Use the simplex method to solve the following maximum problem: Maximize: \( \quad P=4 x_{1}+9 x_{2} \) Subject to the constraints: \( \left\{\begin{array}{l}2 x_{1}+5 x_{2} \leq 9 \\ x_{1}+2 x_{2} \leq 4 \\ x_{1} \geq 0 \\ x_{2} \geq 0\end{array}\right. \) and using your final tableau answer the questions below by entering the correct answer in each blank box. \( x_{1}=\square \) \( x_{2}=\square \)

To solve the given linear programming problem using the simplex method, we first need to set it up properly. We can rewrite the constraints in standard form and introduce slack variables to convert the inequalities into equalities. Maximize: \( P = 4x_1 + 9x_2 \) Subject to: 1. \( 2x_1 + 5x_2 + s_1 = 9 \) 2. \( x_1 + 2x_2 + s_2 = 4 \) 3. \( x_1, x_2, s_1, s_2 \geq 0 \) Where \( s_1 \) and \( s_2 \) are slack variables. Now we want to maximize the objective function. The initial tableau looks like this: \[ \begin{array}{c|cccccc|c} \text{Basis} & s_1 & s_2 & x_1 & x_2 & P \\ \hline s_1 & 2 & 5 & 1 & 0 & 0 & | 9 \\ s_2 & 1 & 2 & 1 & 0 & 0 & | 4 \\ \hline -P & -4 & -9 & 0 & 0 & 1 & | 0 \\ \end{array} \] Now, we follow the simplex method steps until we reach the optimal solution. After performing the required calculations and pivoting, we find the optimal tableau looks as follows: \[ \begin{array}{c|cccccc|c} \text{Basis} & s_1 & s_2 & x_1 & x_2 & P \\ \hline x_2 & 0 & 1 & 2/5 & 1/5 & 0 & | 9/5 \\ x_1 & 1 & 0 & 1/5 & 0 & 0 & | 4/5 \\ \hline -P & 0 & 0 & 0 & 0 & 1 & | 56/5 \\ \end{array} \] From this tableau: - The value of \( x_1 = 4/5 = 0.8 \) - The value of \( x_2 = 9/5 = 1.8 \) So, we enter the following into the blanks: \( x_{1}=0.8 \) \( x_{2}=1.8 \)