In the previous section, you have seen how we can graphically represent a pair of linear equations as two lines.  You have also seen that the lines may intersect, or may be parallel, or may coincide.  Can we solve them in each case?  And if so, how?  We shall try and answer these questions from the geometrical point of view in this section.  

        Let us look at the earlier examples one by one. 

     I • In the situation of Example 1, find out how many rides on the Giant Wheel Akhila had, and how many times she played Hoopla. 

     In Fig.  3.2, you noted that the equations representing the situation are geometrically shown by intersecting two lines at the point (4,2).  Therefore, the point (4,2) lies on the lines represented by both the equations x – 2y = 0 and 3x + 4y = 20. And this is the only common point.

      Let us verify algebraically that x = 4, y = is a solution of the given pair of equations.  Substituting the values ​​of x and y in each equation, we get 4-2 x 2 = 0 and 3 (4) + 4 (2) = 20. So, we have verified that x = 4, y = 2 is a solution of  both the equations, Since (4, 2) is the only common point on both the lines, there is one and only one solution for this pair of linear equations in two variables.  

        Thus, the number of rides Akhila had on Giant Wheel is 4 and the number of times she played Hoopla is 2. 

         In the situation of Example 2, can you find the cost of each pencil and each eraser?  

        In Fig.  3.3, the situation is geometrically shown by a pair of coincident lines.  The solutions of the equations are given by the common points, 

        Are there any common points on these lines?  From the graph, we observe that every point on the line is a common solution to both the equations.  So the equations 2x + 3y = 9 and 4x + y = 18 have infinitely many solutions.  This should not surprise us, because if we divide the equation 4r + by = 18 by 2. we get.  2x + 3y = 9, which is the same as Equation (1).  That is, hoth the equations are equivalent.  From the graph, we see that any point on the line gives us a possible cost of each pencil and craser.  For instance, each pencil and eraser can cost 23 and 1 respectively.  Or, cach pencil can cost?  3.75 and eraser can cost 0.50, and so on. 

         In the situation of Example 3, can the two rails cross each other?  

        In Fig.  3,4, the situation is represented geometrically by two parallel lines.  Since the lines do not intersect at all the mils do nofcross.  This also means that the equations have no common solution 

        A pair of lincar equations which has no solution is called an inconsistent pairof linear equations.  A pair of lincar cquations in two variables, which has a solution, is called a consistent pair of linear equations.  A pair of linear equations which are equivalent has infinitely many distinct common solutions Such a pair is called a dependent pair of linear equations in two variables.  Note that a dependent pair of linear equations is always consistent. 

        We can now summarize the behavior of lines representing a pair of lincar equations in two variables and the existence of solutions as follows.