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Ch. 5 - Systems of Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 5 - Systems of Equations and InequalitiesProblem 1
Chapter 6, Problem 1

Find the value of the objective function at each corner of the graphed region. What is the maximum value of the objective function? What is the minimum value of the objective function? 1. Objective Function z=5x+6y


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1
Identify the corner points of the shaded region from the graph. The points are (3, 4), (4, 8), (7, 7), and (9, 5).
Write down the objective function given: \(z = 5x + 6y\).
Calculate the value of the objective function at each corner point by substituting the coordinates into the function:
For each point \((x, y)\), compute \(z = 5 \times x + 6 \times y\).
Compare the calculated values of \(z\) at all corner points to determine which is the maximum and which is the minimum.

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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 linear expression that you want to maximize or minimize, such as z = 5x + 6y. It assigns a value to each point (x, y) in the feasible region, helping to determine the best solution based on given criteria.
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Feasible Region and Corner Points

The feasible region is the set of all points that satisfy the system of inequalities, often shown as a shaded polygon. The corner points (vertices) of this region are critical because the maximum or minimum values of a linear objective function occur at these points.
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Evaluating the Objective Function at Corner Points

To find the maximum or minimum value of the objective function, substitute the coordinates of each corner point into the function. Comparing these values identifies which corner yields the highest or lowest result, solving the optimization problem.
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Related Practice
Textbook Question

Determine whether the given ordered pair is a solution of the system. (−3,5)(- 3, 5)

{9x+7y=88x−9y=−69\(\begin{cases}\)9x + 7y = 8 \\8x - 9y = -69\(\end{cases}\)

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Textbook Question

Solve each system by the substitution method.

{x+y=2y=x2−4\(\begin{cases}\)x + y = 2 \(\y\) = x^2 - 4\(\end{cases}\)

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Textbook Question

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

{x+4y=142x−y=1\(\begin{cases}\) x + 4y = 14 \\ 2x - y = 1 \(\end{cases}\)

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Textbook Question

Solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

{y=4x+13x+2y=13\(\begin{cases}\) y = 4x + 1 \\ 3x + 2y = 13 \(\end{cases}\)

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Textbook Question

In Exercises 1–18, solve each system by the substitution method. {x+y=2y=x2−4x+4\(\begin{cases}\)x + y = 2 \(\y\) = x^2 - 4x + 4\(\end{cases}\)

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Textbook Question

Ggraph each inequality. x+2y≤8

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