This document provides an introduction and overview of linear equations in two variables. It defines linear equations as equations where each term is a constant or the product of a constant and a single variable. Linear equations in two variables can be represented graphically as a straight line on a coordinate plane, where the solution is the point at which the line crosses the x and y axes. The document discusses how to solve systems of two linear equations in two variables and represents different types of linear equations graphically, including lines parallel to the x and y axes.

1. LINEAR EQUATION IN TWO VARIABLES
Chapter-4
For
Grade-IX
By
Sri. Siddalingeshwara BP M.Sc, B.Ed, (P.hd)
Contact-9590700228
Email-siddu.nandi143@gmail.com

2. Contents:
1.Introduction
2. Linear Equations
3. Solution of a Linear equation
4. Graph of a Linear Equation in Two Variables
5.Equations of Line Parallel to x-axis and y-axis
6. Summary

3. 1. Introduction:
In earlier classes, you have studied linear equation in one variable. Can you write Down a
linear equation in one variable? You may say that 𝑥 + 1 = 0,𝑥 + 2 = 0 and 2𝑦 + 3 = 0 are
examples of linear equation in one variable. You also known That such equations have a
unique (i.e one and only one) solution. And also how to Represent the solution on a number
line. In this chapter, the knowledge of linear equations in one variable shall be recalled and
extended to that of two variables.
“Linear equation is an algebraic equation in which each term is either a constant
or the product of a constant and a single variable. linear equation can have one
are more variable”.
2. Linear Equations:
In y = ax + b, x is called independent variable and y is
called dependent variable. a and b are called constants.
y = 2x + 5 with a = 2 and b = 5, y = -3x + 2 with a = -3 and b = 2,
and y = 4x + - 1 with a = 4 and b = -1 are other examples of linear
equations.
I am thinking of a number. If 1 add 2 to that number, I will get 5. What is the number?
Although it may be fairly easy to guess that the number is 3, you can model the situation
above with a linear equation.
Let x be the number in my mind.
Add 2 to x to get 5.
Adding 2 to x to get 5 means that whatever x is, when I add 2 to x, it has to equal to 5.
The equation is 2 + x = 5
Example:

4. The linear equations in one variable
is an equation which is expressed in the
form of ax+b = 0, where a and b are two
integers, and x is a variable and has only
one solution.
For example, 2x+3=8 is a linear equation
having a single variable in it.
Therefore, this equation has only one
solution,
which is x = 5/2.
And representation of linear
equation in one variable on line

5. An equation is said to be linear equation in two variables if it is written in
the form of ax + by + c=0, where a, b & c are real numbers and the coefficients of x
and y, i.e a and b respectively, are not equal to zero.
For example, 10x+4y = 3 and -x+5y = 2
are linear equations in two variables. The solution for such an equation is a pair of
values, one for x and one for y which further makes the two sides of an equation equal
While solving an equation, you must always keep the following points in mind
1. The same number is added to (or Subtracted from) both the sides of equation.
2. You multiply or divide both the sides of equation by the same non-zero numbers

7. 3. Solution of Linear Equations in Two Variables:
The solution of linear equations in two variables, ax+by = c, is a particular point
in the graph, such that when x-coordinate is multiplied by a and y-coordinate is
multiplied by b, then the sum of these two values will be equal to c.
Basically, for linear equation in two variables, there are infinitely many solutions.
Example 1:
In order to find the solution of Linear equation in 2 variables, two equations
should be known to us.
Consider for Example:
5x + 3y = 30
The above equation has two variables namely x and y.
Graphically this equation can be represented by substituting the variables to zero.
The value of x when y=0 is
5x + 3(0) = 30
⇒ x = 6
and the value of y when x = 0 is,
5 (0) + 3y = 30
⇒ y = 10

8. 2

11. 4.Graph of a Linear Equation in Two Variables:
1.Any linear equation in the standard form ax+by+c=0 has a pair of solutions in
the form (x,y), that can be represented in the coordinate plane.
2.When an equation is represented graphically, it is a straight line that may or may
not cut the coordinate axes
Case-1
1.A linear equation ax+by+c=0 is represented graphically as a straight line.
2.Every point on the line is a solution for the linear equation.
3.Every solution of the linear equation is a point on the line.

12. Example: 1 Draw the graph of x+y=4 in two variable?
Solution:
Scale
X-axis=1unit=1cm
Y-axis=1 unit=1cm

13. Case:2 Lines passing through the origin
1. Certain linear equations exist such that their solution is (0, 0).
Such equations when represented graphically pass through the origin.
2. The coordinate axes namely x-axis and y-axis can be represented as
y=0 and x=0, respectively.
3. The graph of the equation of the form y=kx is a line
which always passes through the origin.
(0,0)
y=kx
origin

14. Example:2

15. Example:3

16. Example:4

17. Example:5

18. 5.Equations of Line Parallel to x-axis and y-axis:
1.Linear equations of the form y=b, when represented graphically are lines
parallel to the x-axis and b is the y-coordinate of the points in that line.
2.Linear equations of the form x=a, when represented graphically are lines
parallel to the y-axis and a is the x-coordinate of the points in that line.
x=a Y=b

19. Examples:1.1

20. Example: 1.2

22. Example:1.1 and 1.2

23. Example: 2.1

24. Example:2.2

26. Example: 2.1 and 2.2

27. 6. Summery

28. References:
1. NCERT Text book
2. Google sources
Thank you