To solve this optimization problem, we first need to define the decision variables and the constraints based on the given information.

Let's denote:
- \(x\) as the number of soccer balls produced.
- \(y\) as the number of basket balls produced.

The objective is to maximize profit. The profit per soccer ball is $4, and the profit per basketball is $5. Thus, the objective function can be written as:
\[ \text{Maximize:} \quad 4x + 5y \]

Now, let's define the constraints:

1. **Manufacturing Machine Time Constraint**: Soccer balls take 5 minutes on the machine, and basket balls take 7 minutes. The total time available for the manufacturing machine is 700 minutes.
\[ 5x + 7y \leq 700 \]

2. **Air Filling Time Constraint**: Soccer balls take 3 minutes to fill with air, and basket balls take 4 minutes. The total time available for filling with air is 500 minutes.
\[ 3x + 4y \leq 500 \]

3. **Non-Negativity Constraints**: The number of soccer balls and basketballs cannot be negative.
\[ x \geq 0, y \geq 0 \]

Given these constraints and the objective function, we can now write the Gurobi code to solve this linear programming problem.

```python
from gurobipy import *

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

# Decision variables
x = m.addVar(lb=0, name="soccer_balls")
y = m.addVar(lb=0, name="basket_balls")

# Objective function: Maximize profit
m.setObjective(4*x + 5*y, GRB.MAXIMIZE)

# Constraints
m.addConstr(5*x + 7*y <= 700, name="manufacturing_time")
m.addConstr(3*x + 4*y <= 500, name="air_filling_time")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Soccer balls to produce: {x.x}")
    print(f"Basketballs to produce: {y.x}")
    print(f"Maximum profit: ${4*x.x + 5*y.x:.2f}")
else:
    print("No optimal solution found. The model is infeasible or unbounded.")
```
```python
```