Question

The feasible region of the linear programming problem is determined by the system of inequalities: \[ x + y \leq 6, \quad x \geq 0, \quad y \geq 0. \] What is the maximum value of \( x + y \) in the feasible region?

• A. \( 6 \)
• B. \( 5 \)
• C. \( 4 \)
• D. \( 3 \)

Hint:

When solving linear programming problems, identify the feasible region and evaluate the objective function at the vertices of the region.

Answer 1

The Correct Option is A

We are asked to find the maximum value of \( x + y \) for the given system of inequalities: \[ x + y \leq 6, \quad x \geq 0, \quad y \geq 0. \] Step 1: Graph the inequalities - The inequality \( x + y \leq 6 \) represents a region below the line \( x + y = 6 \). - The inequalities \( x \geq 0 \) and \( y \geq 0 \) represent the first quadrant of the coordinate plane. Thus, the feasible region is the triangular region formed by the points where the line \( x + y = 6 \) intersects the axes, along with the positive x and y axes. Step 2: Find the vertices of the feasible region The line \( x + y = 6 \) intersects the x-axis at \( (6, 0) \) and the y-axis at \( (0, 6) \). The third vertex is the origin \( (0, 0) \). The vertices of the feasible region are \( (0, 0) \), \( (6, 0) \), and \( (0, 6) \). Step 3: Evaluate \( x + y \) at each vertex - At \( (0, 0) \), \( x + y = 0 + 0 = 0 \) - At \( (6, 0) \), \( x + y = 6 + 0 = 6 \) - At \( (0, 6) \), \( x + y = 0 + 6 = 6 \) Step 4: Conclusion The maximum value of \( x + y \) in the feasible region is \( 6 \), which occurs at the points \( (6, 0) \) and \( (0, 6) \). Answer: The maximum value of \( x + y \) is \( 6 \), so the correct answer is option (1).