The constraints of a problem are listed below. What are the vertices of the feasible region?
x+y <= 7
x-2y <=-2
x >=0
y >=0
To find the vertices of the feasible region, we need to first graph the given constraints, and then identify the points where these constraints intersect. Here are the steps to find the vertices of the feasible region:
Step 1: Graph the first constraint: x + y <= 7
To graph x + y <= 7, we need to convert it into an equation by replacing the inequality sign with an equal sign:
x + y = 7
Now, plot the line x + y = 7 on a coordinate plane.
Step 2: Graph the second constraint: x - 2y <= -2
Similarly, we convert this inequality into an equation by replacing the inequality sign with an equal sign:
x - 2y = -2
Plot the line x - 2y = -2 on the same coordinate plane.
Step 3: Graph the third constraint: x >= 0
This is a simple constraint that represents the x-axis, so plot the x-axis (vertical line passing through x = 0).
Step 4: Graph the fourth constraint: y >= 0
This constraint represents the y-axis, so plot the y-axis (horizontal line passing through y = 0).
Step 5: Identify the intersection points of the lines
From the graph, identify the points where the lines intersect. These intersection points will be the vertices of the feasible region.
Step 6: Determine the coordinates of the vertices
By examining the graph, the vertices of the feasible region are:
Vertex 1: (0, 0) - the point where the x-axis and y-axis intersect.
Vertex 2: (0, 7) - the point where the x-axis and the line x + y = 7 intersect.
Vertex 3: (2, 5) - the point where the lines x + y = 7 and x - 2y = -2 intersect.
Therefore, the vertices of the feasible region are (0, 0), (0, 7), and (2, 5).
To find the vertices of the feasible region, we need to solve the given constraints simultaneously.
Let's start by graphing these inequalities on a coordinate plane:
1. The first constraint, x+y <= 7, can be rewritten as y <= -x + 7.
Plot the line y = -x + 7 by finding two points:
When x = 0, y = 7.
When y = 0, x = 7.
Draw a dashed line through these two points.
2. The second constraint, x - 2y <= -2, can be rewritten as y >= (1/2)x + 1.
Plot the line y = (1/2)x + 1 by finding two points:
When x = 0, y = 1.
When y = 0, x = -2.
Draw a dashed line through these two points.
3. The third constraint, x >= 0, means x values should be non-negative. Draw a vertical line at x = 0.
4. The fourth constraint, y >= 0, means y values should be non-negative. Draw a horizontal line at y = 0.
Now, the feasible region is the shaded region in the intersection of all the shaded regions of the four constraints.
To find the vertices of this region, we need to identify the points where the lines intersect each other.
However, looking at the graph, we can notice that the feasible region is a triangle. The vertices of this triangle occur at the corner points of the lines:
1. The intersection of the lines y = -x + 7 and y = (1/2)x + 1. Solve these two equations simultaneously to find the point (x, y).
2. The intersection of the lines x = 0 and y = -x + 7. Substitute the x-coordinate from the first step into the equation of x = 0 to find the point (x, y).
3. The intersection of the lines x = 0 and y = (1/2)x + 1. Substitute the x-coordinate from the first step into the equation of x = 0 to find the point (x, y).
These three points are the vertices of the feasible region.