the opportunity to solve a problem invented by himself. – G. POLYA v . Introduction In earlier classes, we have discussed systems of linear equations and their applications in day to day problems.
In Class XI, we have studied linear inequalities and systems of linear inequalities in two variables and their solutions by graphical method. Many applications in mathematics involve systems of inequalities/equations. In this chapter, we shall apply the systems of linear inequalities/equations to solve some real life problems of the type as given below: A furniture dealer deals in only two items–tables and chairs. He has Rs , to invest and has storage space of at most pieces.
A table costs Rs and a chair Rs . He estimates that from the sale of one table, he can make a profit of Rs and that from the sale of one chair a profit of Rs . He wants to know how many tables and chairs he should buy from the available money so as to maximise his total profit, assuming that he can sell all the items which he buys. Such type of problems which seek to maximise (or, minimise) profit (or, cost) form a general class of problems called optimisation problems .
Thus, an optimisation problem may involve finding maximum profit, minimum cost, or minimum use of resources etc. A special but a very important class of optimisation problems is linear programming problem. The above stated optimisation problem is an example of linear programming problem. Linear programming problems are of much interest because of their wide applicability in industry, commerce, management science etc.
In this chapter, we shall study some linear programming problems and their solutions by graphical method only, though there are many other methods also to solve such problems.