To solve this linear programming problem graphically, let's go through the steps required to plot the constraints and find the feasible region.

Step 1: Identify and Plot Each Constraint

The problem is:

1. Objective function: Minimize $z = 3x + 2y$
2. Constraints:
   • $x + y \leq 10$
   • $3x + 2y \geq 6$
   • $x \geq 0$
   • $y \geq 0$

We'll graph each of these constraints on the $xy$-plane and shade the feasible region.

Step 2: Graph the Inequalities

1. Plot $x + y \leq 10$

To plot $x + y = 10$:

• When $x = 0$, $y = 10$.
• When $y = 0$, $x = 10$.

This gives us the line $x + y = 10$ passing through points $(0, 10)$ and $(10, 0)$. The inequality $x + y \leq 10$ means shading the area below this line.

2. Plot $3x + 2y \geq 6$

To plot $3x + 2y = 6$:

• When $x = 0$, $y = 3$.
• When $y = 0$, $x = 2$.

This gives us the line $3x + 2y = 6$ passing through points $(0, 3)$ and $(2, 0)$. The inequality $3x + 2y \geq 6$ means shading the area above this line.

3. Plot $x \geq 0$ and $y \geq 0$

These constraints restrict the feasible region to the first quadrant, where both $x$ and $y$ are non-negative.

Step 3: Determine the Feasible Region

The feasible region is where all shaded areas from the inequalities overlap. This region will be bounded by the intersection of the constraints on the $xy$-plane.

Step 4: Identify Corner Points

To solve graphically, find the vertices (corner points) of the feasible region by solving the intersections of the lines:

1. Intersection of $x + y = 10$ and $3x + 2y = 6$.
2. Intersection of $x + y = 10$ with $y = 0$ (which is $x = 10$ and $y = 0$).
3. Intersection of $3x + 2y = 6$ with $x = 0$ (which is $x = 0$ and $y = 3$).

Step 5: Evaluate the Objective Function at Each Vertex

Once you identify the corner points, plug them into the objective function $z = 3x + 2y$ to find the minimum value. The point that gives the smallest value of $z$ will be the solution to the minimization problem.

Would you like me to plot this graph and calculate the intersection points for you?

Here are some further questions to deepen understanding:

1. How do you interpret the objective function in the context of optimization?
2. Why are the corner points of the feasible region important in linear programming?
3. What does it mean if there is no feasible region?
4. How would this problem change if the inequalities were reversed?
5. Can you explain why only the first quadrant is considered for feasible solutions?

Tip: When solving linear programming problems graphically, always start by plotting each constraint carefully, as errors in the feasible region can lead to incorrect solutions.