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.