What is the primary purpose of introducing slack variables in linear programming?

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Q.41) Slack variables are introduced into linear programming constraints to:

• A) Convert inequality constraints into equalities
• B) Increase the number of constraints
• C) Make the objective function non-linear
• D) Eliminate the need for non-negativity

Step 1: Understand the purpose of slack variables. In linear programming, slack variables are non-negative variables added to "less than or equal to" ($\le$) inequality constraints to transform them into equality constraints. This conversion is essential for applying methods like the simplex algorithm.

Step 2: Evaluate the given options based on the definition.

• A) Convert inequality constraints into equalities: This is the primary function of slack variables. For example, $x + y \le 10$ becomes $x + y + s_1 = 10$, where $s_1 \ge 0$ is a slack variable.
• B) Increase the number of constraints: Slack variables increase the number of variables, not the number of constraints. Each inequality constraint is simply rewritten as an equality.
• C) Make the objective function non-linear: Slack variables are added to constraints, not the objective function, and they maintain the linearity of the problem.
• D) Eliminate the need for non-negativity: Slack variables themselves must be non-negative ($s_i \ge 0$), and they do not remove the non-negativity requirements for the original decision variables.

Step 3: Conclude the correct option. Based on the evaluation, option A accurately describes the purpose of introducing slack variables.

The final answer is $\boxed{A}$

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