graph the system of constraints, name the vertices, then find the values of x and y that maximize or minimize the objective function.
x>=2
y>=0
3x+2y>=12
minimum for C = x + 5 y
please help me, im confused on how to start this
graph the three lines
clearly, the vertices are at (2,0), (2,3), (4,0)
now evaluate C at each vertex
To graph the system of constraints, let's start by plotting the lines for each constraint.
1. x >= 2:
This is a vertical line passing through x = 2. Shade the region to the right of this line to represent x >= 2.
2. y >= 0:
This is a horizontal line passing through y = 0. Shade the region above this line to represent y >= 0.
3. 3x + 2y >= 12:
This is a linear inequality. Let's find the line for 3x + 2y = 12 first.
To do this, we can find the points where the line intersects the x and y axes by setting x = 0 and y = 0, respectively.
When x = 0, we have: 3(0) + 2y = 12
Simplifying, we get: 2y = 12
Dividing both sides by 2, we obtain: y = 6
When y = 0, we have: 3x + 2(0) = 12
Simplifying, we get: 3x = 12
Dividing both sides by 3, we obtain: x = 4
Plotting these two points (0, 6) and (4, 0), we can draw the line passing through them.
Since the inequality is "greater than or equal to," we will shade the region above the line.
Now we have three shaded regions: to the right of x = 2, above y = 0, and above 3x + 2y = 12.
To find the vertices (corners) of this feasible region, we need to find the points where the boundary lines intersect. In this case, the vertices are where the lines intersect.
After finding the vertices, we can substitute the coordinate values into the objective function, C = x + 5y, to determine which vertex maximizes or minimizes the function.
Unfortunately, without additional information or a graph, it is not possible for me to determine the exact values of the vertices or the values of x and y that maximize or minimize the objective function.
To graph the system of constraints, we will start by plotting the equations on a coordinate plane. Then, we will shade the feasible region and find the vertices. Finally, we will use those vertices to determine the values of x and y that maximize or minimize the objective function.
1. Plot the equations on a coordinate plane:
- Start by drawing the x and y axes.
- Plot the line x = 2, which is a vertical line passing through 2 on the x-axis.
- Plot the line y = 0, which is a horizontal line passing through 0 on the y-axis.
- To plot the line 3x + 2y = 12, first convert it to slope-intercept form (y = mx + b):
2y = -3x + 12
y = (-3/2)x + 6
Plot the line using its slope of -3/2 (3 units down and 2 units to the right from any point) and its y-intercept of 6 (the point where it crosses the y-axis).
2. Shade the feasible region:
- Shade the region above the line y = 0 (since y must be greater than or equal to 0).
- Shade the region to the right of the line x = 2 (since x must be greater than or equal to 2).
- Shade the region above the line 3x + 2y = 12 (since 3x + 2y must be greater than or equal to 12).
3. Find the vertices:
- The vertices are the points where the shaded regions intersect.
- Each vertex represents a valid solution to the system of constraints.
4. Determine the values of x and y that maximize or minimize the objective function:
- Evaluate the objective function C = x + 5y at each vertex.
- The vertex with the minimum or maximum value of C will be the solution to the optimization problem.
To summarize, start by graphing the system of constraints and shading the feasible region. Then, find the vertices where the shaded regions intersect. Finally, evaluate the objective function at each vertex to find the values of x and y that maximize or minimize the function.