The simplex method is the most basic and logical method of solving an LP problem. In this method, all the relationships are converted into equations as explained in the following sections. LP is concerned with solutions to simultaneous linear equations. These equations are developed on the basis of restrictions on the variables. These restrictions are always expressed as inequalities. The first step is to convert them into equations. This is done by incorporating a new variable known as the slack variable. For example, if the inequality is of the form a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ ? b, this may be converted into a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ + S ₁ = b; the slack variable will take the value that will satisfy the equation. If the inequality is of the form a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ ? b, this will be converted into an equation by subtracting the slack variable, as given by a ₁ x ₁ + a ₂ x ₂ + a ₃ x ₃ - S ₁ = b.

This step is carried out for all the constraints, so that only linear equations will appear in the problem. For example, consider a coal-fired boiler problem, as in the following case study.