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Zorluk: Çok zorSystems of Linear Inequalities in Two Variables

In the xyxy-plane, the region RR consists of all points (x,y)(x, y) that satisfy the system of inequalities below:

3x−4y≥−12y≥2x−7x+2y≥2\begin{aligned} 3x - 4y &\geq -12 \\ y &\geq 2x - 7 \\ x + 2y &\geq 2 \end{aligned}

If (x,y)(x, y) is a point in region RR, what is the maximum possible value of the expression x−2yx - 2y?

  1. A
    2.0
  2. 4.4Cevap
  3. C
    5.0
  4. D
    -5.2

Cevap

4.4
The maximum value of a linear expression over a closed, bounded polygonal region must occur at one of the vertices of that region. By solving the systems of equations for each pair of boundary lines, the vertices of the triangular region are found to be (8,9)(8, 9), (3.2,−0.6)(3.2, -0.6), and (−1.6,1.8)(-1.6, 1.8). Evaluating the expression x−2yx - 2y at each vertex gives −10-10 at (8,9)(8, 9), 4.44.4 at (3.2,−0.6)(3.2, -0.6), and −5.2-5.2 at (−1.6,1.8)(-1.6, 1.8). Comparing these values, 4.44.4 is the maximum possible value.

Adım Adım Çözüm

1
Find the intersection points of the boundary equations by solving the three systems of two linear equations.
System 1 (3x−4y=−123x - 4y = -12 and y=2x−7y = 2x - 7) yields (8,9)(8, 9).
System 2 (y=2x−7y = 2x - 7 and x+2y=2x + 2y = 2) yields (3.2,−0.6)(3.2, -0.6).
System 3 (3x−4y=−123x - 4y = -12 and x+2y=2x + 2y = 2) yields (−1.6,1.8)(-1.6, 1.8).
The boundaries of the region are defined by these lines, so the vertices of the region occur at their intersections.
2
Verify that the region defined by the inequalities is the interior of the triangle formed by these vertices by testing a point inside, such as (2,2)(2, 2).
The point (2,2)(2, 2) satisfies all three inequalities:
3(2)−4(2)=−2≥−123(2) - 4(2) = -2 \geq -12 (True)
2≥2(2)−7=−32 \geq 2(2) - 7 = -3 (True)
2+2(2)=6≥22 + 2(2) = 6 \geq 2 (True)
This confirms that the region is bounded and the vertices are the extreme points of the closed triangular region.
3
Evaluate the expression x−2yx - 2y at each of the three vertices.
At (8,9)(8, 9): 8−2(9)=−108 - 2(9) = -10
At (3.2,−0.6)(3.2, -0.6): 3.2−2(−0.6)=3.2+1.2=4.43.2 - 2(-0.6) = 3.2 + 1.2 = 4.4
At (−1.6,1.8)(-1.6, 1.8): −1.6−2(1.8)=−1.6−3.6=−5.2-1.6 - 2(1.8) = -1.6 - 3.6 = -5.2
A linear expression defined over a closed, bounded polygonal region reaches its maximum and minimum values at the vertices of that region.
4
Compare the evaluated values to identify the maximum value.
The maximum value is 4.44.4.
Among the three candidate values (−10-10, 4.44.4, and −5.2-5.2), 4.44.4 is the greatest.

Anahtar Kavram

Linear programming vertex method for systems of linear inequalities.
Tahmini Süre:3m 0s
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Konular

Systems of Linear Inequalities in Two Variables

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