Define the concept of slack variables in linear programming.

Slack variables are introduced in linear programming to convert inequality constraints into equality constraints.

In linear programming, constraints are usually expressed as inequalities. However, the simplex algorithm, which is used to solve linear programming problems, requires that all constraints be expressed as equalities. To convert an inequality constraint into an equality constraint, a slack variable is introduced.

A slack variable is a non-negative variable that is added to an inequality constraint to make it an equality constraint. The value of the slack variable represents the amount by which the left-hand side of the inequality can be increased to satisfy the constraint. For example, consider the inequality constraint:

2x + 3y ≤ 10

To convert this into an equality constraint, we introduce a slack variable s:

2x + 3y + s = 10

The value of s represents the amount by which the left-hand side of the inequality can be increased to satisfy the constraint. If s = 0, then the inequality is an equality and the constraint is satisfied. If s > 0, then the left-hand side of the inequality can be increased by up to s to satisfy the constraint.

Understanding the function and utility of slack variables is essential for solving optimisation problems in various applications. To explore further, you can review practical examples on optimisation problems.

A-Level Maths Tutor Summary: In linear programming, slack variables let us change inequality constraints (like 'less than' equations) into equality constraints (exact equations) for solving with the simplex algorithm. They are non-negative variables added to the inequality, showing how much we can increase its left side to meet the constraint. They help the simplex algorithm work by turning inequalities into exact equalities we can solve.

Additionally, the concept of slack variables ties into broader mathematical concepts, which you can further explore in types of numbers. Understanding these foundations enhances the application of mathematical theory in practical scenarios, such as those found in applications of differentiation.