7. A linear programming problem in \(x , y\) and \(z\) is described as follows. Maximise \(P = 3 x + 2 y + 2 z\)
subject to $$\begin{aligned} & 2 x + 2 y + z \leqslant 25
& x + 4 y \leqslant 15
& x \geqslant 3 \end{aligned}$$

1. Explain why the Simplex algorithm cannot be used to solve this linear programming problem. The big-M method is to be used to solve this linear programming problem.
2. Define, in this context, what M represents. You must use correct mathematical language in your answer. The initial tableau for a big-M solution to the problem is shown below.

   b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(s _ { 2 }\)	\(s _ { 3 }\)	\(t _ { 1 }\)	Value
   \(s _ { 1 }\)	2	2	1	1	0	0	0	25
   \(s _ { 2 }\)	1	4	0	0	1	0	0	15
   \(t _ { 1 }\)	1	0	0	0	0	-1	1	3
   \(P\)	\(- ( 3 + M )\)	-2	-2	0	0	M	0	\(- 3 M\)

3. Explain clearly how the equation represented in the b.v. \(t _ { 1 }\) row was derived.
4. Show how the equation represented in the b.v. \(P\) row was derived. The tableau obtained from the first iteration of the big-M method is shown below.

   b.v.	\(x\)	\(y\)	\(z\)	\(s _ { 1 }\)	\(s _ { 2 }\)	\(s _ { 3 }\)	\(t _ { 1 }\)	Value
   \(s _ { 1 }\)	0	2	1	1	0	2	-2	19
   \(s _ { 2 }\)	0	4	0	0	1	1	-1	12
   \(x\)	1	0	0	0	0	-1	1	3
   P	0	-2	-2	0	0	-3	\(3 +\) M	9

5. Solve the linear programming problem, starting from this second tableau. You must
   • give a detailed explanation of your method by clearly stating the row operations you use and
   • state the solution by deducing the final values of \(P , x , y\) and \(z\).