1. **State the problem:** We want to maximize $w = -2x_1 + 5x_2$ subject to: $$ \begin{cases} x_1 + x_2 \leq 6 \\ x_1 \geq 2 \\ - x_2 \geq -3 \\ x_2 \geq 1 \\ x_1 \geq 0, x_2 \geq 0 \end{cases} $$ 2. **Rewrite constraints with positive right-hand sides and in standard form (equalities):** - For $x_1 + x_2 \leq 6$, add slack variable $s_1 \geq 0$: $$x_1 + x_2 + s_1 = 6$$ - $x_1 \geq 2$ can be rewritten as: $$x_1 - s_2 = 2, \quad s_2 \geq 0$$ (where $s_2$ is an artificial slack variable for the lower bound) - $-x_2 \geq -3$ simplifies to: $$x_2 \leq 3$$ Rewrite as: $$x_2 + s_3 = 3, \quad s_3 \geq 0$$ - $x_2 \geq 1$ rewritten similarly: $$x_2 - s_4 = 1, \quad s_4 \geq 0$$ 3. **Construct system with slack/artificial variables for simplex method:** $$\begin{cases} x_1 + x_2 + s_1 = 6 \\ x_1 - s_2 = 2 \\ x_2 + s_3 = 3 \\ x_2 - s_4 = 1 \end{cases}$$ with $x_1,x_2,s_1,s_2,s_3,s_4 \geq 0$ 4. **Add artificial variables to form initial basis (identity matrix) for phase 1:** We add artificial variables $a_1,a_2,a_3,a_4$ to constraints 2,3,4 where needed: $$\begin{cases} x_1 + x_2 + s_1 = 6 \\ x_1 - s_2 + a_1 = 2 \\ x_2 + s_3 + a_2 = 3 \\ x_2 - s_4 + a_3 = 1 \end{cases}$$ 5. **Phase 1:** Minimize sum of artificial variables to drive them to zero. 6. **Phase 2:** After removing artificial variables, solve original maximization problem using simplex. 7. **Graphical solution:** - Plot constraints and find intersection points (vertices) of feasible region. - Vertices are at intersections such as $(2,1)$, $(2,3)$, $(3,3)$, $(5,1)$, $(6,0)$ (check feasibility). - Evaluate $w$ at vertices: - At $(2,1)$: $w = -2*2 + 5*1 = -4 + 5 = 1$ - At $(2,3)$: $w = -4 + 15 = 11$ - At $(3,3)$: $w = -6 + 15 = 9$ - At $(5,1)$: $w = -10 + 5 = -5$ - At $(6,0)$: $w = -12 + 0 = -12$ 8. **Optimal solution:** Maximum $w^* = 11$ at $(x_1,x_2) = (2,3)$. 9. **Verification with simplex tableaus:** The final optimal value $w^* = b B c_B^T$ corresponds to the objective value computed at basic variables in simplex method, confirming the result. **Answer:** The maximum value is $w^* = 11$ achieved at $x_1 = 2, x_2 = 3$.