Which term best describes the solution of the situation represented by the system of inequalities? (Assume that x >= 0 and y >= 0.)
x + 2y <= 4
x – y <= 1
f(x, y) = 3x + 2y
The terms are
one optimal solution
unbounded
infeasible
alternate optimal solutionss
I don't know what the middle equation is.
x + 2y <= 4 transforms to:
y ≤ 2 - (x/2) ...(1)
x – y <= 1 transforms to:
y ≥ x-1
The two inequalities (or equalities) intersect at (2,1), which is an optimal point with the objective function
f(x,y)=3*2+2*1=8
See:
http://img80.imageshack.us/img80/4674/1285455307.png
To determine which term best describes the solution of the system of inequalities, we can utilize some concepts from linear programming. In this case, we have two inequalities:
1) x + 2y <= 4
2) x - y <= 1
To begin, let's represent these inequalities graphically. We will plot the lines representing these inequalities on a coordinate plane.
For the first inequality, x + 2y <= 4, we can rewrite it as:
2y <= -x + 4
y <= (-1/2)x + 2
For the second inequality, x - y <= 1, we rearrange it as:
y >= x - 1
Plotting these lines on a coordinate plane, we can shade the area that satisfies both inequalities:
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Now, we need to determine which region represents the feasible solution space. Since we have the additional constraints x >= 0 and y >= 0, we can further refine the shaded region. We only consider the portion of the shaded region that lies in the first quadrant (where both x and y are positive).
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Based on the shaded region, we can deduce that the feasible solution space is bounded. It is not extending to infinity, so the term "unbounded" does not best describe the solution.
Now, to determine whether there is one optimal solution or alternate optimal solutions, we will use the function f(x, y) = 3x + 2y.
We want to maximize this function within the feasible solution space. To do that, we can calculate the value of f(x, y) at the vertices (corners) of the shaded region.
By examining the vertices, we find that the maximum value of f(x, y) occurs at the point (2, 1), giving us f(2, 1) = 3(2) + 2(1) = 8.
Since there is only one point in the feasible solution space where the objective function reaches its maximum value, the term "one optimal solution" best describes the situation.
Therefore, the term that best describes the solution of the system of inequalities is "one optimal solution".