Linear Equation in Two Variables

A linear equation in two variables is an equation of the form:

ax + by + c = 0

where:

  • a, b and c are real numbers.
  • a and b cannot both be zero.
  • x and y are variables.

Examples

  • 2x + 3y = 12
  • x – y = 5
  • 5x + y = 18

Non-Examples

  • x² + y = 8
  • xy = 10
  • √x + y = 5

These are not linear equations because the degree of the variables is greater than 1 or variables are multiplied together.

Pair of Linear Equations

Two linear equations involving the same variables are called a pair of linear equations in two variables.

Example:

2x + y = 5

3x – 2y = 4

These equations together form a pair of linear equations.

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Solution of a Pair of Linear Equations

The solution of a pair of linear equations is an ordered pair (x, y) that satisfies both equations simultaneously.

Example:

x + y = 5

x – y = 1

The solution is:

x = 3

y = 2

Verification:

3 + 2 = 5 ✔

3 – 2 = 1 ✔

Hence, (3, 2) is the solution.

Types of Solutions

1. Unique Solution

When two lines intersect at exactly one point, the pair has one unique solution.

2. No Solution

When two lines are parallel, they never meet.

Hence, the pair has no solution.

3. Infinitely Many Solutions

When both equations represent the same line, every point on the line satisfies both equations.

Hence, there are infinitely many solutions.

Real-Life Applications

Pair of linear equations are used in:

  • Shopping problems
  • Age problems
  • Speed and distance
  • Mixture problems
  • Geometry
  • Business calculations

Key Points

  • A linear equation has degree 1.
  • It contains two variables.
  • Two such equations form a pair.
  • The solution satisfies both equations.
  • A pair can have one, no or infinitely many solutions.

 

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