## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'potatoes' and 'bananas', which we can represent symbolically as $x_1$ and $x_2$ respectively. The objective function to minimize is $6x_1 + 7x_2$. The constraints are:
- $2x_1 + 12x_2 \geq 39$ (at least 39 grams of carbohydrates from potatoes and bananas)
- $2x_1 + 12x_2 \leq 61$ (not more than 61 grams of carbohydrates from potatoes and bananas)
- $-5x_1 + 7x_2 \geq 0$ (specific relationship between potatoes and bananas)
- $x_1$ is an integer (non-fractional amount of potatoes)
- $x_2$ is a real number (no restriction on being an integer for bananas)

## 2: Convert the problem into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ represents 'potatoes'
- $x_2$ represents 'bananas'

The objective function is: $6x_1 + 7x_2$

The constraints are:
- $2x_1 + 12x_2 \geq 39$
- $2x_1 + 12x_2 \leq 61$
- $-5x_1 + 7x_2 \geq 0$

## 3: Express the problem in the required JSON format
```json
{
    'sym_variables': [('x1', 'potatoes'), ('x2', 'bananas')], 
    'objective_function': '6*x1 + 7*x2', 
    'constraints': [
        '2*x1 + 12*x2 >= 39', 
        '2*x1 + 12*x2 <= 61', 
        '-5*x1 + 7*x2 >= 0'
    ]
}
```

## 4: Implement the problem using Gurobi
We will use the Gurobi Python API to model and solve this optimization problem.

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name="potatoes", vtype=gurobi.GRB.INTEGER)  # Integer variable for potatoes
x2 = model.addVar(name="bananas", vtype=gurobi.GRB.CONTINUOUS)  # Continuous variable for bananas

# Define the objective function
model.setObjective(6 * x1 + 7 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(2 * x1 + 12 * x2 >= 39, name="carbohydrates_min")
model.addConstr(2 * x1 + 12 * x2 <= 61, name="carbohydrates_max")
model.addConstr(-5 * x1 + 7 * x2 >= 0, name="potatoes_bananas_relationship")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Potatoes: {x1.varValue}")
    print(f"Bananas: {x2.varValue}")
    print(f"Objective: {model.objVal}")
else:
    print("No optimal solution found.")
```

The final answer is: 
```json 
{
    'sym_variables': [('x1', 'potatoes'), ('x2', 'bananas')], 
    'objective_function': '6*x1 + 7*x2', 
    'constraints': [
        '2*x1 + 12*x2 >= 39', 
        '2*x1 + 12*x2 <= 61', 
        '-5*x1 + 7*x2 >= 0'
    ]
}
```
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name="potatoes", vtype=gurobi.GRB.INTEGER)  # Integer variable for potatoes
x2 = model.addVar(name="bananas", vtype=gurobi.GRB.CONTINUOUS)  # Continuous variable for bananas

# Define the objective function
model.setObjective(6 * x1 + 7 * x2, gurobi.GRB.MINIMIZE)

# Add constraints
model.addConstr(2 * x1 + 12 * x2 >= 39, name="carbohydrates_min")
model.addConstr(2 * x1 + 12 * x2 <= 61, name="carbohydrates_max")
model.addConstr(-5 * x1 + 7 * x2 >= 0, name="potatoes_bananas_relationship")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Potatoes: {x1.varValue}")
    print(f"Bananas: {x2.varValue}")
    print(f"Objective: {model.objVal}")
else:
    print("No optimal solution found.")
```