Define the concept of feasible solution in linear programming.

A feasible solution in linear programming is a solution that satisfies all the constraints of the problem.

In linear programming, a feasible solution is a solution that satisfies all the constraints of the problem. These constraints are usually represented as linear inequalities or equations. For example, consider the following linear programming problem:

Maximize 3x + 4y
Subject to:
2x + y ≤ 10
x + 3y ≤ 12
x, y ≥ 0

A feasible solution to this problem would be any pair of non-negative values of x and y that satisfy both of the constraints. For instance, (2, 4) is a feasible solution because it satisfies both constraints:

2(2) + 4 = 8 ≤ 10
2 + 3(4) = 14 ≤ 12 (not satisfied)

However, (3, 2) is not a feasible solution because it violates the second constraint:

2(3) + 2 = 8 ≤ 10
3 + 3(2) = 9 ≤ 12

Feasible solutions are important in linear programming because they form the basis for finding the optimal solution. The optimal solution is the feasible solution that maximizes or minimizes the objective function (in this case, 3x + 4y). By testing different feasible solutions, we can determine which one is optimal. Understanding feasible solutions is crucial for solving real-world applications where such problems are frequently encountered. Moreover, feasible solutions are the foundation of optimisation problems, helping to achieve the best outcome under given constraints. They are also closely linked to the techniques used in applications of differentiation, which can help further analyse and solve complex problems in linear programming.

A-Level Maths Tutor Summary: In linear programming, a feasible solution meets all the problem's conditions, like linear inequalities or equations. For example, in a problem aiming to maximise 3x + 4y with certain constraints, values of x and y that don't break any rules are feasible solutions. These solutions are vital as they help identify the best outcome that maximises or minimises the objective function.