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 \(x_{3}=\boxed{0}\).