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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 81
Chapter 6, Problem 81

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 10y

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1
Identify the feasible region and its vertices from the given graph or constraints. The maximum and minimum values of a linear objective function over a polygonal feasible region occur at the vertices (corner points) of that region.
List the coordinates of each vertex of the feasible region. These points are where the constraints intersect and define the boundary of the feasible region.
Substitute the y-coordinate of each vertex into the objective function \$10y$ to find the value of the objective function at each vertex.
Compare the values obtained from each vertex to determine which is the maximum and which is the minimum value of the objective function over the feasible region.
State the maximum and minimum values along with the corresponding points in the feasible region where these values occur.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Objective Function

An objective function is a mathematical expression that defines the goal of an optimization problem, such as maximizing or minimizing a value. In this case, the function is 10y, meaning the value depends solely on the variable y. Understanding how to evaluate this function at different points is essential for finding its maximum and minimum.
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Feasible Region

The feasible region is the set of all possible points that satisfy the problem's constraints, often represented graphically as a polygon or area on the coordinate plane. The maximum and minimum values of the objective function must lie within this region, so identifying and understanding its boundaries is crucial.
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Optimization in Linear Programming

Optimization involves finding the highest or lowest value of the objective function within the feasible region. In linear programming, these extrema occur at the vertices (corner points) of the feasible region. Evaluating the objective function at each vertex helps determine the maximum and minimum values.
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Related Practice
Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA.

A=[−101011−1−10],B=[001010100]A = \(\left\)[ \(\begin{matrix}\) -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \(\end{matrix}\) \(\right\)]

Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[0−5−42],B=[3−1−54]A = \(\left\)[ \(\begin{matrix}\) 0 & -5 \\ -4 & 2 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 3 & -1 \\ -5 & 4 \(\end{matrix}\) \(\right\)]

Textbook Question

Find the maximum and minimum values of each objective function over the region of feasible solutions shown at the right. objective function = 3x + 5y

Textbook Question

Find the inverse, if it exists, for each matrix.

[2153]\(\left\)[ \(\begin{matrix}\) 2 & 1 \\ 5 & 3 \(\end{matrix}\) \(\right\)]

Textbook Question

Perform each operation, if possible.

[−2550143−4−1][10−1−10011−1]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \(\end{matrix}\) \(\right\)]

Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA. A=[01−1010001],B=[100010001]A = \(\left\)[ \(\begin{matrix}\) 0 & 1 & -1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \(\end{matrix}\) \(\right\)]