SOLVING SYSTEMS USING MATRICES
Graph the system of constraints and find the value of x and y that maximize the objective function
{x>=0
Y>= 0
Y<=3
Y<=-x+5
Objective function: C=-5x+3y
A:(0,0)
B(2,3
C(5,0)
D(0,3)
Need help!!
so graph the region, then pick the vertex where the function is maximized.
To solve the system of constraints and find the values of x and y that maximize the objective function, you can follow these steps:
1. Graph the System of Constraints:
- On a coordinate plane, plot the points A(0,0), B(2,3), C(5,0), and D(0,3).
- Draw the lines representing the inequalities: x >= 0 (a vertical line passing through x=0), y >= 0 (a horizontal line passing through y=0), y <= 3 (a horizontal line passing through y=3), and y <= -x + 5 (a downward-sloping line passing through the points B(2,3) and C(5,0)).
2. Determine the Feasible Region:
- Shade the region where all the constraints are satisfied, which is the region enclosed by the plotted lines.
3. Find the Vertices of the Feasible Region:
- The vertices of the feasible region are the points where the lines intersect. In this case, the vertices are A(0,0), B(2,3), and D(0,3).
4. Evaluate the Objective Function at Each Vertex:
- Plug in the x and y coordinates of each vertex into the objective function C = -5x + 3y.
- For point A(0,0): C = -5(0) + 3(0) = 0
- For point B(2,3): C = -5(2) + 3(3) = -10 + 9 = -1
- For point D(0,3): C = -5(0) + 3(3) = 0 + 9 = 9
5. Determine the Maximum Value of the Objective Function:
- Compare the values obtained in step 4 to find the highest value. In this case, the maximum value is 9, which occurs at point D(0,3).
Therefore, the values of x and y that maximize the objective function are x = 0 and y = 3.