This is a linear programming (LP) problem where the goal is to maximize the objective function $z = 16x_1 + 8x_2$ subject to a set of constraints. Let’s solve it step-by-step using the graphical method, as it involves only two decision variables ($x_1$ and $x_2$).

Problem Details:

Objective Function:

$z = 16x_1 + 8x_2$

Constraints:

1. $6x_1 + 4x_2 \geq 24$
2. $4x_1 + 2x_2 \leq 16$
3. $3.5x_1 + 3x_2 \leq 21$
4. $x_1, x_2 \geq 0$ (non-negativity constraints)

Steps to Solve:

1. Convert inequalities into equations:

   • Constraint 1: $6x_1 + 4x_2 = 24$
   • Constraint 2: $4x_1 + 2x_2 = 16$
   • Constraint 3: $3.5x_1 + 3x_2 = 21$

2. Plot each line on a graph:

   • Rewrite the equations in terms of $x_2$ (to simplify plotting):
     • For $6x_1 + 4x_2 = 24$: $x_2 = 6 - 1.5x_1$
     • For $4x_1 + 2x_2 = 16$: $x_2 = 8 - 2x_1$
     • For $3.5x_1 + 3x_2 = 21$: $x_2 = 7 - \frac{3.5}{3}x_1 = 7 - 1.1667x_1$

3. Determine feasible region:

   • Identify regions where all constraints are satisfied.
   • Shade the overlapping area that meets all constraints, including $x_1, x_2 \geq 0$.

4. Find corner points of the feasible region:

   • Solve for intersection points of the lines (e.g., solving simultaneous equations for pairs of constraints).
   • Include intersections with the axes.

5. Evaluate the objective function:

   • Calculate $z = 16x_1 + 8x_2$ for each corner point.
   • The point giving the maximum $z$ is the optimal solution.

Next Steps:

Would you like me to:

1. Plot the graph and calculate the feasible region?
2. Solve the corner points algebraically?
3. Provide the solution using another method (e.g., simplex)?
4. Explain any step in greater detail?

Here are some related questions:

1. What is the significance of the feasible region in linear programming?
2. How can graphical solutions be extended to three or more variables?
3. Why are corner points important in solving LP problems?
4. Can you explain duality in linear programming?
5. What are the advantages of the simplex method compared to the graphical method?

Tip: Always check feasibility of the constraints before solving to avoid errors in determining the feasible region.