Let's first analyze the system of inequalities and the corresponding graph:

1. Inequalities:

   -5x + 2y & \leq -40 \\ -2x + 4y & \leq 8 \\ 7x + y & \leq 77 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}$$

2. Graph:

   • We need to find the points of intersection (corners) formed by the boundary lines from these inequalities. These corners represent the vertices of the feasible region.

3. Step 1: Solve the system at the intersections of each pair of lines.

   Intersection 1: $-5x + 2y = -40$ and $-2x + 4y = 8$

   We can solve this system of equations by substitution or elimination.

   Intersection 2: $-5x + 2y = -40$ and $7x + y = 77$

   Intersection 3: $-2x + 4y = 8$ and $7x + y = 77$

   Intersection 4: Check for boundary intersections along the axes ($x = 0$ or $y = 0$)

I'll calculate the coordinates of the corner points based on these steps.Here are the key intersection points (corners) for the system of inequalities:

1. Intersection of $-5x + 2y = -40$ and $-2x + 4y = 8$:

   • $(x, y) = (11, 7.5)$

2. Intersection of $-5x + 2y = -40$ and $7x + y = 77$:

   • $(x, y) = \left(\frac{194}{19}, \frac{105}{19}\right) \approx (10.21, 5.53)$

3. Intersection of $-2x + 4y = 8$ and $7x + y = 77$:

   • $(x, y) = (10, 7)$

Axes Intersections:

• For the line $-5x + 2y = -40$:

  • Intersection with the x-axis: $x = 8$ (when $y = 0$)
  • Intersection with the y-axis: $y = -20$ (when $x = 0$)

• For the line $-2x + 4y = 8$:

  • Intersection with the x-axis: $x = -4$ (when $y = 0$)
  • Intersection with the y-axis: $y = 2$ (when $x = 0$)

• For the line $7x + y = 77$:

  • Intersection with the x-axis: $x = 11$ (when $y = 0$)
  • Intersection with the y-axis: $y = 77$ (when $x = 0$)

Would you like further explanations on how these intersections were found or how to interpret them on the graph?

Here are 5 questions to explore this further:

1. How do you determine if a point lies within the feasible region?
2. Can inequalities have more than one intersection with the axes?
3. What is the significance of each corner in terms of optimization problems?
4. How can you graph these inequalities manually?
5. How does changing the coefficients affect the graph?

Tip: Always check for any points of intersection along the axes as they can form important parts of the feasible region boundaries.