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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 79
Chapter 6, Problem 79

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

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1
Identify the feasible region defined by the constraints given in the problem (usually shown graphically or by inequalities). This region represents all possible values of \(x\) and \(y\) that satisfy the constraints.
List the corner points (vertices) of the feasible region. These points are where the constraint lines intersect and are critical because the maximum and minimum values of a linear objective function over a polygonal region occur at these vertices.
Write down the objective function \(Z = 3x + 5y\) and prepare to evaluate it at each vertex of the feasible region.
Substitute the coordinates of each vertex into the objective function \(Z = 3x + 5y\) to calculate the value of \(Z\) at those points.
Compare the values of \(Z\) obtained at each vertex to determine which is the maximum and which is the minimum value of the objective function over the feasible region.

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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, typically to maximize or minimize its value. In this case, the function 3x + 5y represents a linear combination of variables x and y, whose values we want to optimize over a given region.
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Feasible Region

The feasible region is the set of all possible points (x, y) that satisfy the problem's constraints, often represented as inequalities. This region is usually a polygon or polyhedron in linear programming, and the optimal values of the objective function lie within or on the boundary of this region.
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Linear Programming and Optimization

Linear programming involves finding the maximum or minimum value of a linear objective function subject to linear constraints. The optimal solution for such problems occurs at the vertices (corner points) of the feasible region, so evaluating the objective function at these points determines 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=[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 = 10y

Textbook Question

Use a system of equations to solve each problem. Find an equation of the parabola y = ax2 + bx + c that passes through the points (2, 3), (-1, 0), and (-2, 2).

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Textbook Question

For each pair of matrices A and B, find (a) AB and (b) BA. See Example 7. A=[34−21],B=[605−2]A = \(\left\)[ \(\begin{matrix}\) 3 & 4 \\ -2 & 1 \(\end{matrix}\) \(\right\)], \(\quad\) B = \(\left\)[ \(\begin{matrix}\) 6 & 0 \\ 5 & -2 \(\end{matrix}\) \(\right\)]

Textbook Question

Perform each operation, if possible.

[32−1406][−200231]\(\left\)[ \(\begin{matrix}\) 3 & 2 & -1 \\ 4 & 0 & 6 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) -2 & 0 \\ 0 & 2 \\ 3 & 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 \(\end{matrix}\) \(\right\)]