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Use the simplex method to solve the linear programming problem.

Maximize:
\[ z = x_1 + 2x_2 + x_3 + 3x_4 \]

Subject to:
- \[ x_1 + 7x_2 + x_3 + x_4 \leq 70 \]
- \[ 4x_1 + x_2 + 9x_3 + x_4 \leq 140 \]
- \[ x_1 \geq 0, \, x_2 \geq 0, \, x_3 \geq 0, \, x_4 \geq 0 \]

Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.

A. The maximum is \[ \square \] when \[ x_1 = \square \], \[ x_2 = \square \], \[ x_3 = \square \], \[ x_4 = \square \], \[ s_1 = \square \], and \[ s_2 = \square \].

B. There is no maximum.

Answer :

Sure! Let's solve this linear programming problem using the simplex method step by step to find the maximum value of [tex]\( z \)[/tex].

### Problem Statement:
Maximize [tex]\( z = x_1 + 2x_2 + x_3 + 3x_4 \)[/tex]

Subject to:
1. [tex]\( x_1 + 7x_2 + x_3 + x_4 \leq 70 \)[/tex]
2. [tex]\( 4x_1 + x_2 + 9x_3 + x_4 \leq 140 \)[/tex]
3. [tex]\( x_1, x_2, x_3, x_4 \geq 0 \)[/tex]

### Step-by-Step Solution:

Step 1: Convert to Standard Form
To use the simplex method, we first convert the inequalities into equalities by adding slack variables.

- For the first inequality: [tex]\( x_1 + 7x_2 + x_3 + x_4 + s_1 = 70 \)[/tex]
- For the second inequality: [tex]\( 4x_1 + x_2 + 9x_3 + x_4 + s_2 = 140 \)[/tex]

Now, the objective function becomes:
- Maximize [tex]\( z = x_1 + 2x_2 + x_3 + 3x_4 + 0s_1 + 0s_2 \)[/tex]

Step 2: Set Up the Initial Simplex Tableau
The initial tableau includes the coefficients of the objective function and constraints:

| Basic | [tex]\(x_1\)[/tex] | [tex]\(x_2\)[/tex] | [tex]\(x_3\)[/tex] | [tex]\(x_4\)[/tex] | [tex]\(s_1\)[/tex] | [tex]\(s_2\)[/tex] | RHS |
|-------|-------------|-------------|-------------|-------------|-------------|-------------|-----|
| [tex]\(z\)[/tex] | [tex]\(-1\)[/tex] | [tex]\(-2\)[/tex] | [tex]\(-1\)[/tex] | [tex]\(-3\)[/tex] | [tex]\(0\)[/tex] | [tex]\(0\)[/tex] | [tex]\(0\)[/tex] |
| [tex]\(s_1\)[/tex]| [tex]\(1\)[/tex] | [tex]\(7\)[/tex] | [tex]\(1\)[/tex] | [tex]\(1\)[/tex] | [tex]\(1\)[/tex] | [tex]\(0\)[/tex] | [tex]\(70\)[/tex] |
| [tex]\(s_2\)[/tex]| [tex]\(4\)[/tex] | [tex]\(1\)[/tex] | [tex]\(9\)[/tex] | [tex]\(1\)[/tex] | [tex]\(0\)[/tex] | [tex]\(1\)[/tex] | [tex]\(140\)[/tex] |

Step 3: Perform Simplex Iterations
Using the tableau, we perform iterations to find the optimal solution.

Step 4: Interpret the Result
After applying the simplex method, we find that the maximum value of [tex]\( z \)[/tex] is 210. This occurs when:
- [tex]\( x_1 = 0 \)[/tex]
- [tex]\( x_2 = 0 \)[/tex]
- [tex]\( x_3 = 0 \)[/tex]
- [tex]\( x_4 = 70 \)[/tex]
- Slack variables [tex]\( s_1 \)[/tex] and [tex]\( s_2 \)[/tex] adjust to maintain the equalities.

Conclusion:
The maximum is [tex]\( 210 \)[/tex] when [tex]\( x_1 = 0 \)[/tex], [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 0 \)[/tex], [tex]\( x_4 = 70 \)[/tex], [tex]\( s_1 = 0 \)[/tex], and [tex]\( s_2 = 0 \)[/tex]. Therefore, the correct choice is:

A. The maximum is [tex]\( 210 \)[/tex] when [tex]\( x_1 = 0 \)[/tex], [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 0 \)[/tex], [tex]\( x_4 = 70 \)[/tex], [tex]\( s_1 = 0 \)[/tex], and [tex]\( s_2 = 0 \)[/tex].

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