Consider the feasible region in R3 defined by the inequalities

Question

Consider the feasible region in R3 defined by the inequalities

x1 + 2x2 − x3 ≥ 3,

x2 + 3x3 ≤ 2, and

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.

(a) Introduce non-negative slack variables x4 and x5 and write down the corresponding linear system.

(b) Find the basic solution obtained by making x1 and x2 basic variables. Is this basic solution feasible?

(c) Find the basic solution obtained by making x1 and x3 basic variables. Is this basic solution feasible?

Answer 1

(a) To introduce non-negative slack variables to the system of inequalities, we add the slack variables to the inequalities and make sure that their coefficients are non-negative. This gives us the following system of equations:

x1 + 2x2 − x3 + x4 = 3 x2 + 3x3 + x5 = 2 x1 ≥ 0 x2 ≥ 0 x3 ≥ 0 x4 ≥ 0 x5 ≥ 0

(b) To find the basic solution obtained by making x1 and x2 basic variables, we set x4 and x5 to 0 and solve the system of equations. This gives us:

x1 = 3 x2 = 0 x3 = 0 x4 = 0 x5 = 2

This solution is feasible because all of the variables are non-negative.

(c) To find the basic solution obtained by making x1 and x3 basic variables, we set x4 and x5 to 0 and solve the system of equations. This gives us:

x1 = 3 x2 = −1 x3 = 1 x4 = 0 x5 = 3

This solution is not feasible because x2 is negative.