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

Let's define the variables:
- $x_1$ as the number of scoops of Alpha powder Jane buys.
- $x_2$ as the number of scoops of Beta powder Jane buys.

The objective function is to minimize the total cost, which can be represented as:
\[ \text{Minimize} \quad 1x_1 + 2x_2 \]

Given that a scoop of Alpha contains 5 grams of iron and 20 grams of biotin, and a scoop of Beta contains 10 grams of iron and 3 grams of biotin, we can formulate the constraints as follows:
- For iron: $5x_1 + 10x_2 \geq 50$ (since Jane needs at least 50 grams of iron).
- For biotin: $20x_1 + 3x_2 \geq 40$ (since Jane needs at least 40 grams of biotin).

Additionally, since the number of scoops cannot be negative:
- $x_1 \geq 0$
- $x_2 \geq 0$

So, the symbolic representation is:

```json
{
    'sym_variables': [('x1', 'number of scoops of Alpha powder'), ('x2', 'number of scoops of Beta powder')],
    'objective_function': 'Minimize 1*x1 + 2*x2',
    'constraints': ['5*x1 + 10*x2 >= 50', '20*x1 + 3*x2 >= 40', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this linear programming problem using Gurobi in Python, we use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Jane_Supplement_Problem")

# Define variables
x1 = m.addVar(name='Alpha_scoops', vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name='Beta_scoops', vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function to minimize cost
m.setObjective(1*x1 + 2*x2, GRB.MINIMIZE)

# Add constraints for iron and biotin requirements
m.addConstr(5*x1 + 10*x2 >= 50, name='Iron_requirement')
m.addConstr(20*x1 + 3*x2 >= 40, name='Biotin_requirement')

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Alpha scoops: {x1.x}")
    print(f"Beta scoops: {x2.x}")
    print(f"Total cost: ${m.objVal}")
else:
    print("No optimal solution found")
```