To solve the linear programming problem using the simplex method, we will:

Problem:

Maximize $P = -x_1 + 2x_2$

Subject to the constraints: $-x_1 + x_2 \leq 2$ $-x_1 + 3x_2 \leq 14$ $x_1 - 4x_2 \leq 6$ with $x_1 \geq 0$ and $x_2 \geq 0$.

Step 1: Convert inequalities to equalities by introducing slack variables.

We introduce slack variables $s_1, s_2, s_3$ to convert the inequalities into equalities:

1. $-x_1 + x_2 + s_1 = 2$
2. $-x_1 + 3x_2 + s_2 = 14$
3. $x_1 - 4x_2 + s_3 = 6$

The objective function becomes:
$P = -x_1 + 2x_2$ We aim to maximize $P$.

Step 2: Set up the initial simplex tableau.

The initial simplex tableau with the slack variables is:

& x_1 & x_2 & s_1 & s_2 & s_3 & P & \text{RHS} \\ \hline \text{Constraint 1} & -1 & 1 & 1 & 0 & 0 & 0 & 2 \\ \text{Constraint 2} & -1 & 3 & 0 & 1 & 0 & 0 & 14 \\ \text{Constraint 3} & 1 & -4 & 0 & 0 & 1 & 0 & 6 \\ \hline \text{Objective} & 1 & -2 & 0 & 0 & 0 & 1 & 0 \\ \end{array}$$ ### Step 3: Perform the simplex algorithm. We need to find the entering and exiting variables to pivot the tableau. The entering variable is the one with the most negative coefficient in the objective row, which is $$x_2$$ (since its coefficient is -2). Now, we'll determine the exiting variable by computing the ratios of the right-hand side (RHS) values to the corresponding values in the $$x_2$$ column: - For the first row: $$\frac{2}{1} = 2$$ - For the second row: $$\frac{14}{3} \approx 4.67$$ - For the third row: $$\frac{6}{-4}$$ is negative, so we ignore it. The minimum ratio is 2, so the exiting variable is $$s_1$$, and we pivot on the element in row 1, column $$x_2$$. ### Step 4: Pivot and update the tableau. We perform the pivot operation and update the tableau. After this pivot, the basic solution changes, and we repeat the process until there are no negative coefficients in the objective row. After completing the pivot steps, we will get the final tableau where the optimal solution is reached. ### Step 5: Interpret the final solution. After completing the simplex iterations, the optimal solution for the problem will be identified in terms of the values of $$x_1$$ and $$x_2$$ that maximize $$P$$. Let me calculate the optimal solution by performing the necessary simplex method steps. The optimal solution to the linear programming problem is: - $$x_1 = 4$$ - $$x_2 = 6$$ The maximum value of $$P$$ is: $$P = -x_1 + 2x_2 = -4 + 2(6) = 8$$ Thus, the maximum value of $$P$$ is $$P = 8$$ when $$x_1 = 4$$ and $$x_2 = 6$$. Would you like more details or clarifications on any steps? Here are some related questions to explore: 1. How does the simplex method work step by step? 2. What is the significance of slack variables in linear programming? 3. Can linear programming be applied to non-linear optimization problems? 4. How does the dual simplex method differ from the regular simplex method? 5. What are the limitations of the simplex method? **Tip:** In linear programming, always ensure that your constraints and objective function are correctly represented before applying the simplex method.