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To help you with this problem, let's walk through how to convert the given linear program (LP) into its standard equality-constrained form. Here's the plan:

### Step 1: Identify the elements of the given LP
You are given the LP:

**Objective function:**
\[
\text{maximize } 3z_1 - z_2
\]
This needs to be converted into a minimization problem, as standard LPs require.

**Constraints:**
\[
\begin{aligned}
-z_1 + 6z_2 - z_3 + z_4 &\geq -3 \\
7z_2 + z_4 &= 5 \\
z_3 + z_4 &\leq 2 \\
-1 \leq z_2 \leq 5, \quad -1 \leq z_3 \leq 5, \quad -2 \leq z_4 \leq 2
\end{aligned}
\]
These constraints will need to be converted into equality constraints with non-negative variables.

### Step 2: Convert to a minimization problem
To convert the objective function from maximization to minimization, multiply the entire objective by -1:

\[
\text{minimize } -(3z_1 - z_2) = -3z_1 + z_2
\]

### Step 3: Convert inequalities to equalities
To convert inequalities into equalities, we introduce slack and surplus variables where necessary. Let's go through each constraint:

1. **For the first constraint:** 
   \[
   -z_1 + 6z_2 - z_3 + z_4 \geq -3
   \]
   Convert this into an equality by introducing a surplus variable \( s_1 \geq 0 \):
   \[
   -z_1 + 6z_2 - z_3 + z_4 - s_1 = -3
   \]

2. **For the second constraint:**
   \[
   7z_2 + z_4 = 5
   \]
   This is already an equality, so no changes are needed here.

3. **For the third constraint:**
   \[
   z_3 + z_4 \leq 2
   \]
   Convert this into an equality by introducing a slack variable \( s_2 \geq 0 \):
   \[
   z_3 + z_4 + s_2 = 2
   \]

4. **Bounded constraints for \( z_2, z_3, z_4 \):**
   These constraints can be rewritten as inequalities and handled by introducing auxiliary variables to ensure non-negativity:
   - For \( -1 \leq z_2 \leq 5 \), introduce \( z_2' = z_2 + 1 \), so that \( 0 \leq z_2' \leq 6 \), and similarly for \( z_3 \) and \( z_4 \).

### Step 4: Assemble the standard form
Now we assemble everything into the matrix form \( c^T x \), \( A x = b \), and \( x \geq 0 \).

Would you like me to proceed with further details on the matrices \( A \), \( b \), and \( x \), or clarify any particular steps?

Transform the given linear program (LP) into the equality-constrained standard form of LP. Identify the elements A, b, c, and x, and explain how the decision variables of the transformed LP relate to those of the original LP.

Learn how to transform a given linear program with inequalities into the standard form with equality constraints. This problem involves converting the objective function from maximization to minimization and introducing slack and surplus variables to handle inequality constraints.

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