## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Laura', 'hours worked by George', 'hours worked by Dale', 'hours worked by Jean', 'hours worked by Peggy']. Let's denote them as $x_1, x_2, x_3, x_4, x_5$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to minimize is $6.78x_1 + 6.02x_2 + 2.49x_3 + 5.75x_4 + 5.93x_5$.

## Step 3: List the constraints in symbolic notation
1. $4x_1 + 9x_5 \geq 35$
2. $4x_1 + 14x_3 \geq 62$
3. $14x_3 + 12x_4 \geq 37$
4. $4x_2 + 14x_3 \geq 38$
5. $4x_1 + 12x_4 \geq 55$
6. $14x_3 + 12x_4 + 9x_5 \geq 53$
7. $4x_2 + 14x_3 + 12x_4 \geq 53$
8. $4x_1 + 4x_2 + 9x_5 \geq 53$
9. $4x_1 + 4x_2 + 12x_4 \geq 53$
10. $4x_1 + 12x_4 + 9x_5 \geq 53$
11. $14x_3 + 12x_4 + 9x_5 \geq 61$
12. $4x_2 + 14x_3 + 12x_4 \geq 61$
13. $4x_1 + 4x_2 + 9x_5 \geq 61$
14. $4x_1 + 4x_2 + 12x_4 \geq 61$
15. $4x_1 + 12x_4 + 9x_5 \geq 61$
16. $14x_3 + 12x_4 + 9x_5 \geq 64$
17. $4x_2 + 14x_3 + 12x_4 \geq 64$
18. $4x_1 + 4x_2 + 9x_5 \geq 64$
19. $4x_1 + 4x_2 + 12x_4 \geq 64$
20. $4x_1 + 12x_4 + 9x_5 \geq 64$
21. $14x_3 + 12x_4 + 9x_5 \geq 70$
22. $4x_2 + 14x_3 + 12x_4 \geq 70$
23. $4x_1 + 4x_2 + 9x_5 \geq 70$
24. $4x_1 + 4x_2 + 12x_4 \geq 70$
25. $4x_1 + 12x_4 + 9x_5 \geq 70$
26. $4x_1 + 4x_2 + 14x_3 + 12x_4 + 9x_5 \geq 70$
27. $7x_1 - 7x_4 \geq 0$
28. $5x_1 - 2x_3 \geq 0$
29. $12x_4 + 9x_5 \leq 158$
30. $x_5$ is an integer.

## Step 4: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Laura'), 
        ('x2', 'hours worked by George'), 
        ('x3', 'hours worked by Dale'), 
        ('x4', 'hours worked by Jean'), 
        ('x5', 'hours worked by Peggy')
    ], 
    'objective_function': '6.78*x1 + 6.02*x2 + 2.49*x3 + 5.75*x4 + 5.93*x5', 
    'constraints': [
        '4*x1 + 9*x5 >= 35',
        '4*x1 + 14*x3 >= 62',
        '14*x3 + 12*x4 >= 37',
        '4*x2 + 14*x3 >= 38',
        '4*x1 + 12*x4 >= 55',
        '14*x3 + 12*x4 + 9*x5 >= 53',
        '4*x2 + 14*x3 + 12*x4 >= 53',
        '4*x1 + 4*x2 + 9*x5 >= 53',
        '4*x1 + 4*x2 + 12*x4 >= 53',
        '4*x1 + 12*x4 + 9*x5 >= 53',
        '14*x3 + 12*x4 + 9*x5 >= 61',
        '4*x2 + 14*x3 + 12*x4 >= 61',
        '4*x1 + 4*x2 + 9*x5 >= 61',
        '4*x1 + 4*x2 + 12*x4 >= 61',
        '4*x1 + 12*x4 + 9*x5 >= 61',
        '14*x3 + 12*x4 + 9*x5 >= 64',
        '4*x2 + 14*x3 + 12*x4 >= 64',
        '4*x1 + 4*x2 + 9*x5 >= 64',
        '4*x1 + 4*x2 + 12*x4 >= 64',
        '4*x1 + 12*x4 + 9*x5 >= 64',
        '14*x3 + 12*x4 + 9*x5 >= 70',
        '4*x2 + 14*x3 + 12*x4 >= 70',
        '4*x1 + 4*x2 + 9*x5 >= 70',
        '4*x1 + 4*x2 + 12*x4 >= 70',
        '4*x1 + 12*x4 + 9*x5 >= 70',
        '4*x1 + 4*x2 + 14*x3 + 12*x4 + 9*x5 >= 70',
        '7*x1 - 7*x4 >= 0',
        '5*x1 - 2*x3 >= 0',
        '12*x4 + 9*x5 <= 158'
    ]
}
```

## Step 5: Gurobi code for the optimization problem
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name='hours_worked_by_Laura', lb=0)
    x2 = model.addVar(name='hours_worked_by_George', lb=0)
    x3 = model.addVar(name='hours_worked_by_Dale', lb=0)
    x4 = model.addVar(name='hours_worked_by_Jean', lb=0)
    x5 = model.addVar(name='hours_worked_by_Peggy', lb=0, integrality=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(6.78 * x1 + 6.02 * x2 + 2.49 * x3 + 5.75 * x4 + 5.93 * x5, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(4 * x1 + 9 * x5 >= 35)
    model.addConstr(4 * x1 + 14 * x3 >= 62)
    model.addConstr(14 * x3 + 12 * x4 >= 37)
    model.addConstr(4 * x2 + 14 * x3 >= 38)
    model.addConstr(4 * x1 + 12 * x4 >= 55)
    model.addConstr(14 * x3 + 12 * x4 + 9 * x5 >= 53)
    model.addConstr(4 * x2 + 14 * x3 + 12 * x4 >= 53)
    model.addConstr(4 * x1 + 4 * x2 + 9 * x5 >= 53)
    model.addConstr(4 * x1 + 4 * x2 + 12 * x4 >= 53)
    model.addConstr(4 * x1 + 12 * x4 + 9 * x5 >= 53)
    model.addConstr(14 * x3 + 12 * x4 + 9 * x5 >= 61)
    model.addConstr(4 * x2 + 14 * x3 + 12 * x4 >= 61)
    model.addConstr(4 * x1 + 4 * x2 + 9 * x5 >= 61)
    model.addConstr(4 * x1 + 4 * x2 + 12 * x4 >= 61)
    model.addConstr(4 * x1 + 12 * x4 + 9 * x5 >= 61)
    model.addConstr(14 * x3 + 12 * x4 + 9 * x5 >= 64)
    model.addConstr(4 * x2 + 14 * x3 + 12 * x4 >= 64)
    model.addConstr(4 * x1 + 4 * x2 + 9 * x5 >= 64)
    model.addConstr(4 * x1 + 4 * x2 + 12 * x4 >= 64)
    model.addConstr(4 * x1 + 12 * x4 + 9 * x5 >= 64)
    model.addConstr(14 * x3 + 12 * x4 + 9 * x5 >= 70)
    model.addConstr(4 * x2 + 14 * x3 + 12 * x4 >= 70)
    model.addConstr(4 * x1 + 4 * x2 + 9 * x5 >= 70)
    model.addConstr(4 * x1 + 4 * x2 + 12 * x4 >= 70)
    model.addConstr(4 * x1 + 12 * x4 + 9 * x5 >= 70)
    model.addConstr(4 * x1 + 4 * x2 + 14 * x3 + 12 * x4 + 9 * x5 >= 70)
    model.addConstr(7 * x1 - 7 * x4 >= 0)
    model.addConstr(5 * x1 - 2 * x3 >= 0)
    model.addConstr(12 * x4 + 9 * x5 <= 158)

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Laura: ', x1.varValue)
        print('Hours worked by George: ', x2.varValue)
        print('Hours worked by Dale: ', x3.varValue)
        print('Hours worked by Jean: ', x4.varValue)
        print('Hours worked by Peggy: ', x5.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```