The problem you've provided is a linear programming problem. Here's the formulation:

Objective:

Maximize $Z = 4x + 6y$

Subject to the constraints:

1. $2x + y \leq 8$
2. $x + 2y \leq 10$
3. $x \geq 0$
4. $y \geq 0$

This problem aims to find the values of $x$ and $y$ that maximize $Z = 4x + 6y$ while satisfying the given constraints.

Solution process:

1. Graph the constraints: Plot the inequalities $2x + y \leq 8$, $x + 2y \leq 10$, along with the non-negativity conditions $x \geq 0$ and $y \geq 0$. These inequalities form a feasible region.

2. Find the corner points: The maximum value of the objective function will occur at one of the corner points of the feasible region. We'll determine these points by solving the system of inequalities.

3. Evaluate the objective function at each corner point to find the maximum value of $Z = 4x + 6y$.

Let's go through this systematically:

Step 1: Find the corner points

Solve the system of equations formed by the intersection of the boundary lines of the constraints:

1. $2x + y = 8$
2. $x + 2y = 10$

• Intersection of $2x + y = 8$ and $x + 2y = 10$:
  • Solve the first equation for $y$: $y = 8 - 2x$.
  • Substitute into the second equation: $x + 2(8 - 2x) = 10 \Rightarrow x + 16 - 4x = 10 \Rightarrow -3x = -6 \Rightarrow x = 2.$
  • Substitute $x = 2$ into $y = 8 - 2x$: $y = 8 - 4 = 4.$ So, the intersection point is $(2, 4)$.

Other corner points come from the intersections with the axes:

• Intersection with $x$-axis: Set $y = 0$ in $2x + y = 8$, yielding $x = 4$. So, one corner point is $(4, 0)$.

• Intersection with $y$-axis: Set $x = 0$ in $x + 2y = 10$, yielding $y = 5$. So, another corner point is $(0, 5)$.

Step 2: Evaluate the objective function at each corner point

• At $(4, 0)$: $Z = 4(4) + 6(0) = 16$.
• At $(0, 5)$: $Z = 4(0) + 6(5) = 30$.
• At $(2, 4)$: $Z = 4(2) + 6(4) = 8 + 24 = 32$.

Step 3: Conclusion

The maximum value of $Z = 32$ occurs at the point $(2, 4)$.

Would you like further details on any part of the solution?

Here are some related questions to expand understanding:

1. What are the graphical interpretations of linear programming constraints?
2. How do you know which points in the feasible region will yield the maximum or minimum value?
3. Can a linear programming problem have more than one optimal solution?
4. What happens if the feasible region is unbounded?
5. How would the solution change if the objective function was to minimize $Z$?

Tip: In linear programming, the optimal solution always lies at a vertex (corner point) of the feasible region.