I Would like to proceed with the graphical solution

Great! Let's proceed with the graphical solution for maximizing the profit based on the problem provided.

### Step 1: Define the Constraints
We start by identifying the constraints given in the problem. These constraints are based on the available assembly and finishing hours.

1. **Assembly Constraint:**  
   \( 8x_1 + 6x_2 \leq 480 \)

2. **Finishing Constraint:**  
   \( 4x_1 + 2x_2 \leq 200 \)

3. **Non-negativity Constraint:**  
   \( x_1 \geq 0 \), \( x_2 \geq 0 \)

Where:
- \( x_1 \) is the number of elegant tables.
- \( x_2 \) is the number of ordinary tables.

### Step 2: Convert Constraints to Equalities
To graph these constraints, we first convert the inequalities into equalities to find the boundary lines:

1. **Assembly Constraint (Line Equation):**  
   \( 8x_1 + 6x_2 = 480 \)

2. **Finishing Constraint (Line Equation):**  
   \( 4x_1 + 2x_2 = 200 \)

### Step 3: Find Intercepts for the Constraints
To plot these lines, we can find the intercepts with the axes.

**1. Assembly Line Intercepts:**
- **For \( x_1 = 0 \) (Find \( x_2 \)):**  
  \( 8(0) + 6x_2 = 480 \Rightarrow x_2 = \frac{480}{6} = 80 \)
- **For \( x_2 = 0 \) (Find \( x_1 \)):**  
  \( 8x_1 + 6(0) = 480 \Rightarrow x_1 = \frac{480}{8} = 60 \)

**2. Finishing Line Intercepts:**
- **For \( x_1 = 0 \) (Find \( x_2 \)):**  
  \( 4(0) + 2x_2 = 200 \Rightarrow x_2 = \frac{200}{2} = 100 \)
- **For \( x_2 = 0 \) (Find \( x_1 \)):**  
  \( 4x_1 + 2(0) = 200 \Rightarrow x_1 = \frac{200}{4} = 50 \)

### Step 4: Plot the Lines on a Graph
Now, we can plot the lines on a graph based on the intercepts:

- The Assembly Line will connect the points \( (0, 80) \) and \( (60, 0) \).
- The Finishing Line will connect the points \( (0, 100) \) and \( (50, 0) \).

### Step 5: Identify the Feasible Region
The feasible region is the area where both constraints overlap, taking into account the non-negativity constraints (\( x_1 \geq 0 \) and \( x_2 \geq 0 \)). This region will be a polygon bounded by the lines and the axes.

### Step 6: Find the Vertices of the Feasible Region
The vertices of the feasible region are where the constraint lines intersect each other or the axes. To find these, we solve the equations where the lines intersect:

1. **Intersection of the Assembly and Finishing Lines:**
   Solve \( 8x_1 + 6x_2 = 480 \) and \( 4x_1 + 2x_2 = 200 \) simultaneously.

   \[
   8x_1 + 6x_2 = 480 \quad \text{(1)}
   \]
   \[
   4x_1 + 2x_2 = 200 \quad \text{(2)}
   \]

   Multiply equation (2) by 3 to align coefficients:

   \[
   12x_1 + 6x_2 = 600 \quad \text{(3)}
   \]

   Subtract equation (1) from (3):

   \[
   4x_1 = 120 \Rightarrow x_1 = 30
   \]

   Substitute \( x_1 = 30 \) into equation (2):

   \[
   4(30) + 2x_2 = 200 \Rightarrow 120 + 2x_2 = 200 \Rightarrow 2x_2 = 80 \Rightarrow x_2 = 40
   \]

   So, the intersection point is \( (30, 40) \).

Other vertices are:
- Intersection with \( x_1 = 0 \) and \( x_2 = 80 \) (from assembly constraint)
- Intersection with \( x_1 = 0 \) and \( x_2 = 100 \) (from finishing constraint)
- Intersection with \( x_2 = 0 \) and \( x_1 = 60 \) (from assembly constraint)
- Intersection with \( x_2 = 0 \) and \( x_1 = 50 \) (from finishing constraint)

### Step 7: Evaluate the Objective Function at Each Vertex
Evaluate the profit function \( P = 700x_1 + 500x_2 \) at each vertex:

1. \( (0, 80) \): \( P = 700(0) + 500(80) = 40,000 \)
2. \( (0, 100) \): \( P = 700(0) + 500(100) = 50,000 \)
3. \( (60, 0) \): \( P = 700(60) + 500(0) = 42,000 \)
4. \( (50, 0) \): \( P = 700(50) + 500(0) = 35,000 \)
5. \( (30, 40) \): \( P = 700(30) + 500(40) = 21,000 + 20,000 = 51,000 \)

### Step 8: Determine the Maximum Profit
The maximum profit occurs at the vertex \( (30, 40) \), where the profit is ₱51,000.

### Conclusion:
To maximize the profit, the shop should produce 30 elegant tables and 40 ordinary tables. This combination yields the maximum profit of ₱51,000.

Would you like a visual representation of the graph or have any further questions?

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### Relative Questions:
1. How would the solution change if the profit per table for either type was different?
2. What happens to the feasible region if one of the constraints becomes stricter?
3. Can the graphical method be applied to more than two variables, and if not, why?
4. How does the intersection of constraints impact the optimal solution in linear programming?
5. How could the problem be solved if the resources (assembly or finishing hours) were increased?

### Tip:
In linear programming problems, always double-check the calculations at each vertex of the feasible region, as a mistake in evaluating the objective function can lead to selecting a suboptimal solution.

Learn how to maximize profit using the graphical method in linear programming. Solve a problem involving constraints and objective functions to find the optimal production quantities of elegant and ordinary tables. This problem covers key concepts such as linear programming, graphical method, feasible region, and objective function evaluation.

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