To solve this optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of soccer balls as $x_1$ and the number of basketballs as $x_2$. The objective is to maximize profit, which can be represented by the objective function: $4x_1 + 5x_2$. 

The constraints are:
- Manufacturing machine time constraint: $5x_1 + 7x_2 \leq 700$
- Air filling time constraint: $3x_1 + 4x_2 \leq 500$
- Non-negativity constraints: $x_1, x_2 \geq 0$

This problem can be symbolically represented as:
```json
{
    'sym_variables': [('x1', 'number of soccer balls'), ('x2', 'number of basketballs')],
    'objective_function': '4*x1 + 5*x2',
    'constraints': ['5*x1 + 7*x2 <= 700', '3*x1 + 4*x2 <= 500', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem, we will use Gurobi in Python. Here is the code to find the optimal solution:
```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="soccer_balls", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="basketballs", lb=0)

# Set the objective function
m.setObjective(4*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x1 + 7*x2 <= 700, "machine_time")
m.addConstr(3*x1 + 4*x2 <= 500, "air_filling")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```