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Difficulty: Very hardSystems of Linear Inequalities in Two Variables

In the xyxy-plane, the solution set to the system of inequalities below is a bounded region.

y≤−2x+21y≤−12x+9y≥−x+5x≥2y≥1\begin{aligned} y &\leq -2x + 21 \\ y &\leq -\frac{1}{2}x + 9 \\ y &\geq -x + 5 \\ x &\geq 2 \\ y &\geq 1 \end{aligned}

What is the maximum possible value of 3x+2y3x + 2y for a point (x,y)(x, y) in this region?

  1. A
    22
  2. B
    31
  3. C
    32
  4. 34Answer

Answer

34
The correct value is 34. The solution set to the system of inequalities is a bounded pentagonal region. Evaluating the expression 3x+2y3x + 2y at each vertex of this region yields: 1212 at (2,3)(2, 3), 2222 at (2,8)(2, 8), 3434 at (8,5)(8, 5), 3232 at (10,1)(10, 1), and 1414 at (4,1)(4, 1). Comparing these values shows that the maximum value is 34.

Step-by-Step Solution

1
Understand the behavior of the objective function over the region
The maximum value of a linear expression 3x+2y3x + 2y over a closed, bounded polygonal region must occur at one of the vertices (corner points) of the region.
This is a fundamental theorem of linear programming, which simplifies the search for the maximum value to only the boundary intersections of the system.
2
Determine the vertices of the feasible region
The boundary lines are y=−2x+21y = -2x + 21, y=−12x+9y = -\frac{1}{2}x + 9, y=−x+5y = -x + 5, x=2x = 2, and y=1y = 1. Finding their intersections that satisfy all inequalities yields five vertices: (2,3)(2, 3), (2,8)(2, 8), (8,5)(8, 5), (10,1)(10, 1), and (4,1)(4, 1).
Solving the pairwise equations of lines that bound the shaded region determines the exact coordinates of all corner points.
3
Evaluate the expression 3x+2y3x + 2y at each of the five vertices
At (2,3)(2, 3): 3(2)+2(3)=123(2) + 2(3) = 12
At (2,8)(2, 8): 3(2)+2(8)=223(2) + 2(8) = 22
At (8,5)(8, 5): 3(8)+2(5)=343(8) + 2(5) = 34
At (10,1)(10, 1): 3(10)+2(1)=323(10) + 2(1) = 32
At (4,1)(4, 1): 3(4)+2(1)=143(4) + 2(1) = 14
Calculating the value at each candidate vertex allows us to compare and find the absolute maximum.
4
Compare the evaluated values
The maximum value is 3434, occurring at the vertex (8,5)(8, 5).
Comparing all calculated values shows that 34 is the largest possible value.

Key Concept

Linear Programming and Bounded Systems of Inequalities
Estimated Time:3m 0s
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Topics

Systems of Linear Inequalities in Two Variables

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