(Solved): Exercise 2 (simplex method without complications). Consider the following linear programming proble ...

Exercise 2 (simplex method without complications). Consider the following linear programming problem \\[ \\text { maximize } \\quad z=6 x_{1}+8 x_{2}+5 x_{3}+9 x_{4} \\] subject to \\[ \\begin{array}{l} 2 x_{1}+x_{2}+x_{3}+3 x_{4} \\leq 5 \\\\ x_{1}+3 x_{2}+x_{3}+2 x_{4} \\leq 3 \\\\ x_{1}, x_{2}, x_{3}, x_{4} \\geq 0 \\end{array} \\] (a) Solve the LP problem using the simplex algorithm. After introducing slack variables to convert to standard equational form, indicate your choice of starting basic set and then each pivot step in your solution. You do not have to be systematic about the choice of pivot rule in each pivot step. (b) Using your final simplex tableau in (a), explain why the solution is unique, i.e. there is exactly one choice of \\( x_{1}, x_{2}, x_{3}, x_{4} \\) at which the objective function is maximized.