Methods of solving LPP: a graphical method

You have three linear programming problems:

A.

$$Maximize \ z = 4x_1 + 3x_2$$

Subject to the following constraints:

$$i) \ x_1 + x_2 \le 60 \\ ii)\ 2x_1 + x_2 \le 80 \\iii)\ 3x_1 + x_2 \le 120 \\ x_1, x_2 \ge 0$$

B.

$$Maximize \ z = x_1 + x_2$$Subject to the following constraints:

$$iv) \ x_1 + 3x_2 \le 50 \\ v)\ 2x_1 + 3x_2 \le 90 \\vi)\ x_1 + 2x_2 \le 45 \\ x_1, x_2 \ge 0$$

C.

$$Minimize \ z = x_1 + x_2$$Subject to the following constraints:

$$vii) \ 1.125x_1 + x_2 \ge 22.5 \\ viii)\ 0.5x_1 + 1.5x_2 \ge 30 \\ix)\ 3x_1 + 4x_2 \ge 90 \\ x_1, x_2 \ge 0$$Determine the redundant constraints (in ascending roman numeral) from the graphical plots.

Output format:

i ii iii iv