This document discusses linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written as ax + by = c, where a, b, and c are real numbers and a and b are not both equal to zero. It also explains that a linear equation in two variables has infinitely many solutions and that the graph of a linear equation is a straight line. The document provides examples of linear equations and their graphical representations.

1. Linear Equations In
Two Variables
Class-09
FA-3 (2014-2015)

2. Linear Equations In
Two Variables
“ The principal use of the analytic art is
to bring mathematical problem to
equations and to exhibit those
equations in the most simple terms
that can be .”

3. Contents
• Introduction
• Linear equations
• Points for solving a linear equation
• Solution of a linear equation
• Graph of a linear equation in two variables
• Equations of lines parallel to x-axis and
y-axis
• Examples and solutions
• Summary

4. Introduction
• An excellent characteristic
of equations in two
variables is their adaptability
to graphical analysis. The
rectangular coordinate
system is used in analyzing
equations graphically. This
system of horizontal and
vertical lines, meeting each
other at right angles and
thus forming a rectangular
grid, is called the Cartesian
coordinate system.
Cartesian Plane

5. Introduction
• A simple linear equation is
an equality between two
algebraic expressions
involving an unknown
value called the variable.
In a linear equation the
exponent of the variable is
always equal to 1. The
two sides of an equation
are called Right Hand Side
(RHS) and Left-Hand Side
(LHS). They are written on
either side of equal sign.
Equation LHS RHS
4x + 3 = 5 4x + 3 5
2x + 5y = 0 2x + 5y 0
-2x + 3y = 6 -2x + 3y 6

6. Cont…
• A linear equation in
two variables can be
written in the form of
ax + by = c, where a,
b, c are real numbers,
and a, b are
equal to zero.
Equation a b c
2x+3y=9 2 3 -9
X+y/4-4=0 1 1/4 -4
5=2x 2 0 5
Y-2=0 0 1 -2
2+x/3=0 1/3 0 2

7. Linear equation :-
• A 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 equations can have one or more
variables. Linear equations occur with great regularity in
applied mathematics. While they arise quite naturally
when modeling many phenomena, they are particularly
useful since many non-linear equations may be reduced
to linear equations by assuming that quantities of interest
vary to only a small extent from some "background" state
-3 -2 -1 0 1 2 3
X + 2 = 0
X = -2

8. Solution of a linear equation
Every linear equation has a unique
solution as there is a single
variable in the equation to be
solved but in a linear equation
involving two variables in the
equation, a solution means a pair
of values, one for x and one for y
which satisfy the given equation
Example- p (x)=2x+3y (1)
y in terms of x
If x=3
2x + 3y = (2x3) + (3xy) = 12
6 + 3y = 12
y = 2
therefore the solution is (3,2)
(2)If x = 2
2x + 3y = (2x2) + (3xy) = 12
4 + 3y = 12
y= 8/3
therefore the solution is (2,8/3)
Similarly many another
solutions can be taken
out from this single
equation. That is ,a linear
equation in two variables
has infinitely many
solutions.

9. Graph of a linear equation is
representation of the linear
equation .
Observations on a graph
Every point whose
coordinates satisfy the
equation lies on the line.
Every point on the line gives
a solution of the equation.
Any point, which does not lie
on the line is not a solution
of equation. X+2Y=6
Graph of a linear equation in two
variables

10. A linear equation in one variable
represents a point on a number
line and a straight line parallel to
any of the axes in a coordinate
plane.

11. • Example 1:
Represent the equation 2 + 3 = 0 graphically
on the number line in the Cartesian plane
• Solution:
(i) The equation 2 + 3 = 0 has a unique solution .
Now, the geometrical representation of 2 + 3 =
0 i.e., = 1.5 on a number line is as follows.
Equations of lines parallel to x-axis

12. • (ii) The equation 2 + 3 = 0
can be written as . Thus,
represents a straight line in
the Cartesian plane parallel
to -axis and at a distance
of i.e., −1.5 from -axis. The
graph of this equation has
been shown in the following
figure

13. Equations of lines parallel to y-axis
• Equations of lines parallel to
y-axis The graph of x=a is a
straight line parallel to the
y-axis
• In two variables, 2 + 9 = 0
represents a straight line
passing through point
(−4.5, 0) and parallel
to -axis. It is a collection of
all points of the plane,
having their -coordinate as
4.5.

14. Summary
• An equation of the form ax +by + c =0,wherea,b and c are real
numbers, such that a and b are not both zero, is called a linear
equation in two variables.
• A linear equation in two variables has infinitely many solutions.
• The graph of every linear equation in two variables is a straight line.
• X=0 is the equation of the y-axis and y=0 is the equation of the x-axis
• The graph of x=a is a straight line parallel to the y-axis.
• The graph of y=a is a straight line parallel to the x-axis.
• An equation of the type y=mx represents a line passing through the
origin.
• Every point on the graph of a linear equation in two variables is a
solution of the linear equation.
• Every solution of the linear equation is a point on the graph of the
linear equation.

15. Thank You
Done by: IX
MANOJ GOWDA.M.L