graphical method to find the solution of pair of linear equation in two variables

Graphical Method to find the Solution of Pair of Linear Equation in Two Variables

•Sketching straight lines

•Solving Pair of linear equations in two variables by straight line graphs Graphical Method

To draw the graph of a straight line we must find the coordinates of some points which lie on the line.We do this by forming a table of values . Give the x coordinate a value and find the corresponding y coordinate for several points

Table of values

x

y=2x+1

Make a table of values for the equation y = 2x + 1

0 1 2 3

1 3 5 7

Now plot the points and join them up

So (0,1) (1,3) (2,5) and (3,7) all lie on the line with equation y=2x + 1

•Sketching straight lines

x

y

0 1 2 3 4

7

6

5

4

3

2

1

Plot the points (0,1) (1,3) (2,5) and (3,7) on the grid

..

..

Now join then up to give a straight line

All the points on the line satisfy the equation y = 2x + 1

Sketching lines by finding where the lines cross the x axis and the y axis A quicker method

Straight lines cross the x axis when the value of y = o

Straight lines cross the y axis when the value of x =0

Sketch the line 2x + 3y = 6

Line crosses x axis when y = 0

2x + 0 =6

2x =6

x =3

at ( 3,0)

Line crosses y axis when x = 0

0 + 3y = 6

3y =6

y = 2

at ( 0,2)

Plot 0,2) and (3,0) and join them up with a straight line

x

y

0 1 2 3 4

7

6

5

4

3

2

1.

.

Now join then up to give a straight line

All the points on the line satisfy the equation 2x + 3y = 6

•Solving Pair of linear equations in two variables by straight line graphs (Graphical Method)

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

Graphical representation of pair of linear Equations

as two lines.

y = 2x - 8

y = x - 1

2x – y = 8

x – y = 1

PlotExample 1

X 0 5

Y -8 2

X 0 3

Y -1 2

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

Graphical representation of pair of linear Equations

as two lines.

y = 2x + 1y = (4x + 2)/2

-2x - y = -1-4x + 2y =

2

Solve

Example 2

X 0 2

Y 1 5

X -1 4

Y -1 9

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

Graphical representation of pair of linear Equations as

two lines.

y = x + 3

y = x + 6

-x + y = 3

-x + y = 6

Question

x 0 4

y 3 7

x 0 2

y 6 8

We have seen that the lines may intersect or may be parallel or may coincide.

Can we find the solution of the pair of equations from the lines drawn that is

solution from the geometrical point of view?

Let us look at the earlier example one by one.

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

(7,6) is the only common point for both the intersecting lines

The co-ordinates of the point of intersection of lines give the solutions

to the equations.

y = 2x - 8

y = x - 1

2x – y = 8

x – y = 1

SolveExample 1

Solutions

x = 7, y = 6

(7,6)

(7,6) is the one and only one solution for the given pair of linear equations.

The pair of equations has a unique solution is called consistent pair of linear

equations.

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

Here graph geometrically represent a pair of coincident lines.

Every point on the line is a common solution for the equations given

The pair of equations has infinite many

solution s.

y = 2x + 1y = (4x +

2)/2

-2x - y = -1-4x + 2y =

22

SolveExample 2

Solutions

Infinite many solutions

Coincident lines

A pair of linear equation which are equivalent has

infinite many distinct common solutions are

called dependent pair of linear equations

A dependent pair of linear equation s is always

consistent

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Graphical Solution of pair of linear equations in two variables

The equations have no common solution.

Lines do not intersect at all

Here graph geometrically represent a pair of parallel lines.

y = x + 3

y = x + 6

-x + y = 3

-x + y = 6

SolveQuestion

Solutions

Parallel lines

No solution

The pair of linear equations which has no

solution is called an inconsistent pair of linear

equation.

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

Home Work

-2x + y =

1x + y = 10

Solve

Example 2(3,7)

Solutions

x = 3, y = 7

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-x + y = 32x + y =

6

SolveQuestion

(1,4)

Solutions

x = 1, y = 4

0 1 2 3 4 5 6 7 8 9 10-9 -8 -7 -6 -5 -4 -3 -2 -1-10 x

y

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

-9

-10

-2x + y = 1x + y =

10

Solve

Example 2

(3,7)Solutions

x = 3, y = 7

Presented by

Nikhilesh Shrivastava

K.V.KankerTGT(Maths)