- The document discusses linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c. - It describes how linear equations in two variables have infinitely many solutions, represented by pairs of x and y values. The graph of a linear equation in two variables is a straight line. - The document also discusses how equations of lines parallel to the x-axis or y-axis can be represented. The graph of an equation of the form x = a is a line parallel to the y-axis, while an equation of the form y = a graphs as a line parallel to the x-axis.

1. Linear Equations
In Two Variables
Class 9
Prepared by:
Sharda Chauhan
TGT Mathematics

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 not 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 :-
•
-3 -2 -1 0 1 2 3
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
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.

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

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

19. 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, 2x + 9 = 0
represents a straight line
passing through point
(−4.5, 0) and parallel to y-axis.
It is a collection of all points of
the plane, having their x-
coordinate as 4.5.

20. 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.

21. Question Bank
• 1) Express the following linear equations in the form ax + by + c = 0 and
indicate the values of a, b, c in each case:
i) x = 3y
1x − 3y + 0 = 0
Comparing this equation with ax + by + c = 0,
a = 1, b = −3, c = 0
ii) 2x = −5y
2x + 5y + 0 = 0
Comparing this equation with ax + by + c = 0,
a = 2, b = 5, c = 0
iii) 3x + 2 = 0
3x + 0.y + 2 = 0
Comparing this equation with ax + by + c = 0,
a = 3, b = 0, c = 2

22. • 2) Write 4solutions for the following equation :
2x + y = 7
y in terms of x
For x = 0,
2(0) + y = 7
y = 7
Therefore, (0, 7) is a solution of this equation.
For x = 1,
2(1) + y = 7
y = 5
Therefore, (1, 5) is a solution of this equation.
For x = −1,
2(−1) + y = 7
y = 9
Therefore, (−1, 9) is a solution of this equation.
For x = 2,
2(2) + y = 7
y = 3
Therefore, (2, 3) is a solution of this equation.

23. 3) Find the value of k, if x = 2, y = 1 is a solution
of the equation 2x + 3y = k.
Putting x = 2 and y = 1 in the given equation,
2x + 3y = k
2(2) + 3(1) = k
4 + 3 = k
k = 7
Therefore, the value of k is 7.
•

24. 4) Draw the graph of the following linear
equations in two variables:
i) x + y = 4
y in terms of x
y = 4 – x
If x = 0
y = 4
(0,4) is a solution.
If x = 4
y = 0
(4,0) is a solution.
ii) y = 3x
y in terms of x
y = 3x
If x = -1
y = -3
(-1,-3)is a solution.
If x = 1
y = 3
(1,3) is a solution.
x 0 4
y 4 0
x -1 1
y -3 3

25. Thank You
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