To solve this linear programming problem using the graphical solution method, we need to follow these steps:

1. Objective Function:

The objective is to maximize the function: $Z = -X + 2Y$

2. Constraints:

The constraints are given by the following inequalities:

1. $6X - 2Y \leq 3$
2. $-2X + 3Y \leq 6$
3. $X + Y \leq 3$
4. $X \geq 0$
5. $Y \geq 0$

3. Convert the inequalities to equalities (for plotting):

To find the points of intersection, we'll first rewrite the inequalities as equalities:

1. $6X - 2Y = 3$ → $Y = 3X - \frac{3}{2}$
2. $-2X + 3Y = 6$ → $Y = \frac{2}{3}X + 2$
3. $X + Y = 3$ → $Y = 3 - X$

4. Plot the lines and determine the feasible region:

We plot these lines on the $XY$-plane, taking into account that $X \geq 0$ and $Y \geq 0$, and identify the feasible region where all the constraints overlap.

5. Find the corner points (vertices):

By solving the system of equations where these lines intersect, we find the corner points (vertices) of the feasible region.

6. Evaluate the objective function at the vertices:

We substitute the corner points into the objective function $Z = -X + 2Y$ and check which vertex maximizes the value of $Z$.

Corner points:

1. Intersection of $6X - 2Y = 3$ and $X + Y = 3$
2. Intersection of $-2X + 3Y = 6$ and $X + Y = 3$
3. Intersection of $X + Y = 3$ and the axes $X = 0$ and $Y = 0$

7. Optimal solution for $Y$:

After solving and evaluating the objective function at each vertex, we find that the optimal solution for $Y$ is 2.

Final Answer:

The optimal value of $Y$ is 2.

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Would you like further details on any of these steps?

Related Questions:

1. How do you find the intersection points of two linear equations?
2. What are the steps to graphically solve a linear programming problem?
3. How do you check feasibility of corner points in linear programming?
4. Can the objective function always be maximized at a corner point?
5. What happens when there's no feasible solution in a linear programming problem?

Tip: Always check the feasibility of the corner points before evaluating the objective function at those points.