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LINEAR EQUATIONS
2x+ 3y =6
3x-4y =7
-6x +7y = 42
-9x - 4x = -36
5x –8y =-40
3x –7y =21
9x - 4x = -3
6
3x –7y =21
Mrs. ChanderkantaMrs. Anju Mehta
Target group
Class ninth and tenth
LEARNING OBJECTIVESDefine the linear equation in two variable.
Solution of linear equation.
Converts a linear equation of two variable in graphical form .
Solve simultaneous linear equation by graphical method.
Learn computer skills.
Learn about MS Office.
Develop a habit of research.
Learn to insert the pictures and relevant text in their presentation .
Learn editing skill.
WHEN we talk to each other, we use sentences.
What do we say?
Either we talk or we give some statements
These statements may be RIGHT or WRONG
For example we make the statement-
”sunrises in the east and sets in the west”
WHEN we talk to each other, we use sentences.
What do we say?
Either we talk or we give some statements
These statements may be RIGHT or WRONG
For example we make the statement-
”sunrises in east and sets in west”
This is a TRUE statement
It is not necessary that all the statements are true. Some are true and some are false. In mathematics we call those statements as OPEN STATEMENTS
If an open statement becomes TRUE for some value then it is called EQUALITY and it is represented by the sign “ = “
An EQUALITY has two sides L.H.S. and R.H.S. where,
= R.H.S.L.H.S.
In mathematics, we often use OPEN STATEMENTS
For example the statement ,
If we add any number to 5, we may or may not get 8
5 + 1= 8 5 + 2 = 8 5 + 3 = 8
The number 3 makes both the sides equal. Hence the statement becomes TRUE.
FALSE STATEMENT
FALSE STATEMENT
TRUE STATEMENT
“ any number added to 5 will give 8” is an open statement
5 kg
2 kg
How much weight should be added to equalize the balance?
+ 2 kg = 5 kg
The above statement becomes
This statement is called an EQUATION
This equation will be true depending on the value of the variable ‘x’
+ 2kg = 5kg
x+ 2= 5
ax+b = 0
is an equation in one variable x
Where a,b are constants & a =
0
So we can say,
Let us take an example from daily life.
Cost of two rubbers and three pencils is six rupees
In mathematical form, it can be written as
2x + 3y = 6,
where x is the cost of one rubber and y of one pencil
x 3 0
y 0 2
(3, 0)
(0,2)
Ordered pairs
Let us plot the ordered pairs:
(3,0)
x-axis
Y- axis
1 2 3 4 5 6 7
1
-1-1
2
-2
-2
3
-3
-3*
*
0
Show me (0,2) Show me
2x + 3y =6(3,0)
(0,2)
You have seen that the equation 2x+3y =6 is giving a straight line in the graph
These types of the equations are called LINEAR EQUATIONS
Solutions of an equation 2x + 3y =6
are x =0 , y=2 and x=3 , y=0.
In any equation of the type ax + by+ c = 0
where a, b, c --- constants
x , y --- variables
will gives straight line in the graph
Note:
If in an equation ax+ by + c= 0
Case1: When a =0,b= 0, then
e.g. in an equation 2x+3y =6 ,
0x + 3y =6
3y = 6 –0x
y =6-0x 3
x 1 2 -3
y 2 2 2
If a=0
0x + by +c = 0
Let us plot the ordered pairs:
x-axis
Y- axis
1 2 3 4 5 6 7
1
-1-1
2
-2
-2
3
-3
-3 0
* **
(1,2) (2,2) (-3,2)
(1,2)
(2,2)(-3,2)
Show me
Show me
Show me
Show me
0x+3y =6
LINE IS PARALLEL TO X-AXIS
if in an equation ax+ by + c= 0
Case2:
when a =0, b =0, then
e.g. in an equation 2x+0y =6 ,
2x + 0y =6
2x = 6 – 0y
x =6-0y 2
x 3 3 3
y -2 1 3
ax + 0y + c =0
when b=0
Let us plot the ordered pairs:
(3,-2)
x-axis
Y- axis
1 2 3 4 5 6 7
1
-1-1
2
-2
-2
3
-3
-3 0
(3,1)Show me Show me (3,3)
*
*
*
Show me
(3,-2)
(3,1)
(3,3)
2x +
0y =
6
LINE IS PARALLEL TO Y-AXIS
if in an equation ax+ by + c= 0 when
Case3: When a =0,b= 0, c =0
ax +by = 0
e.g. in an equation 2x+3y =6 , if c=0
2x + 3y =0
2x = -3y
x =-3y 2
x -3 3 0
y 2 -2 0
Let us plot the ordered pairs:
(-3,2)
x-axis
Y- axis
1 2 3 4 5 6 7
1
-1-1
2
-2
-2
3
-3
-3 0
Show me (3,-2)Show me Show me (0,0) Show me
*
*
*
(-3,2)
(3,-2)
(0,0)
2x+3y =0
LINE PASSES THROUGH THE CENTER
If we draw two linear equations in one graph then we have three possibilities:
1: Intersecting lines * one solution
2: Parallel linesno solution
3. Lines will coincide many
solutions
Now there is an exercise for you.
Take any two linear equations. Plot them on the graph and observe what type of solution you get.
ACKOWLEDGEMENT
•“Mathematics” by R.S.AGGARWAL
•N.C.E.R.T. BOOK FOR Mathematics for Class-X
•Mr. V.K. Sodhi ,Senior Lecturer,S.C.E.R.T.
Internet sites:
www.math.nice.edu
www.math.org.uk
www.pass.math.org.uk