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

2. Sketching straight lines Solving Pair of linear equations in two variables by straight line graphs Graphical Method

3. 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 01 23 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

4. x y 0 1 2 3 4 76543217654321 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

5. 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)

6. Plot 0,2) and (3,0) and join them up with a straight line x y 0 1 2 3 4 76543217654321.. Now join then up to give a straight line All the points on the line satisfy the equation 2x + 3y = 6

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

8. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 8 9 10 -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  Plot Example 1 X05 Y-82 X03 Y2

9. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -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 + 1 y = (4x + 2)/2 -2x - y = -1 -4x + 2y = 2  Solve Example 2 X02 Y15 X4 Y 9

10. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -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 x04 y37 x02 y68

11. 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.

12. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -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  Solve Example 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.

13. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -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 + 1 y = (4x + 2)/2 -2x - y = -1 -4x + 2y = 2 22 Solve Example 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

14. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -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  Solve Question Solutions Parallel lines No solution The pair of linear equations which has no solution is called an inconsistent pair of linear equation.

15. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 Home Work -2x + y = 1 x + y = 10  Solve Example 2 (3,7) Solutions x = 3, y = 7

16. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 -x + y = 3 2x + y = 6  Solve Question (1,4) Solutions x = 1, y = 4

17. 0 1234567 8 910 -9-8 -7 -6 -5 -4-3-2 -10 x y 1 2 3 4 5 6 7 8 9 10 -2 -3 -4 -5 -6 -7 -8 -9 -10 -2x + y = 1 x + y = 10  Solve Example 2 (3,7) Solutions x = 3, y = 7

18. Presented by Nikhilesh Shrivastava K.V.Kanker TGT(Maths)