as well as pursuing a career in Data Science, Actuarial Science, and Business Analytics.

XI. Important Notes:

  • Regular practice is key to mastering the concepts of linear equations.
  • Keep revising your key formulas and methods.
  • Refer to the official CBSE website and NCERT books for the latest updates.

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