Question

By graphical method solve the LPP under the following constraints: \[ x + 2y \geq 10, \] \[ 3x + 4y \leq 24, \] \[ x \geq 0, \, y \geq 0, \] then find the minimum value of \( z = 200x + 500y \).

Hint:

To solve LPPs graphically, plot the constraints and find the feasible region. Then evaluate the objective function at the corner points of the region.

Answer 1

Step 1: Graph the inequalities.
To solve the LPP graphically, plot the following lines: \[ x + 2y = 10 \text{(Line 1)} \] \[ 3x + 4y = 24 \text{(Line 2)} \] and the boundaries \( x = 0 \) and \( y = 0 \). Find the intercepts for each line: - For Line 1, when \( x = 0 \), \( 2y = 10 \), so \( y = 5 \). When \( y = 0 \), \( x = 10 \). - For Line 2, when \( x = 0 \), \( 4y = 24 \), so \( y = 6 \). When \( y = 0 \), \( x = 8 \). Plot the lines on a graph.

Step 2: Identify the feasible region.
The feasible region is the area where all constraints overlap.

Step 3: Evaluate the objective function at the corner points.
From the graph, the corner points are identified, and the objective function \( z = 200x + 500y \) is evaluated at each corner.

Step 4: Find the minimum value of \( z \).
By evaluating \( z \) at each corner point, we find the minimum value of \( z = 200x + 500y \). Conclusion: The minimum value of \( z \) is found at one of the corner points of the feasible region.