## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by George', 'hours worked by Bill', 'hours worked by Hank', and 'hours worked by Jean', which can be represented symbolically as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Convert the objective function into a symbolic representation
The objective function to minimize is $4 \times (\text{hours worked by George}) + 9 \times (\text{hours worked by Bill}) + 5 \times (\text{hours worked by Hank}) + 3 \times (\text{hours worked by Jean})$. In symbolic terms, this is $4x_1 + 9x_2 + 5x_3 + 3x_4$.

## 3: List the constraints in symbolic notation
1. $5x_1 \leq 89$ and $5x_1 \geq 0$ is not needed as $x_0$ is given for George, implying $5x_1$ is his score, so $5x_1 = 5$.
2. $8x_2 \leq 89$ and $8x_2 \geq 0$ is not needed as $x_1$ is given for Bill, implying $8x_2$ is his score, so $8x_2 = 8$.
3. $8x_3 \leq 89$ and $8x_3 \geq 0$ is not needed as $x_2$ is given for Hank, implying $8x_3$ is his score, so $8x_3 = 8$.
4. $5x_4 \leq 89$ and $5x_4 \geq 0$ is not needed as $x_3$ is given for Jean, implying $5x_4$ is her score, so $5x_4 = 5$.
5. $8x_2 + 5x_4 \geq 18$.
6. $8x_3 + 5x_4 \geq 19$.
7. $5x_1 + 8x_3 \geq 15$.
8. $5x_1 + 8x_3 + 5x_4 \geq 17$.
9. $8x_2 + 8x_3 + 5x_4 \geq 17$.
10. $5x_1 + 8x_3 + 5x_4 \geq 19$.
11. $8x_2 + 8x_3 + 5x_4 \geq 19$.
12. $5x_1 + 8x_2 + 8x_3 + 5x_4 \geq 19$.
13. $8x_1 - 2x_2 \geq 0$.
14. $-3x_1 + x_3 \geq 0$.
15. $-x_2 + 7x_4 \geq 0$.
16. $5x_1 + 8x_2 \leq 77$.
17. $8x_2 + 8x_3 \leq 55$.
18. $5x_1 + 8x_3 \leq 64$.
19. $8x_2 + 8x_3 + 5x_4 \leq 53$.
20. $5x_1 + 8x_2 + 8x_3 \leq 67$.
21. $5x_1 + 8x_2 + 5x_4 \leq 44$.
22. $5x_1 + 8x_3 + 5x_4 \leq 35$.

## 4: Correcting and Simplifying Constraints Based on Given Information
Given that $5x_1 = 5$, $8x_2 = 8$, $8x_3 = 8$, and $5x_4 = 5$, we can directly substitute these into our constraints:
- $x_1 = 1$, $x_2 = 1$, $x_3 = 1$, and $x_4 = 1$.

## 5: Applying Given Values to Constraints
Applying these values:
- The constraints that are directly satisfied by the given scores are not needed as separate constraints.

## 6: Formulate the Optimization Problem in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi as gp

# Define the model
m = gp.Model("optimization_problem")

# Define the variables
x1 = m.addVar(name="hours_worked_by_George", lb=0)
x2 = m.addVar(name="hours_worked_by_Bill", lb=0)
x3 = m.addVar(name="hours_worked_by_Hank", lb=0)
x4 = m.addVar(name="hours_worked_by_Jean", lb=0)

# Objective function
m.setObjective(4*x1 + 9*x2 + 5*x3 + 3*x4, gp.GRB.MINIMIZE)

# Constraints
m.addConstr(5*x1 == 5)
m.addConstr(8*x2 == 8)
m.addConstr(8*x3 == 8)
m.addConstr(5*x4 == 5)

m.addConstr(8*x2 + 5*x4 >= 18)
m.addConstr(8*x3 + 5*x4 >= 19)
m.addConstr(5*x1 + 8*x3 >= 15)
m.addConstr(5*x1 + 8*x3 + 5*x4 >= 17)
m.addConstr(8*x2 + 8*x3 + 5*x4 >= 17)
m.addConstr(5*x1 + 8*x3 + 5*x4 >= 19)
m.addConstr(8*x2 + 8*x3 + 5*x4 >= 19)
m.addConstr(5*x1 + 8*x2 + 8*x3 + 5*x4 >= 19)

m.addConstr(8*x1 - 2*x2 >= 0)
m.addConstr(-3*x1 + x3 >= 0)
m.addConstr(-x2 + 7*x4 >= 0)

m.addConstr(5*x1 + 8*x2 <= 77)
m.addConstr(8*x2 + 8*x3 <= 55)
m.addConstr(5*x1 + 8*x3 <= 64)
m.addConstr(8*x2 + 8*x3 + 5*x4 <= 53)
m.addConstr(5*x1 + 8*x2 + 8*x3 <= 67)
m.addConstr(5*x1 + 8*x2 + 5*x4 <= 44)
m.addConstr(5*x1 + 8*x3 + 5*x4 <= 35)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by George: ", x1.varValue)
    print("Hours worked by Bill: ", x2.varValue)
    print("Hours worked by Hank: ", x3.varValue)
    print("Hours worked by Jean: ", x4.varValue)
else:
    print("The model is infeasible")
```

## 7: Symbolic Representation
The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'hours worked by George'), ('x2', 'hours worked by Bill'), ('x3', 'hours worked by Hank'), ('x4', 'hours worked by Jean')],
    'objective_function': '4*x1 + 9*x2 + 5*x3 + 3*x4',
    'constraints': [
        '5*x1 == 5',
        '8*x2 == 8',
        '8*x3 == 8',
        '5*x4 == 5',
        '8*x2 + 5*x4 >= 18',
        '8*x3 + 5*x4 >= 19',
        '5*x1 + 8*x3 >= 15',
        '5*x1 + 8*x3 + 5*x4 >= 17',
        '8*x2 + 8*x3 + 5*x4 >= 17',
        '5*x1 + 8*x3 + 5*x4 >= 19',
        '8*x2 + 8*x3 + 5*x4 >= 19',
        '5*x1 + 8*x2 + 8*x3 + 5*x4 >= 19',
        '8*x1 - 2*x2 >= 0',
        '-3*x1 + x3 >= 0',
        '-x2 + 7*x4 >= 0',
        '5*x1 + 8*x2 <= 77',
        '8*x2 + 8*x3 <= 55',
        '5*x1 + 8*x3 <= 64',
        '8*x2 + 8*x3 + 5*x4 <= 53',
        '5*x1 + 8*x2 + 8*x3 <= 67',
        '5*x1 + 8*x2 + 5*x4 <= 44',
        '5*x1 + 8*x3 + 5*x4 <= 35'
    ]
}
```