problem 2,3 and 4

With Problem 2: (20 marks) Show without using the Simplex Method that I optimal solution to the following linear programming problem: (5/26,5/2,27/26) is an max 9x1 + 1432 + 7:33 s.t. 2.1 + x2 + 3.03 c => 0. Prove that if x' satisfies Ar" = c and x > 0, then x' is an optimal solution. Problem 4: (20 marks) Let A RMXn and c E R". Prove that Ax 0. (Hint: You can construct a pair of LP problems to assist your proof.) Page 1 / 11 + Problem 2: (20 marks) Show without using the Simplex Method that = optimal solution to the following linear programming problem: (5/26,5/2.27/26) is an max 9.3 + 14:12 + 7,13 s.t. 2.1 +32 +33 5 6 5.01 +4r2 + 13 S 12 212 55 31. 12. rz unrestricted. Problem 3: (20 marks) Let A be a symmetric square matrix. Consider the linear programming problem min c Arzo 10. Prove that if satisfies Ar = c and r 20, then or is an optimal solution. Problem 4: (20 marks) Let AE R and c ER". Prove that Ar so implies or so if and only if A'y=c for some y > 0. (Hint: You can construct a pair of LP problems to assist your proof)