Determine the 5 vertices of the feasible region, given the constraints below. 3x+5y≤30 x≤9 y≤3 x≥0 y≥0
To determine the vertices of the feasible region, we will graph the system of inequalities and find the points where the lines intersect.
The first inequality, 3x + 5y ≤ 30, can be rewritten as 5y ≤ -3x + 30, which is equivalent to y ≤ (-3/5)x + 6.
The second inequality, x ≤ 9, represents a vertical line passing through x = 9.
The third inequality, y ≤ 3, represents a horizontal line passing through y = 3.
The last two inequalities, x ≥ 0 and y ≥ 0, represent the x-axis and y-axis, respectively.
We can start by graphing the last two inequalities, which form the axes:
The x-axis is the line y = 0 and the y-axis is the line x = 0.
Now let's graph the remaining inequalities:
First, plot the line y ≤ (-3/5)x + 6. To do this, plot the point (0, 6) on the y-axis and use the slope -3/5 to obtain a second point. From (0, 6), you can move 5 units to the right and 3 units down to plot a second point at (5, 3). Draw a line through these two points and shade the region below the line (including the line itself) since it represents y ≤ (-3/5)x + 6.
Next, graph the line x ≤ 9 by plotting a vertical line passing through x = 9. Shade the region to the left of the line since it represents x ≤ 9.
Finally, graph the line y ≤ 3 by plotting a horizontal line passing through y = 3. Shade the region below the line since it represents y ≤ 3.
The feasible region is the shaded region where all the shaded regions overlap. The vertices of this region are the points where the lines intersect.
Analyzing the graph, we find the following five vertices of the feasible region:
1) (0, 0)
2) (0, 3)
3) (5, 3)
4) (9, 0)
5) (9, 1.8)
These five points represent the vertices of the feasible region.
To determine the vertices of the feasible region, we need to solve the system of inequalities created by the given constraints:
1. 3x + 5y ≤ 30
2. x ≤ 9
3. y ≤ 3
4. x ≥ 0
5. y ≥ 0
Let's solve the system step by step:
Step 1: Graph the individual constraints:
Graph the line 3x + 5y = 30:
To graph this line, we can find two points on it. Let's assume x = 0 and solve for y:
3(0) + 5y = 30
5y = 30
y = 6.
So, one point is (0, 6).
Assuming y = 0 and solving for x:
3x + 5(0) = 30
3x = 30
x = 10.
So, another point is (10, 0).
Graphing these two points and connecting them gives us the line 3x + 5y = 30.
Graph the line x = 9:
This is a vertical line passing through x = 9.
Graph the line y = 3:
This is a horizontal line passing through y = 3.
Plot the points A(0, 0), B(9, 0), C(9, 3), and D(0, 3) on the graph.
Step 2: Determine the feasible region by shading:
Shade the region that satisfies all the constraints. Since x ≥ 0 and y ≥ 0 are non-negativity constraints, we shade the region that lies in the first quadrant.
Shade the region below the line 3x + 5y ≤ 30.
Shade the region below and to the left of the line x ≤ 9.
Shade the region below and to the left of the line y ≤ 3.
Step 3: Identify the vertices of the feasible region:
The vertices of the feasible region are the points where the lines intersect.
By examining the graph, we find the following vertices:
Vertex 1: Point A(0, 0)
Vertex 2: Point E(0, 6)
Vertex 3: Point D(0, 3)
Vertex 4: Point C(9, 3)
Vertex 5: Point B(9, 0)
Therefore, the five vertices of the feasible region are A(0, 0), E(0, 6), D(0, 3), C(9, 3), and B(9, 0).
To determine the vertices of the feasible region, we need to solve the system of linear inequalities formed by the given constraints.
The first constraint is 3x + 5y ≤ 30. To graph this inequality, we need to convert it to the form y ≤ mx + b, where m is the slope and b is the y-intercept.
First, subtract 3x from both sides: 5y ≤ -3x + 30.
Then, divide both sides by 5 to isolate y: y ≤ (-3/5)x + 6.
The second constraint is x ≤ 9. This is a simple inequality that represents a vertical line at x = 9.
The third constraint is y ≤ 3. This is a simple inequality that represents a horizontal line at y = 3.
The fourth constraint is x ≥ 0, which represents a vertical line at x = 0.
The fifth constraint is y ≥ 0, which represents a horizontal line at y = 0.
To find the vertices, we need to identify the points where the lines formed by these inequalities intersect. Let's find them step by step:
1. Start with the point (0, 0), which is the intersection of the lines x = 0 and y = 0.
2. Move along the line y = 0 towards the right until you reach the line x = 9. This intersection point is (9, 0).
3. Move along the line x = 9 upwards until you reach the line y = 3. This intersection point is (9, 3).
4. Move along the line y = 3 towards the left until you reach the line y = (-3/5)x + 6. This intersection point is ((30/5), 3), which simplifies to (6, 3).
5. Finally, move along the line y = (-3/5)x + 6 downwards until you reach the point where it intersects with the line x = 0. This intersection point is (0, 6).
Therefore, the five vertices of the feasible region are:
(0, 0), (9, 0), (9, 3), (6, 3), and (0, 6).