1. Introduction

A linear equation in two variables is an equation that can be written in the form:


ax + by + c = 0
  • and are variables,
  • are real numbers, and
  • and are not both zero.

When two such equations are given together, they form a pair of linear equations in two variables.

2. General Form of a Pair of Linear Equations in Two Variables

A system of two linear equations in two variables can be written as:


a_1x + b_1y + c_1 = 0

a_2x + b_2y + c_2 = 0 ] where are real numbers, and , .

3. Methods to Solve a Pair of Linear Equations

There are several methods to solve a system of linear equations:

i. Graphical Method

  • Plot both equations as straight lines on a graph.
  • The point where they intersect is the solution .
  • The nature of solutions:
    • If lines intersect, they have one unique solution (consistent and independent).
    • If lines are parallel, they have no solution (inconsistent).
    • If lines coincide, they have infinitely many solutions (consistent and dependent).

ii. Substitution Method

  1. Express one variable in terms of the other using one equation.
  2. Substitute this in the second equation.
  3. Solve for one variable, then use it to find the second variable.

iii. Elimination Method

  1. Multiply equations if necessary to make coefficients of one variable the same.
  2. Add or subtract the equations to eliminate one variable.
  3. Solve for the remaining variable and substitute back to find the other.

iv. Cross Multiplication Method

Used when the system is given in standard form. The solution formulas are:


x = \frac{b_1c_2 - b_2c_1}{a_1b_2 - a_2b_1}, \quad y = \frac{c_1a_2 - c_2a_1}{a_1b_2 - a_2b_1}

v. Determinant (Matrix) Method

Uses determinants to solve the system of equations using Cramer's Rule.

4. Types of Solutions (Consistency of Equations)

  • Unique solution: (lines intersect)
  • No solution: (parallel lines)
  • Infinitely many solutions: (coincident lines)

5. Applications in Real Life

Linear equations in two variables are used in:

  • Business (cost-profit analysis)
  • Physics (motion and forces)
  • Economics (supply and demand)
  • Geometry (finding intersection points)

6. Examples

Example 1: Solve by Substitution Method


2x + 3y = 12

4x - y = 5 ]

  • Express in terms of :

  y = \frac{12 - 2x}{3}


  4x - \frac{12 - 2x}{3} = 5

Example 2: Solve by Elimination Method


3x + 2y = 5

2x - 3y = -4 ]

  • Multiply to make coefficients equal, then add/subtract.

7. Summary

  • Linear equations in two variables represent straight lines.
  • The solution is the intersection point of the two lines.
  • Methods include graphical, substitution, elimination, and cross multiplication.
  • The number of solutions depends on the relation between the coefficients...