For the following linear programming problem, determine the optimal solution for "y" using the graphical solution method.


Max	−X + 2Y
s.t.	  6X − 2Y ≤ 3
 	−2X + 3Y ≤ 6
 	    X +   Y ≤ 3
 	        X, Y ≥ 0
Group of answer choices

0

2

2.4

1

1.875

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**.

---

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.

For the following linear programming problem, determine the optimal solution for 'y' using the graphical solution method.
Max -X + 2Y
s.t.
6X − 2Y ≤ 3
-2X + 3Y ≤ 6
X + Y ≤ 3
X, Y ≥ 0
Group of answer choices
0
2
2.4
1
1.875

Learn how to solve a linear programming problem using the graphical solution method. The objective function is -X + 2Y, with constraints 6X - 2Y ≤ 3, -2X + 3Y ≤ 6, and X + Y ≤ 3. This step-by-step guide explains how to find the feasible region and maximize the objective function at corner points. Suitable for college-level students.

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