class 10 linear equation in two variables

LINEAR EQUATIONS IN TWO VARIABLES
– CLASS 10 NOTES

INTRODUCTION
In mathematics, an equation that can be written in the
form ax + by + c = 0, where a, b, and c are real numbers
and both a and b are not zero, is called a linear equation
in two variables. These types of equations represent
straight lines when plotted on a Cartesian plane.
They are called “linear” because the highest degree of
the variables x and y is 1.
Such equations have infinitely many solutions, and each
solution is an ordered pair (x, y).


GENERAL FORM AND EXAMPLES
The general form of a linear equation in two variables is:
ax + by + c = 0
Where:
 x and y are the variables
 a, b, and c are real numbers
 a and b are not both zero
Examples:
 3x + 2y = 6
 x-y+4=0
 2x + 5y - 7 = 0
Each of these represents a straight line on a graph.

, SOLUTION OF A LINEAR EQUATION
A solution of a linear equation in two variables is a pair of
values (x, y) that satisfies the equation.

For example, in the equation 2x + 3y = 12:
 If x = 0 → y = 4 → (0, 4) is a solution
 If y = 0 → x = 6 → (6, 0) is a solution
There are infinitely many such solutions.


GRAPHICAL REPRESENTATION
To graph a linear equation:
1. Choose at least two values for x, find corresponding
y values.

2. Plot the ordered pairs (x, y) on the Cartesian plane.
3. Join the points using a straight line.
Example:
For the equation x + y = 5:
 x = 0 → y = 5 → point (0, 5)
 y = 0 → x = 5 → point (5, 0)
Join these points to draw the graph.


PAIR OF LINEAR EQUATIONS IN TWO
VARIABLES
Two linear equations taken together form a system of linear
equations:

a1x + b1y + c1 = 0
a2x + b2y + c2 = 0


METHODS OF SOLVING LINEAR EQUATIONS