Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 5 - Systems and Matrices

Problem 77

Chapter 6, Problem 77

The graphs show regions of feasible solutions. Find the maximum and minimum values of each objective function. objective function = 3x + 5y

Verified step by step guidance

Identify the feasible region on the graph, which is the set of all points (x, y) that satisfy the given constraints. This region is typically a polygon formed by the intersection of linear inequalities.

List the coordinates of all the vertices (corner points) of the feasible region. These points are where the boundary lines intersect and are critical because the maximum and minimum values of a linear objective function occur at these vertices.

Write down the objective function, which is given as $3x + 5y$. This function will be evaluated at each vertex to find the maximum and minimum values.

Substitute the coordinates of each vertex into the objective function $3x + 5y$ to calculate the value of the objective function at each vertex.

Compare the calculated values from the previous step to determine which vertex gives the maximum value and which gives the minimum value of the objective function within the feasible region.

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Key Concepts

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

Feasible Region

The feasible region is the set of all points that satisfy the given constraints in a linear programming problem. It is typically represented as a polygonal area on a graph where all inequalities overlap. Understanding this region is crucial because the optimal solutions for the objective function lie within or on the boundary of this area.

Objective Function

An objective function is a linear expression, such as 3x + 5y, that we aim to maximize or minimize. It represents a quantity of interest, like profit or cost, depending on the problem context. Evaluating this function at points in the feasible region helps identify the best possible values.

Corner Point Method

The corner point method involves evaluating the objective function at each vertex (corner point) of the feasible region. Since the maximum or minimum values of a linear objective function occur at these vertices, this method efficiently finds optimal solutions without checking every point in the region.

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Related Practice

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x² - 4

How is the graph of y = | x - 1 | obtained by transforming the graph of y = | x |?

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

x + 2y + 3z = 4

4x + 3y + 2z = 1

-x - 2y - 3z = 0

Textbook Question

Use a system of equations to solve each problem. See Example 8. Find an equation of the line y = ax + b that passes through the points (-2, 1) and (-1, -2).

Textbook Question

Perform each operation, if possible.

$[\begin{matrix} - 1 & 0 \\ 2 & 5 \end{matrix}] [\begin{matrix} - 3 & 4 \\ 2 & 8 \end{matrix}]$

Textbook Question

Use Cramer's rule to solve each system of equations. If D = 0, then use another method to determine the solution set. See Examples 5–7.

-2x - 2y + 3z = 4

5x + 7y - z = 2

2x + 2y - 3z = -4

Textbook Question

Consider the following nonlinear system. Work Exercises 75 –80 in order.

y = | x - 1 |

y = x² - 4

Use the definition of absolute value to write y = | x - 1 | as a piecewise-defined function.

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