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The problem involves maximizing the objective function \( Z = 4x + y \), subject to a set of linear constraints, and identifying the corner points of the feasible region on the graph.

### Given Constraints:
1. \( x + y \leq 50 \)
2. \( 3x + y \leq 90 \)
3. \( x \geq 0 \) (non-negativity constraint)
4. \( y \geq 0 \) (non-negativity constraint)

The feasible region is bounded by these inequalities, and the vertices (corner points) of this region represent potential solutions for optimization.

### Steps to find the corner points:
We need to find the intersection points of the boundary lines defined by the constraints. Based on the graph, we can identify the corner points:

- **Point \( O \)**: Origin, \( (0,0) \), is clear.
- **Point \( A \)**: Intersection of the line \( x + y = 50 \) with the \( x \)-axis (where \( y = 0 \)).
  
  For \( y = 0 \) in \( x + y = 50 \):
  \[
  x + 0 = 50 \quad \Rightarrow \quad x = 50
  \]
  So, \( A(50, 0) \).

- **Point \( B \)**: Intersection of the lines \( x + y = 50 \) and \( 3x + y = 90 \). To find this, we solve the system of equations:
  \[
  x + y = 50
  \]
  \[
  3x + y = 90
  \]
  Subtract the first equation from the second:
  \[
  (3x + y) - (x + y) = 90 - 50
  \]
  \[
  2x = 40 \quad \Rightarrow \quad x = 20
  \]
  Substitute \( x = 20 \) back into \( x + y = 50 \):
  \[
  20 + y = 50 \quad \Rightarrow \quad y = 30
  \]
  So, \( B(20, 30) \).

- **Point \( C \)**: Intersection of the line \( 3x + y = 90 \) with the \( y \)-axis (where \( x = 0 \)).

  For \( x = 0 \) in \( 3x + y = 90 \):
  \[
  3(0) + y = 90 \quad \Rightarrow \quad y = 90
  \]
  So, \( C(0, 90) \).

### Summary of Corner Points:
- \( A(50, 0) \)
- \( B(20, 30) \)
- \( C(0, 90) \)
- \( O(0, 0) \)

Now that we have the corner points, we can substitute these into the objective function \( Z = 4x + y \) to find which point maximizes \( Z \).

Would you like to see the detailed calculations for maximizing the objective function, or do you have any other questions about the process?

Here are some related questions for practice:
1. What is the maximum value of \( Z = 4x + y \) at the corner points?
2. How would you graphically represent the feasible region for a different set of inequalities?
3. Can you explain why the optimization process only considers the corner points?
4. How do the non-negativity constraints affect the feasible region?
5. What would happen if one of the inequalities were non-binding?

**Tip:** When solving systems of linear inequalities for optimization, always verify each corner point because the optimal solution will occur at one of these points.

Maximize Z = 4x + y, subject to the constraints: x + y ≤ 50, 3x + y ≤ 90, x ≥ 0, y ≥ 0.

Learn how to solve the linear programming problem: maximize Z = 4x + y subject to a system of inequalities. Explore key optimization concepts and solve for the corner points of the feasible region.

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