The image contains a linear programming problem in a graphical method. Here’s how we can break it down:

Given Problem:

Minimize the objective function:
$Z = 4x_1 + 6x_2$ Subject to the constraints:
$3x_1 + 2x_2 \leq 18 \quad \text{(1)}$
$x_1 + 4x_2 \leq 16 \quad \text{(2)}$
$x_1 \geq 0, \quad x_2 \geq 0 \quad \text{(3)}$

Steps to Solve:

1. Graphing the Constraints:

   • Plot the lines for each constraint. For the inequalities, shade the feasible region (where all constraints are satisfied).

2. Finding the Corner Points:

   • Identify the points where the lines intersect. These points are potential candidates for the optimal solution.

3. Evaluating the Objective Function:

   • Substitute the coordinates of each corner point into the objective function $Z = 4x_1 + 6x_2$ and choose the point that gives the minimum value.

Would you like me to walk through the graphical solution in detail, or would you prefer to explore another aspect of this problem? Let me know!

Related Questions:

1. How do you graph the constraints for a linear programming problem?
2. What are the steps for solving a linear programming problem using the graphical method?
3. How do you find the feasible region in a linear programming problem?
4. How can you evaluate the objective function at each corner point?
5. What other methods exist for solving linear programming problems besides the graphical method?

Tip:

When graphing linear constraints, always plot the lines corresponding to the equalities (i.e., set the inequalities as equalities), and remember that shading indicates the feasible region!