Algebra 2
Algebra 2
1st Edition
ISBN: 9780078884825
Author: McGraw-Hill/Glencoe
Publisher: Glencoe/McGraw-Hill School Pub Co
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Chapter 3.4, Problem 5CYU
To determine

To calculate: The coordinates of the vertices of the feasible region formed by graphing the given inequalities −3≤y≤7, 4y≥4x−8 and 6y+3x≤12 . Also maximum and minimum values of the function f(x,y)=−12x+9y in this region.

Expert Solution & Answer
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Answer to Problem 5CYU

The graph of the given system of inequalities is,

  Algebra 2, Chapter 3.4, Problem 5CYU , additional homework tip 1

The coordinates of the feasible region are A(−10,7), B(−1,−3) and C(2.667,0.667).

The maximum value of the function is either unknown or does not exist and minimum value of the function is −26.001 at (2.667,0.667).

Explanation of Solution

Given information:

The inequalities −3≤y≤7, 4y≥4x−8 and 6y+3x≤12 and the function f(x,y)=−12x+9y.

Formula used:

Linear programming is a technique to find the maximum and the minimum value of a given function over a given system of some inequalities, with each inequality representing a constraint. Graph the inequalities and obtain the vertices of the feasible region (solution set). Substitute the coordinates of the feasible region in the function and determine the maximum and the minimum value.

Calculation:

Consider the provided system of inequalities −3≤y≤7, 4y≥4x−8 and 6y+3x≤12.

Recall that linear programming is a technique to find the maximum and the minimum value of a given function over a given system of some inequalities, with each inequality representing a constraint. Graph the inequalities and obtain the vertices of the feasible region (solution set).

Graph the given inequalities and locate the vertices of the feasible region.

  Algebra 2, Chapter 3.4, Problem 5CYU , additional homework tip 2

Since, it is an unbounded region, the vertices of the region cannot be accurately determined. There can be infinite many points in this unbounded region, so, take the corner points where the system of inequalities cut each other, A(−10,7), B(−1,−3) and C(2.667,0.667).

Now, to find the maximum and minimum value of the function, substitute the coordinates of the feasible region in the function and determine the maximum and the minimum value.

So, substitute A(−10,7), B(−1,−3) and C(2.667,0.667) in the function f(x,y)=−12x+9y and evaluate the maximum and minimum value as,

  (x,y)−12x+9yf(x,y)(−10,7)−12(−10)+9(7)176(−1,−3)−12(−1)+9(−3)−15(2.667,0.667)−12(2.667)+9(0.667)−26.001

From the above table, it is observed that the minimum value of the function is −26.001 at (2.667,0.667) because there are no more points below which can give minimum value of the function.

The maximum value of function is unknown or does not exist because there are infinite many points in the unbounded region which can give different values more than 176 at different points. So, the maximum value of function is either unknown or does not exist.

Thus, graph of the given system of inequalities is,

  Algebra 2, Chapter 3.4, Problem 5CYU , additional homework tip 3

The coordinates of the feasible region are A(−10,7), B(−1,−3) and C(2.667,0.667) and the maximum value of the function is either unknown or does not exist and minimum value of the function is −26.001 at (2.667,0.667) .

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Chapter 3 Solutions

Algebra 2

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