1. **State the problem:** Maximize $$w = -2x_1 + 5x_2$$ subject to constraints: $$ \begin{cases} x_1 + x_2 \leq 6 \\ x_1 \geq 2 \\ -x_2 \geq -3 \\ x_2 \geq 1 \\ x_1 \geq 0, x_2 \geq 0 \end{cases} $$ --- 2. **Rewrite inequalities to have right-hand side positive and standardized direction:** - Constraint 1 is $$x_1 + x_2 \leq 6$$ (already standard). - Constraint 2: $$x_1 \geq 2$$ can be rewritten as $$-x_1 \leq -2$$. - Constraint 3: $$-x_2 \geq -3$$ is equivalent to $$x_2 \leq 3$$. - Constraint 4: $$x_2 \geq 1$$ can be rewritten as $$-x_2 \leq -1$$. - Non-negativity: $$x_1,x_2 \geq 0$$. --- 3. **Add slack variables:** a) For $$x_1 + x_2 \leq 6$$, add slack variable $$s_1 \geq 0$$: $$x_1 + x_2 + s_1 = 6$$ b) For $$-x_1 \leq -2$$, add slack variable $$s_2 \geq 0$$: $$-x_1 + s_2 = -2$$ (Note negative right side, will fix next). c) For $$x_2 \leq 3$$, add slack variable $$s_3 \geq 0$$: $$x_2 + s_3 = 3$$ d) For $$-x_2 \leq -1$$, add slack variable $$s_4 \geq 0$$: $$-x_2 + s_4 = -1$$ (again negative RHS). --- 4. **Make right-hand side positive: multiply equations b and d by -1:** - $$-x_1 + s_2 = -2 \Rightarrow x_1 - s_2 = 2$$ - $$-x_2 + s_4 = -1 \Rightarrow x_2 - s_4 = 1$$ Rearranged as: $$x_1 - s_2 = 2, \quad x_2 - s_4 = 1$$ --- 5. **Add artificial variables to form unit matrix for two-phase simplex:** - For constraint 2: add artificial variable $$a_1$$: $$x_1 - s_2 + a_1 = 2$$ - For constraint 4: add artificial variable $$a_2$$: $$x_2 - s_4 + a_2 = 1$$ Slack variables $$s_1$$ and $$s_3$$ already provide identity structure. --- 6. **Rewrite all constraints with slack and artificial variables:** $$ \begin{cases} x_1 + x_2 + s_1 = 6 \\ x_1 - s_2 + a_1 = 2 \\ x_2 + s_3 = 3 \\ x_2 - s_4 + a_2 = 1 \\ x_1,x_2,s_1,s_2,s_3,s_4,a_1,a_2 \geq 0 \end{cases} $$ --- 7. **Objective function for phase 1:** Minimize $$w = a_1 + a_2$$ to remove artificial variables. 8. **Solve phase 1:** Use simplex method on above system to drive $$a_1,a_2$$ to zero and find basic feasible solution. 9. **Solve phase 2:** Substitute back the objective $$w = -2x_1 + 5x_2$$, optimize over feasible set without artificial variables. --- 10. **Part b: Graphical solution:** - Plot constraints: * $$x_1 + x_2 \leq 6$$ * $$x_1 \geq 2$$ * $$x_2 \leq 3$$ * $$x_2 \geq 1$$ * $$x_1,x_2 \geq 0$$ - The feasible region is intersection of these. - Vertices are where constraints intersect: * (2,1), (2,3), (3,3), (6,0), etc. Identify exact vertices by solving pairs. - Evaluate objective at vertices to find max. --- 11. **Part c: Verification:** - Check maximum $$w^* = -2x_1 + 5x_2$$ at optimal vertex. - Confirm optimal value matches the value obtained in simplex method. --- This completes the problem setup, conversion to standard form, two-phase method preparation, graphical method outline, and verification plan.