Question

The maximum value of \( Z = 3x - y \) subject to the constraints \[ x + y \leq 8, \quad x \geq 0, \quad y \geq 0 \text{ is}. \]

• A. -8
• B. 24
• C. 16
• D. 8

Hint:

When maximizing or minimizing a linear objective function under constraints, focus on the boundary points of the feasible region, as they often provide the optimal solutions.

Answer 1

The Correct Option is B

We are given the constraint \( x + y \leq 8 \), with \( x \geq 0 \) and \( y \geq 0 \). To maximize \( Z = 3x - y \), we need to maximize the value of \( x \) while minimizing \( y \). Since \( x + y = 8 \), the maximum value of \( x \) occurs when \( y = 0 \). Substituting \( y = 0 \) into the equation \( x + y = 8 \), we get: \[ x = 8. \] Thus, the maximum value of \( Z = 3x - y \) occurs at \( x = 8 \) and \( y = 0 \), which gives: \[ Z = 3(8) - 0 = 24. \] Therefore, the maximum value of \( Z \) is 24.