To formulate a linear programming (LP) model for the seafood factory's packaging problem, we need to define the decision variables, objective function, and constraints based on the given information.

Let:
- \(x_1\) be the number of packages of fish meat.
- \(x_2\) be the number of packages of shrimp meat.

The objective is to maximize profit. Given that a package of fish meat generates $7 and a package of shrimp generates $3, the objective function can be written as:
\[ \text{Maximize} \quad 7x_1 + 3x_2 \]

Constraints are based on the availability of machines:
- Each package of fish meat requires 3 minutes in the weight checking machine.
- Each package of shrimp meat requires 1.5 minutes in the weight checking machine.
- The weight checking machine is available for at most 1200 minutes per week.
  This gives us our first constraint: \(3x_1 + 1.5x_2 \leq 1200\).

- Each package of fish meat requires 15 minutes in the packaging inspection machine.
- Each package of shrimp meat requires 7 minutes in the packaging inspection machine.
- The packaging inspection machine is available for at most 1200 minutes per week.
  This gives us our second constraint: \(15x_1 + 7x_2 \leq 1200\).

Additionally, we know that \(x_1\) and \(x_2\) must be non-negative since they represent the number of packages.

Now, let's translate this into Gurobi code in Python:

```python
from gurobipy import *

# Create a model
m = Model("Seafood_Factory")

# Decision variables
x1 = m.addVar(lb=0, name="fish_packages")
x2 = m.addVar(lb=0, name="shrimp_packages")

# Objective function
m.setObjective(7*x1 + 3*x2, GRB.MAXIMIZE)

# Constraints
m.addConstr(3*x1 + 1.5*x2 <= 1200, "weight_checking_machine")
m.addConstr(15*x1 + 7*x2 <= 1200, "packaging_inspection_machine")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Fish packages: {x1.x}")
    print(f"Shrimp packages: {x2.x}")
    print(f"Maximum profit: ${7*x1.x + 3*x2.x:.2f}")
else:
    print("No optimal solution found")
```