x>=0
y<=8
y>=x
y>=-1/2x+6
What is the minimum value for z=3x-1/2y over the feasibility region defined by the constraints shown above?%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A–4%0D%0A–4%0D%0A%0D%0A–5%0D%0A–5%0D%0A%0D%0A–3%0D%0A–3%0D%0A%0D%0A–6
To find the minimum value of z=3x-1/2y, we need to evaluate z at the corner points of the feasibility region.
The corner points of the feasibility region can be found by solving the system of linear inequalities:
x >= 0
y <= 8
y >= x
y >= -(1/2)x + 6
The corner points are:
(0, 8)
(0, 6)
(4, 6)
Let's evaluate z=3x-1/2y at these corner points:
1. For (0, 8):
z = 3(0) - 1/2(8) = 0 - 4 = -4
2. For (0, 6):
z = 3(0) - 1/2(6) = 0 - 3 = -3
3. For (4, 6):
z = 3(4) - 1/2(6) = 12 - 3 = 9
The minimum value of z=3x-1/2y over the feasibility region is -4.
To find the minimum value for z=3x-1/2y over the feasibility region defined by the given constraints, we need to follow these steps:
1. Graph the feasible region defined by the constraints:
- Start by graphing the line x = 0, which represents the constraint x >= 0. This line divides the xy-plane into two regions, with x values greater than or equal to 0 on one side.
- Next, graph the line y = 8, which represents the constraint y <= 8. This line divides the xy-plane into two regions, with y values less than or equal to 8 on one side.
- Then, graph the line y = x, which represents the constraint y >= x. This line forms a diagonal line from the origin (0,0) to the point (8,8), separating the xy-plane further.
- Finally, graph the line y = -(1/2)x + 6, which represents the constraint y >= -(1/2)x + 6. This line has a y-intercept of 6 and a slope of -1/2. It intersects the y-axis at (0,6) and the x-axis at (12,0).
The feasible region is the intersection of all these graphs. Shade or mark the region where all the constraints are satisfied.
2. Find the vertices of the feasible region:
The vertices are the points where the lines intersect. To find these points, solve the equations of each pair of intersecting lines. The vertices can be found at the points of intersection.
3. Substitute the x and y values from each vertex into the equation for z=3x-1/2y:
Calculate the value of z for each vertex by substituting the x and y values into the equation. The vertex with the smallest value of z is the minimum value for z over the feasibility region.
After going through these steps, you will obtain the values for z at each vertex. Identify the vertex with the smallest value, and that will be the minimum value for z over the feasibility region.
To find the minimum value of z=3x-1/2y over the feasibility region, we need to determine the feasible values of x and y and then substitute them into the objective function.
The given constraints are:
x >= 0 (Constraint 1)
y <= 8 (Constraint 2)
y >= x (Constraint 3)
y >= -1/2x+6 (Constraint 4)
First, let's analyze the given constraints to find the feasible region:
Constraint 1 (x >= 0) implies that x can take any nonnegative value.
Constraint 2 (y <= 8) restricts y to values less than or equal to 8.
Constraint 3 (y >= x) ensures that y is always greater than or equal to x.
Constraint 4 (y >= -1/2x+6) represents a linear inequality. To find the boundary line, we set y = -1/2x+6 and solve for x:
y = -1/2x + 6
2y = -x + 12
x = 12 - 2y
Plotting this line on a graph, we can see that it has a negative slope and passes through points (6, 0) and (0, 12).
Now, let's determine the feasible region by considering the boundaries defined by the above constraints:
1. x-axis: x >= 0
2. y-axis: y <= 8
3. Line: y = -1/2x + 6
4. y >= x
Considering these constraints, we find that the feasible region is the shaded area enclosed by the x-axis, y-axis, and the line y = -1/2x + 6.
Next, we substitute the corner points of the feasible region into the objective function z=3x-1/2y to find its minimum value.
Let's find the corner points:
1. (0, 0)
2. (0, 8)
3. (6, 0)
4. (8, 8)
Substituting these points into the objective function, we get:
1. z = 3(0) - 1/2(0) = 0
2. z = 3(0) - 1/2(8) = -4
3. z = 3(6) - 1/2(0) = 18
4. z = 3(8) - 1/2(8) = 20
The minimum value for z=3x-1/2y over the feasibility region is -4.