Question

Find the minimum value of  ( z = x + 3y ) under the following constraints: 

• x + y ≤ 8
• 3x + 5y ≥ 15
• x ≥ 0, y ≥ 0

Hint:

For linear programming, identify feasible region by plotting constraint lines, then evaluate the objective function at extreme points.

Answer 1

Step 1: Convert constraints into a feasible region. Solving for boundary points: - \( x = 0 \Rightarrow y = 8 \)
- \( y = 0 \Rightarrow x = 8 \) For \( 3x + 5y = 15 \): - \( x = 0 \Rightarrow y = 3 \)
- \( y = 0 \Rightarrow x = 5 \) 

Step 2: Evaluate \( z \) at extreme points. Using intersection points of constraints: \[ \text{Minimum } z = 3. \]