Class 10 Notes - Pair of Linear Equations in Two Variables

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Pair of Linear Equations in Two Variables

In this chapter, you will learn how to solve problems involving two linear equations with two variables. These equations represent straight lines, and their solutions correspond to the points where the lines intersect.

Key Concepts

  • Linear Equation: An equation of the form ax + by + c = 0, where a, b, and c are real numbers, and x and y are variables.
  • Pair of Linear Equations: Two equations each of degree one, involving the same two variables.
  • Solution: The values of x and y that satisfy both equations simultaneously.
  • Consistent System: Has at least one solution (intersecting or coincident lines).
  • Inconsistent System: Has no solution (parallel lines).

Methods of Solving

  1. Graphical Method: Plot both equations on a graph. The intersection point(s) give the solution(s).
  2. Substitution Method: Solve one equation for one variable and substitute in the other.
  3. Elimination Method: Add or subtract equations to eliminate one variable, then solve for the other.
  4. Cross-Multiplication Method: Use the cross-multiplication formula to directly find the values of x and y.

Types of Solutions

  • Unique Solution: Lines intersect at one point (consistent and independent).
  • Infinitely Many Solutions: Lines coincide (consistent and dependent).
  • No Solution: Lines are parallel (inconsistent).

Example

Solve:

2x + 3y = 13
x + 2y = 8

Using Elimination Method:

  1. Multiply the second equation by 2:
    2x + 4y = 16
  2. Subtract the first equation from this result:
    (2x + 4y) - (2x + 3y) = 16 - 13
    y = 3
  3. Substitute y = 3 in the second equation:
    x + 2×3 = 8 ⇒ x = 2

Solution: x = 2, y = 3

Applications

  • Solving real-life problems involving two unknowns (age, distance, cost, etc.).
  • Finding intersection points of lines in geometry.
  • Solving word problems in algebra and coordinate geometry.

Practice Questions

  1. Solve the following pair of equations using the substitution method:
    x + y = 10
    x - y = 4
  2. Determine whether the following pair of equations is consistent, inconsistent, or dependent:
    3x - 2y = 6
    6x - 4y = 12
  3. Solve graphically:
    x + 2y = 7
    2x - y = 4

Summary

  • Pair of linear equations in two variables can be solved by graphical and algebraic methods.
  • The nature of the solution depends on the relationship between the two lines.
  • These concepts are widely used in real-life problem solving.