We need to use the non-standard simplex method whenever there is a negative constant to the right of one or more ≤ constraints.
Alternatively, set up the initial simplex tableau. If there are one or more negative constants above the horizontal line, then we need to use the non-standard simplex method.
Answer
If one (or more) constraints are an equation (rather than an inequality), we create two inequalities out of the equation.
For example, if we have the equation 2x + y = 7, then this equation is true when both 2x + y ≤ 7 and 2x + y ≥ 7. Since the second inequality is going "in the wrong direction," we multiply both sides by -1 to "flip" the inequality. This gives us: -2x - y ≤ -7.
In this way, we turn the equation 2x + y = 7 into two inequalities:
2x + y ≤ 7
-2x - y ≤ -7
(A consequence of this is that we always need to use the non-standard simplex method whenever solving a linear programming problem with an equality constraint.)