You are dealing with a linear programming problem, where the goal is to maximize the objective function $Z = 40x_1 + 30x_2 + 30x_3$, subject to the constraints:

1. $6x_1 + 2x_2 + 5x_3 \leq 200$
2. $4x_1 + 5x_2 + 3x_3 \leq 150$
3. $x_1, x_2, x_3 \geq 0$

To solve this, we typically use the Simplex Method or software like Excel, Python, or other optimization tools. Here's the step-by-step process conceptually:

Step 1: Write the Standard Form

Convert the inequalities into equalities by introducing slack variables. The constraints become:

$6x_1 + 2x_2 + 5x_3 + s_1 = 200$ $4x_1 + 5x_2 + 3x_3 + s_2 = 150$ Where $s_1$ and $s_2$ are slack variables representing unused resources and $s_1, s_2 \geq 0$.

The new objective function to maximize is still $Z = 40x_1 + 30x_2 + 30x_3$.

Step 2: Initial Simplex Tableau

Set up the initial tableau for the simplex method, including coefficients of variables $x_1, x_2, x_3, s_1, s_2$ from both constraints and the objective function.

Step 3: Iterations of Simplex Method

By applying pivot operations, we can iteratively find the values of $x_1, x_2, x_3$ that maximize $Z$.

Would you like a detailed solution using the Simplex method, or would you prefer I solve it using Python for a quicker answer?

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5 Related Questions:

1. What are the properties of linear programming problems?
2. How does the Simplex method work in solving optimization problems?
3. What are slack variables and how are they used in linear programming?
4. Can linear programming be solved graphically for more than two variables?
5. What is the difference between the Simplex method and interior-point methods?

Tip:

Linear programming models can be solved efficiently using software like Python (SciPy), Excel (Solver), or specialized optimization tools like LINGO or Gurobi.