X>0 y>0-x+3>y y<1/3x+1
Use the following constraints and objective function to answer the next 3 questions. Objective function: C=5x−4y Question 1 Using a graphing calculator, find and select all the vertices for the feasible region.(1 point) Responses (0,3) (0,3) (1,0) (1,0) (-3,0) (-3,0) (3,0) (3,0) (1.5,1.5) (1.5,1.5) (0,1) (0,1) (0,0)
To find the vertices of the feasible region, we need to find the points where the lines representing the constraints intersect.
First, let's find the intersection of the lines y = 0 - x + 3 and y = 1/3x + 1:
0 - x + 3 = 1/3x + 1
Multiply both sides of the equation by 3 to eliminate the fraction:
0 - 3x + 9 = x + 3
Add 3x to both sides:
9 = x + 3x + 3
Combine like terms:
9 = 4x + 3
Subtract 3 from both sides:
6 = 4x
Divide by 4:
x = 3/2 = 1.5
Now let's substitute this value of x into one of the equations to find y:
y = 0 - x + 3 = 0 - 1.5 + 3 = 1.5
So the first vertex is (1.5, 1.5).
Next, let's find the intersection of the lines y = 0 - x + 3 and y = 0, with x > 0:
0 - x + 3 = 0
Solve for x:
x = 3
Substitute this value of x into y = 0 - x + 3:
y = 0 - 3 + 3 = 0
So the second vertex is (3, 0).
Finally, let's find the intersection of the lines y = 1/3x + 1 and y = 0, with x > 0:
1/3x + 1 = 0
Subtract 1 from both sides:
1/3x = -1
Multiply both sides by 3 to eliminate the fraction:
x = -3
Substitute this value of x into y = 1/3x + 1:
y = 1/3(-3) + 1 = -1 + 1 = 0
So the third vertex is (-3, 0).
Therefore, the vertices of the feasible region are (1.5, 1.5), (3, 0), and (-3, 0).
To find the vertices of the feasible region, we need to solve the system of inequalities:
1) y > 0 - x + 3
2) y < (1/3)x + 1
Let's solve this system step-by-step:
Step 1: Graph the boundary lines for each inequality.
For the first inequality, y > 0 - x + 3, let's find the boundary line by replacing the inequality with an equal sign:
y = 0 - x + 3
Simplifying the equation, we get:
y = -x + 3
For the second inequality, y < (1/3)x + 1, let's find the boundary line by replacing the inequality with an equal sign:
y = (1/3)x + 1
Step 2: Shade the region that satisfies each inequality.
For the first inequality, y > 0 - x + 3, we need to shade the region above the boundary line y = -x + 3.
For the second inequality, y < (1/3)x + 1, we need to shade the region below the boundary line y = (1/3)x + 1.
Step 3: Find the points of intersection between the two boundary lines.
To find the vertices of the feasible region, we need to find the intersection points of the two boundary lines.
Solving the system of equations:
y = -x + 3
y = (1/3)x + 1
Setting the equations equal to each other, we get:
-x + 3 = (1/3)x + 1
Multiply through by 3 to eliminate the fraction:
-3x + 9 = x + 3
Combine like terms:
-4x = -6
Divide by -4:
x = (3/2) or 1.5
Substituting this value back into either equation, let's use the first equation:
y = -x + 3
y = -(3/2) + 3
y = (1/2)
So, the intersection point is (1.5, 0.5).
Step 4: Find the remaining vertices.
To find the remaining vertices, we need to determine the points of intersection between the vertices of the shaded regions and the boundary lines.
From the graph, it appears that the remaining vertices are (0, 3), (0, 1), and (3, 0).
Therefore, the vertices of the feasible region are (0, 3), (0, 1), (3, 0), and (1.5, 0.5).
Please let me know if you need any further clarification.
To find the vertices of the feasible region, we need to solve the system of inequalities:
1) x > 0
2) y > 0 - x + 3
3) y < (1/3)x + 1
We can start by graphing the equations on a graphing calculator to visualize the feasible region and identify its vertices.
1) Plot the line x = 0 (vertical line passing through the y-axis).
2) Plot the line y = 0 - x + 3 (a downward sloping line passing through the point (3, 0)).
3) Plot the line y = (1/3)x + 1 (a upward sloping line passing through the point (0, 1)).
The region where all three inequalities are satisfied is the feasible region. The vertices of the feasible region are the points where the lines intersect.
It appears that the feasible region has the following vertices:
1) (0, 3)
2) (1, 0)
3) (-3, 0)
4) (3, 0)
5) (1.5, 1.5)
6) (0, 1)
7) (0, 0)
So the correct vertices for the feasible region are:
(0, 3), (1, 0), (-3, 0), (3, 0), (1.5, 1.5), (0, 1), and (0, 0).