To solve this linear programming problem using the simplex method, we will first convert the inequalities into equations by introducing slack variables. Here's the problem breakdown:

Objective Function: $z = 4x_1 + 2x_2$

Subject to constraints:

1. $3x_1 + x_2 \leq 22$
2. $3x_1 + 4x_2 \leq 34$
3. $x_1 \geq 0$
4. $x_2 \geq 0$

Let's introduce slack variables $s_1$ and $s_2$ for the first and second constraints, respectively, to convert them into equalities:

1. $3x_1 + x_2 + s_1 = 22$
2. $3x_1 + 4x_2 + s_2 = 34$

Here, $s_1$ and $s_2$ are the slack variables that represent the unused resources in each constraint.

Initial Simplex Tableau:

We set up the initial simplex tableau with the objective function and the constraints:

& x_1 & x_2 & s_1 & s_2 & z & \text{Solution} \\ \hline s_1 & 3 & 1 & 1 & 0 & 0 & 22 \\ s_2 & 3 & 4 & 0 & 1 & 0 & 34 \\ z & -4 & -2 & 0 & 0 & 1 & 0 \\ \end{array}$$ Next, we'll proceed with the simplex method iterations to find the optimal solution. The pivot column is chosen based on the most negative coefficient in the objective row, and the pivot row is determined by the smallest positive ratio of the solution column to the pivot column. Let's calculate the iterations to find the optimal solution. **First Iteration:** 1. **Pivot Column:** The most negative coefficient in the objective function row is $$-4$$ under $$x_1$$, so the pivot column is $$x_1$$. 2. **Pivot Row:** Calculate the ratios: - $$\frac{22}{3} = 7.33$$ - $$\frac{34}{3} \approx 11.33$$ The smallest ratio is $$7.33$$, so the pivot row is $$s_1$$. **Pivot Element:** The pivot element is $$3$$ in row $$s_1$$ and column $$x_1$$. **Updated Tableau:** After pivoting, the updated tableau is: $$\begin{array}{c|ccccc|c} & x_1 & x_2 & s_1 & s_2 & z & \text{Solution} \\ \hline x_1 & 1 & \frac{1}{3} & \frac{1}{3} & 0 & 0 & \frac{22}{3} \\ s_2 & 0 & \frac{11}{3} & -1 & 1 & 0 & \frac{16}{3} \\ z & 0 & \frac{2}{3} & \frac{4}{3} & 0 & 1 & \frac{88}{3} \\ \end{array}$$ **Second Iteration:** 1. **Pivot Column:** The most negative coefficient in the objective function row is now 0, so no further iterations are needed. **Optimal Solution:** The optimal solution occurs when all the coefficients in the objective row (excluding the solution column) are non-negative. **Solution:** - $$x_1 = \frac{22}{3} \approx 7.33$$ - $$x_2 = 0$$ - $$z = \frac{88}{3} \approx 29.33$$ Therefore, the maximum value of $$z$$ is approximately 29.33 when $$x_1 = 7.33$$ and $$x_2 = 0$$. Would you like more detailed steps or have any questions? Here are some related questions: 1. What is the role of slack variables in the simplex method? 2. How do you determine the pivot column and pivot row in the simplex method? 3. Can the simplex method handle non-linear objective functions? 4. What is the significance of a tableau in the simplex method? 5. How do you interpret the final tableau in the simplex method? 6. What are the conditions for an optimal solution in the simplex method? 7. Can the simplex method be used for minimization problems? 8. How do degeneracy and unboundedness affect the simplex method? **Tip:** When solving linear programming problems, always check if the solution is feasible by ensuring that all constraints are satisfied with the obtained values of variables.