1. B.J.P.S Samiti’s
M.V.HERWADKAR ENGLISH MEDIUM HIGH SCHOOL
CLASS 9th: LINEAR EQUATIONS IN TWO VARIABLES
Program:
Semester:
Course: NAME OF THE COURSE
Staff Name: VINAYAK PATIL 1
2. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 2
• All of you have studied about linear equations in earlier classes.
• x + 1 = 0 , x -3 = 0 , y - √2 = 0 , √2 y + √3 = 0
are examples of linear equations in one variable . These have unique
solution and all of you know how to represent solution of linear
equation on number line.
Here, in this chapter , we will extend your knowledge of linear
equation in one variable into linear equation of two variables.
INTRODUCTION
3. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 3
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.
X + 2 = 0
X = -2
-3 -2 -1 0 1 2 3
4. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 4
EXERCISE 10.1
• 1) The cost of a notebook is twice the cost of the pen write a linear
equation in two variables to represent this statement?
5. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 5
SOLUTIONS OF A LINEAR EQUATIONS
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
6. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 6
EXERCISE 10.2
1) Find four solutions of the equations:
(i)2x + y = 7 , (ii) x = 4y .
2) Find the value of k , if x = 2 and y = 1 is the
solution of the equation 2 x + 3 y = k
7. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 7
GRAPH OF A LINEAR EQUATION IN
TWO VARIABLES
• A linear equation in two variables is
represented geometrically by a line whose
points make up the collection of solutions of
the equation. This is called the GRAPH of the
linear equation.
• So, to obtain the graph of a linear equation in
two variables , it is enough to plot two points
corresponding to the two solutions and join
them by a line. However Plotting of more
points can immediately check the correctness
of the graph.
8. EXERCISE 10.3
1) Draw the graph of i) x + y = 4 ii) y = 3x
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 8
9. Equations of Lines Parallel to the x-axis
and y-axis
• Well students , all of you know that the points (2,0) ,
(-3,0 ) , (4,0) , (n,0) ,for any real number n , lie in the
Cartesian plane ,even they all lie on x-axis. Because
on the x-axis , the ycoordinate of each point is zero.
In fact , every point on the x-axis is of the form (x,0).
SO, the equation of x-axis is y = 0 which can be
expressed as 0.x + 1.y = 0
• Similarly , equation of y- axis is given by x = 0
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 9
10. M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 10
1) Solve the equation 2x + 1 = x – 3 and
represent the solution(s) on
(i) number line (ii) the cartesian plane .
Sol. 2 x + 1 = x – 3 gives us 2x – x = -3 – 1
i.e. x = - 4
12. EXERCISE 10.4
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 12
Give the geometric representations of
2 x + 9 = 0 as an equation
(i) in one variable
(ii) in two varaibles.
13. HOMEWORK
1) Define linear equation in two variables with
example
M.V.HERWADKAR ENGLISH MEDIUM SCHOOL 13
