Set up the simplex matrix used to solve the linear programming problem. Assume all variables are nonnegative.

Maximize 
f = 5x + 9y
 subject to
8x	 + 	5y	 ≤ 	200
x	 + 	6y	 ≤ 	250.

To solve this linear programming problem using the simplex method, we first need to set up the **simplex tableau**. Here are the steps to convert the linear programming problem into a form suitable for the simplex method:

### Step 1: Convert inequalities into equalities
We introduce **slack variables** to turn the inequalities into equalities. The slack variables represent the unused resources and allow us to write the constraints as equations.

Given:
\[
8x + 5y \leq 200
\]
\[
x + 6y \leq 250
\]

We introduce slack variables \(s_1\) and \(s_2\) to convert these inequalities to equalities:

1. \(8x + 5y + s_1 = 200\)
2. \(x + 6y + s_2 = 250\)

### Step 2: Write the objective function
The objective function to maximize is:
\[
f = 5x + 9y
\]

We can rewrite this in terms of a function that equals zero:
\[
f - 5x - 9y = 0
\]

Thus, our objective row becomes:
\[
-Z + 5x + 9y = 0
\]
where \(Z\) represents the objective value (profit or cost in this case).

### Step 3: Set up the initial simplex tableau
Now we create a tableau that includes the constraints, slack variables, and the objective function.

| Basic Variable | \(x\) | \(y\) | \(s_1\) | \(s_2\) | RHS  |
|:--------------:|:-----:|:-----:|:-------:|:-------:|:----:|
| \(s_1\)        | 8     | 5     | 1       | 0       | 200  |
| \(s_2\)        | 1     | 6     | 0       | 1       | 250  |
| \(-Z\)         | -5    | -9    | 0       | 0       | 0    |

### Explanation of the Tableau:
- The **basic variables** column contains the variables that are currently in the solution (slack variables \(s_1\) and \(s_2\)).
- The coefficients of \(x\) and \(y\) represent the constraints in the system.
- The **RHS** (right-hand side) column represents the available resources (200 and 250).
- The bottom row is the objective function (in maximization form, with negative coefficients for the variables in the objective function).

This tableau can now be used to apply the simplex algorithm to find the optimal solution.

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Would you like me to continue with the steps of the simplex method? Let me know if you want details on performing the simplex iterations, or if you have any other questions.

Here are 5 related questions to explore further:

1. What is the role of slack variables in the simplex method?
2. How do we determine when to stop the simplex algorithm?
3. What is the significance of the pivot element in the simplex tableau?
4. How does the simplex method handle unbounded solutions?
5. Can the simplex method be applied to minimization problems as well?

**Tip**: Always ensure your objective function is in maximization form when applying the standard simplex method. If it's a minimization problem, you'll need to adjust accordingly!

Set up the simplex matrix used to solve the linear programming problem. Maximize f = 5x + 9y subject to 8x + 5y ≤ 200 and x + 6y ≤ 250. Assume all variables are nonnegative.

Learn how to set up the simplex matrix for a linear programming problem. We maximize f = 5x + 9y subject to constraints, introducing slack variables and creating a tableau. This guide covers the initial steps of the simplex method, suitable for advanced high school and undergraduate students.

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