Question

Find the maximum value of the linear objective function \( Z = 3x + 4y \) subject to the system constraints: \( x + y \le 4 \), \( x \ge 0 \), and \( y \ge 0 \).

• A. \( 12 \)
• B. \( 16 \)
• C. \( 24 \)
• D. \( 0 \)

Hint:

To locate the maximizing corner point quickly without full math checks, compare the variable coefficients in the objective function. Since \( y \) carries a larger weight coefficient (4) than \( x \) (3), maximizing the \( y \)-coordinate budget at \( (0,4) \) will naturally yield the highest value.

Answer 1

The Correct Option is B

Concept: According to the Corner Point Theorem of Linear Programming, the optimal maximum or minimum value of an objective function always occurs at one of the corner vertices of the bounded feasible region.

Step 1: Identify and map the corner vertices of the feasible region.
The non-negativity constraints (\( x \ge 0 \), \( y \ge 0 \)) restrict the feasible region to the first quadrant. Find where the primary constraint line \( x + y = 4 \) intersects both axes:

• Setting \( y = 0 \) gives the x-intercept at \( (4,0) \).
• Setting \( x = 0 \) gives the y-intercept at \( (0,4) \).

The boundary lines meet at the origin, forming a closed triangular feasible region with corner vertices at: \( (0,0) \), \( (4,0) \), and \( (0,4) \).

Step 2: Evaluate the objective function \( Z \) at each vertex point.
Plug the coordinate numbers for each corner point into the objective equation \( Z = 3x + 4y \):

• At vertex \( (0,0) \): \( Z = 3(0) + 4(0) = 0 \)
• At vertex \( (4,0) \): \( Z = 3(4) + 4(0) = 12 \)
• At vertex \( (0,4) \): \( Z = 3(0) + 4(4) = 16 \)

Step 3: Identify the absolute maximum value.
Comparing our results shows that the absolute maximum value is 16, which occurs at the corner coordinate position \( (0,4) \).