Question

The corner points of the feasible region of the LPP: Minimize \( Z = -50x + 20y \) subject to \( 2x - y \geq -5 \), \( 3x + y \geq 3 \), \( 2x - 3y \leq 12 \), and \( x, y \geq 0 \) are:

• A. \( (0, 5), (0, 6), (1, 0), (6, 0) \)
• B. \( (0, 3), (0, 5), (3, 0), (6, 0) \)
• C. \( (0, 3), (0, 5), (1, 0), (6, 0) \)
• D. \( (0, 5), (0, 6), (1, 0), (3, 0) \)

Hint:

Graph the constraints to find the feasible region. Then, solve for the corner points where the constraints intersect. These points will help in determining the optimal solution for the objective function.

Answer 1

The Correct Option is C

Given the objective function and the constraints:
1. \( 2x - y \geq -5 \)
2. \( 3x + y \geq 3 \)
3. \( 2x - 3y \leq 12 \)
4. \( x \geq 0, y \geq 0 \)
We need to graph these inequalities to find the feasible region and then determine the corner points of this region.
- From the first constraint: \( 2x - y = -5 \), we can rearrange it to express in slope-intercept form:
\( y = 2x + 5 \).
- For the second constraint: \( 3x + y = 3 \), rearranging gives: \( y = -3x + 3 \).
- For the third constraint: \( 2x - 3y = 12 \), we rearrange to express as: \( y = \frac{2x - 12}{3} \).
After graphing these three lines along with the \( x \geq 0 \) and \( y \geq 0 \) conditions, we identify the corner points of the feasible region. These points are the vertices where the constraints intersect. By solving the system of equations of the lines and checking which points lie in the feasible region, we determine that the corner points are: \[ (0, 3), (0, 5), (1, 0), (6, 0) \]