1) A constraint of the type ≤ (≥) can be converted to an equation by adding a slack variable to (subtracting a surplus variable form) the left side of the constraint.
For illustration, in the constraint: XI + 2X2 ≤ 6 we add a slack SI ≥ 0 to the left side to obtain the equation: XI + 2X2 + SI = 6, SI ≥ 0.
If the constraint represents the limit on the usage of a resource, SI will represent the slack or unused amount of the resource.
After that consider the constraint: 3XI + 2X2 - 3X3 ≥ 5, we subtract a surplus variable S2 ≥ 0 from the left side to obtain the equation.3XI + 2X2 - 3X3 - S2 = 5, S2 ≥ 02) The right side of an equation can always be made non-negative by multiplying both sides by -1.
For illustration, 2XI + 3X2 - 7X3 = -5 is mathematically equivalent to -2XI - 3X2 + 7X3 = +5.3) The direction of an equation is reversed when both sides are multiplied by -1. For illustration, whereas 2 < 4, -2 > -4. Thus the inequality 2XI - X2 ≤-5 can be replaced by -2XI + X2 ≥ 5.