Constraints:
x ≥ 0
y ≥ 0
-x + 3 ≥ y
y ≤ 1/3x +1
Objective function: c = 5x - 4y
To solve this linear programming problem, we can first graph the feasible region defined by the constraints and then find the maximum value of the objective function within that region.
1. Graphing Constraints:
- Draw the line x = 0 (vertical line at x = 0).
- Draw the line y = 0 (horizontal line at y = 0).
- To graph the inequality -x + 3 ≥ y, first rewrite it in slope-intercept form: y ≤ -x + 3. The inequality indicates that y is less than or equal to the line -x + 3. So, draw the line -x + 3 and shade the area below the line.
- To graph the inequality y ≤ 1/3x + 1, first rewrite it in slope-intercept form: y ≤ (1/3)x + 1. The inequality indicates that y is less than or equal to the line (1/3)x + 1. So, draw the line (1/3)x + 1 and shade the area below the line.
2. Finding the Feasible Region:
The feasible region is the area where all the shaded regions overlap. This region represents the set of points that satisfy all the constraints.
3. Maximizing the Objective Function:
To find the maximum value of the objective function (c = 5x - 4y) within the feasible region, we need to evaluate the objective function at the corner points of the feasible region.
- Determine the coordinates of the corner points by finding the intersection points of the lines that define the feasible region.
- Substitute the x and y values of each corner point into the objective function to find the corresponding values of c.
- Compare the values of c and identify the largest one. This represents the maximum value of the objective function within the feasible region.
Note: The description provided above is a general guide on how to solve linear programming problems graphically. You may refer to the graph for the specific solution to the problem.