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The problem you've uploaded involves solving a linear programming problem. Here's how we can break it down:

### Given Information:
- The student earns **$10 per hour** for tutoring and **$7 per hour** as a teacher's aide.
- He can work a **maximum of 20 hours per week** total.
- The tutoring center requires **at least 3 hours** per week tutoring but **no more than 8 hours**.
- You are tasked with determining how many hours he should work at each job to **maximize his earnings**.

### Define the Variables:
- Let **x** be the number of hours spent tutoring.
- Let **y** be the number of hours spent as a teacher's aide.

### Objective Function:
The objective is to maximize earnings, given by the equation:
\[
\text{Profit} = 10x + 7y
\]
where:
- $10x$ is the income from tutoring (at $10 per hour),
- $7y$ is the income from being a teacher's aide (at $7 per hour).

### Constraints:
The problem provides several constraints:
1. **Total hours constraint**: The total hours worked per week cannot exceed 20 hours:
   \[
   x + y \leq 20
   \]
2. **Tutoring hours constraints**: The student must work at least 3 hours and no more than 8 hours tutoring:
   \[
   3 \leq x \leq 8
   \]
3. **Non-negative constraint**: The number of hours worked cannot be negative:
   \[
   x \geq 0, \quad y \geq 0
   \]

### Solving the Problem:
To solve this linear programming problem, we need to evaluate the objective function at the vertices (corner points) of the feasible region defined by the constraints. The constraints form a bounded region, and the maximum profit will occur at one of the vertices.

Let’s calculate the corner points:
- When $x = 3$: From $x + y \leq 20$, we have $3 + y \leq 20$, so $y \leq 17$.
  \[
  \text{Point: } (3, 17)
  \]
- When $x = 8$: From $x + y \leq 20$, we have $8 + y \leq 20$, so $y \leq 12$.
  \[
  \text{Point: } (8, 12)
  \]
- When $y = 0$ (no hours as teacher's aide): From $x + y \leq 20$, we have $x \leq 20$.
  \[
  \text{Point: } (8, 0) \text{ since tutoring is restricted by the upper bound 8.}
  \]

Now let's compute the profit at each point:
1. For $x = 3, y = 17$:
   \[
   \text{Profit} = 10(3) + 7(17) = 30 + 119 = 149
   \]
2. For $x = 8, y = 12$:
   \[
   \text{Profit} = 10(8) + 7(12) = 80 + 84 = 164
   \]
3. For $x = 8, y = 0$:
   \[
   \text{Profit} = 10(8) + 7(0) = 80
   \]

### Conclusion:
- The maximum profit is **$164**, which occurs when the student works **8 hours tutoring** and **12 hours as a teacher's aide**.

Would you like more details on the solution process? Here are some additional questions:

1. How would the solution change if the tutoring rate were different?
2. What would happen if the student could work more than 20 hours per week?
3. How can you graph the constraints and the feasible region?
4. How do you solve a linear programming problem using the simplex method?
5. What if the student was allowed to work fewer than 3 hours tutoring—how would that affect the outcome?

**Tip:** When solving linear programming problems, always look at the corner points of the feasible region to find the optimal solution.

A student earns $10 per hour for tutoring and $7 per hour as a teacher's aide. To have enough free time for studies, he can work no more than 20 hours per week. The tutoring center requires that each tutor spends at least three hours per week tutoring, but no more than eight hours per week. How many hours should he work to maximize his earnings?

Solve a linear programming problem where a student earns $10 per hour tutoring and $7 per hour as a teacher's aide, with a 20-hour per week limit. Learn how to maximize the student's earnings while adhering to constraints on hours worked.

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