We solve the given Linear Programming (LP) problem using the Graphical Method.

Given LP Problem:

Objective Function:
$\text{Maximize } Z = 9X_1 + 6X_2$

Constraints:
$X_1 + X_2 \leq 4$ $2X_1 + X_2 \leq 6$ $X_1, X_2 \geq 0$

Step 1: Find the Feasible Region

Solve for intercepts of each constraint:

1. For $X_1 + X_2 = 4$:

   • If $X_1 = 0$, then $X_2 = 4$ → Point $(0,4)$
   • If $X_2 = 0$, then $X_1 = 4$ → Point $(4,0)$

2. For $2X_1 + X_2 = 6$:

   • If $X_1 = 0$, then $X_2 = 6$ → Point $(0,6)$
   • If $X_2 = 0$, then $2X_1 = 6 \Rightarrow X_1 = 3$ → Point $(3,0)$

Step 2: Find the Intersection of Two Lines

Solve for the intersection of
$X_1 + X_2 = 4$ $2X_1 + X_2 = 6$

Subtract the first equation from the second: $(2X_1 + X_2) - (X_1 + X_2) = 6 - 4$

$X_1 = 2$

Substituting $X_1 = 2$ in $X_1 + X_2 = 4$:

$2 + X_2 = 4$

$X_2 = 2$

Thus, the intersection point is (2,2).

Step 3: Evaluate the Objective Function at Each Vertex

1. At (0,4):
   $Z = 9(0) + 6(4) = 24$

2. At (4,0):
   $Z = 9(4) + 6(0) = 36$

3. At (3,0):
   $Z = 9(3) + 6(0) = 27$

4. At (2,2):
   $Z = 9(2) + 6(2) = 18 + 12 = 30$

Step 4: Determine the Maximum Value

The maximum value of $Z$ is 30 at $(2,2)$.

Final Answer:

$\boxed{30}$

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Would you like a graphical explanation or further clarification?

Here are 5 related questions to expand your understanding:

1. How would you solve this problem using the Simplex Method?
2. What happens if the constraint $2X_1 + X_2 \leq 6$ changes to $2X_1 + X_2 \leq 8$?
3. How does the feasible region change if we add a constraint $X_1 \geq 1$?
4. How do we determine whether a given LP problem has multiple optimal solutions?
5. What are slack variables, and how do they help in solving LP problems?

Tip:

When solving LP problems graphically, always identify corner points and evaluate the objective function at those points to find the optimal solution.