LINEAR EQUATIONS

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