Use the simplex method to solve the linear programming problem.
\[
\begin{array}{ll}
\text { Maximize } & z=8 x_{1}+3 x_{2}+x_{3} \\
\text { subject to: } & x_{1}+3 x_{2}+5 x_{3} \leq 101 \\
& x_{1}+3 x_{2}+7 x_{3} \leq 213 \\
\text { with } & x_{1} \geq 0, x_{2} \geq 0, x_{3} \geq 0
\end{array}
\]
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The maximum is $\square$ when $x_{1}=\square x_{2}=\square$ and $x_{3}=\square$. (Simplify your answers.)
B. There is no maximum.

Step 1 ：We are given a linear programming problem where we are asked to maximize the objective function \(z=8x_1+3x_2+x_3\) subject to the constraints \(x_1+3x_2+5x_3 \leq 101\), \(x_1+3x_2+7x_3 \leq 213\), and \(x_1 \geq 0, x_2 \geq 0, x_3 \geq 0\).

Step 2 ：To solve this problem, we can use the simplex method. This is an iterative algorithm that starts from a feasible solution and moves towards the optimal solution by improving the objective function at each step.

Step 3 ：First, we need to convert the inequalities into equalities by introducing slack variables. This allows us to set up the initial simplex tableau.

Step 4 ：Next, we perform the simplex algorithm to find the optimal solution.

Step 5 ：The optimal value of the objective function is 808 and the corresponding values of the variables are \(x_1 = 101\), \(x_2 = 0\), and \(x_3 = 0\).

Step 6 ：Final Answer: The maximum is \(\boxed{808}\) when \(x_{1}=\boxed{101}\), \(x_{2}=\boxed{0}\) and \(x_{3}=\boxed{0}\).