To solve this problem, we first need to convert the natural language description into a symbolic representation of the optimization problem. Let's denote:

- $x_1$ as the number of 1st generation motherboards produced.
- $x_2$ as the number of 2nd generation motherboards produced.

The objective function is to maximize profit, which can be represented algebraically as:
\[100x_1 + 125x_2\]

Given the constraints:

1. Silicon availability: $4x_1 + 6x_2 \leq 450$
2. Assembly time availability: $20x_1 + 30x_2 \leq 900$
3. Soldering time availability: $30x_1 + 40x_2 \leq 500$
4. Minimum production of 1st generation motherboards: $x_1 \geq 6$
5. Minimum production of 2nd generation motherboards: $x_2 \geq 7$

The symbolic representation of the problem in JSON format is:
```json
{
    'sym_variables': [('x1', 'number of 1st generation motherboards'), ('x2', 'number of 2nd generation motherboards')],
    'objective_function': '100*x1 + 125*x2',
    'constraints': [
        '4*x1 + 6*x2 <= 450',
        '20*x1 + 30*x2 <= 900',
        '30*x1 + 40*x2 <= 500',
        'x1 >= 6',
        'x2 >= 7'
    ]
}
```

Now, let's implement this optimization problem using Gurobi in Python:

```python
from gurobipy import *

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

# Add variables
x1 = m.addVar(lb=6, vtype=GRB.INTEGER, name="x1")
x2 = m.addVar(lb=7, vtype=GRB.INTEGER, name="x2")

# Set the objective function
m.setObjective(100*x1 + 125*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(4*x1 + 6*x2 <= 450, "silicon")
m.addConstr(20*x1 + 30*x2 <= 900, "assembly")
m.addConstr(30*x1 + 40*x2 <= 500, "soldering")

# Optimize the model
m.optimize()

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