(a)
(1) True
(2) True
(b)
Step 2: Identify the feasible region by shading the area that satisfies all constraints simultaneously.
Step 3: Graph the objective function for a specific value (e.g., set Q to a constant) to obtain an initial level line.
Step 4: Move this level line parallel in the direction of increasing (for maximization) or decreasing (for minimization) values of the objective function, while staying within the feasible region.
Step 5: The optimal solution occurs at the last point where the level line touches the feasible region, typically at a vertex.
(c)
The system of constraints is:
x + y ≤ 6
2x + y ≥ 8
y ≥ 0
Graphing these constraints reveals no feasible region. The first constraint defines a region below the line x + y = 6, while the second constraint defines a region above the line 2x + y = 8. These two regions do not overlap for any point satisfying y ≥ 0. Therefore, there is no point (x, y) that satisfies all constraints simultaneously.
The linear programming problem has no feasible solution.
Parent Tip: Review the logic above to help your child master the concept of linear programming problems and solutions.
