Solve the linear programming problem by the simplex method.

Maximize

*P* = *x* + 4*y* − 2*z*

subject to

3*x*

+

*y*

−

*z*

≤

100

2*x*

+

*y*

−

*z*

≤

50

−*x*

+

*y*

+

*z*

≤

100

*x* ≥ 0, *y* ≥ 0, *z* ≥ 0

The maximum is *P* =   at

(*x*, *y*, *z*) =

.

To solve this linear programming problem using the simplex method, we’ll first convert the constraints into equations by introducing slack variables and then proceed with the simplex algorithm steps.

### Problem Statement
Maximize:  
\[
P = x + 4y - 2z
\]

Subject to:
1. \(3x + y - z + s_1 = 100\)
2. \(2x + y - z + s_2 = 50\)
3. \(-x + y + z + s_3 = 100\)
4. \(x, y, z, s_1, s_2, s_3 \geq 0\)

where \(s_1\), \(s_2\), and \(s_3\) are slack variables added to convert inequalities into equalities.

### Steps in the Simplex Method

1. **Set up the initial tableau**: The objective function and constraints are written in tableau form.
2. **Identify pivot column and row**: Based on the most negative entry in the objective row (for maximization).
3. **Perform pivot operations**: Adjust the tableau to maintain feasibility and improve the objective function.
4. **Repeat until optimality**: Continue until there are no negative indicators in the objective row, indicating an optimal solution.

I’ll work through these steps and provide the solution.The solution to the linear programming problem is as follows:

- The maximum value of \( P \) is **250**.
- This maximum occurs at:
  \[
  (x, y, z) = (0, 75, 25)
  \]

Would you like further explanation on the simplex steps?

Here are related questions you might consider:
1. What are the general steps of the simplex method?
2. How can we verify if the solution is unique or if alternative optimal solutions exist?
3. How does introducing slack variables convert inequalities to equalities?
4. Why is it necessary to transform a maximization problem for certain solvers?
5. What other methods could solve this problem if not using the simplex method?

**Tip:** When formulating a linear programming problem, always ensure all constraints and variables are clearly defined and consistent with each other.

Solve the linear programming problem by the simplex method.

Maximize

P = x + 4y − 2z

subject to

3x + y − z ≤ 100

2x + y − z ≤ 50

−x + y + z ≤ 100

x ≥ 0, y ≥ 0, z ≥ 0

The maximum is P = at (x, y, z) = .

Step-by-step solution to a linear programming problem using the simplex method. Maximize P = x + 4y - 2z subject to given constraints. Includes setting up slack variables, forming the initial tableau, and finding the optimal solution for P, with (x, y, z) = (0, 75, 25). Ideal for introductory courses in linear programming.

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