Maths Notes - CH04: Pair of Linear Equations in Two Variables

CBSE Class 9th Maths Notes

Chapter 4: Pair of Linear Equations in Two Variables

Introduction:

➥ For any linear equation, each solution (x, y) corresponds to a point on the line.

➥ General form: ax + by + c = 0.

➥ The graph of a linear equation is a straight line.

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

➥ General form of a pair of linear equations:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

where a₁, a₂, b₁, b₂, c₁, and c₂ are real numbers, such that a₁ + b₁ ≠ 0 and a₂ + b₂ ≠ 0.

Solution of Linear Equations:

➥A pair of values of variables x and y which satisfy both the equations in the given system of equations is said to be a solution of the simultaneous pair of linear equations.

⟹A pair of linear equations in two variables can be represented and solved by:

1. Graphical Method
2. Algebraic Method

Graphical Method:

The graph of a pair of linear equations in two variables is represented by two lines.

Algebraic Methods:

1. Substitution Method
2. Elimination Method
3. Cross-Multiplication Method

Consistency of System:

1. Consistent System: A system of linear equations is said to be consistent if it has at least one solution.
2. Inconsistent System: A system of linear equations is said to be inconsistent if it has no solution.

Conditions for Consistency:

Let the two equations be:

a₁x + b₁y + c₁ = 0
a₂x + b₂y + c₂ = 0

The conditions for consistency are:

Note: Practice solving linear equations using both graphical and algebraic methods to strengthen your understanding.