IN T WO V ARIABLES Let us try this approach. Denote the number of rides that Akhila had by x , and the number of times she played Hoopla by y . Now the situation can be represented by the two equations: y = x x + y = Can we find the solutions of this pair of equations? There are several ways of finding these, which we will study in this chapter.
. Graphical Method of Solution of a Pair of Linear Equations A pair of linear equations which has no solution, is called an inconsistent pair of linear equations. A pair of linear equations 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 summarise the behaviour of lines representing a pair of linear equations in two variables and the existence of solutions as follows: (i) the lines may intersect in a single point. In this case, the pair of equations has a unique solution (consistent pair of equations).
(ii) the lines may be parallel. In this case, the equations have no solution (inconsistent pair of equations). (iii) the lines may be coincident. In this case, the equations have infinitely many solutions [dependent (consistent) pair of equations].
Consider the following three pairs of equations. (i) x – y = and x + y – = (The lines intersect) (ii) x + y – = and x + y – = (The lines coincide) (iii) x + y – = and x + y – = (The lines are parallel) Let us now write down, and compare, the values of , in all the three examples. Here, a , b , c