To solve this problem, we first need to define the decision variables and the objective function. Let's denote the number of Mexican burritos as \(M\) and the number of Korean burritos as \(K\).

The objective is to maximize profit, which can be calculated as $7\(M\) + $4.5\(K\), since each Mexican burrito brings in $7 of profit and each Korean burrito brings in $4.5 of profit.

Next, we need to define the constraints based on the availability of ingredients:

1. Cheese constraint: Each Mexican burrito requires 7 units of cheese, and each Korean burrito requires 5 units of cheese. Given that there are 70 units of cheese available, the constraint can be written as \(7M + 5K \leq 70\).

2. Beans constraint: Since only Mexican burritos require beans (8 units per burrito) and there are 60 units of beans available, this constraint is \(8M \leq 60\).

3. Onions constraint: Only Korean burritos require onions (5 units per burrito), with 45 units available, leading to the constraint \(5K \leq 45\).

4. Non-negativity constraints: Since we cannot produce a negative number of burritos, both \(M \geq 0\) and \(K \geq 0\).

With these definitions, we can now formulate the linear programming problem:

Maximize: \(7M + 4.5K\)

Subject to:
- \(7M + 5K \leq 70\)
- \(8M \leq 60\)
- \(5K \leq 45\)
- \(M, K \geq 0\)

Here's how we can implement this using Gurobi in Python:

```python
from gurobipy import *

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

# Define the decision variables
M = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Mexican_Burritos")
K = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="Korean_Burritos")

# Set the objective function
m.setObjective(7*M + 4.5*K, GRB.MAXIMIZE)

# Add constraints
m.addConstr(7*M + 5*K <= 70, "Cheese_Constraint")
m.addConstr(8*M <= 60, "Beans_Constraint")
m.addConstr(5*K <= 45, "Onions_Constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Mexican Burritos: {M.x}")
    print(f"Korean Burritos: {K.x}")
    print(f"Maximum Profit: ${7*M.x + 4.5*K.x:.2f}")
else:
    print("No optimal solution found.")
```