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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 56
Chapter 6, Problem 56

Find the value of the objective function z = 2x + 3y at each corner of the graphed region shown. What is the maximum value of the objective function? What is the minimum value of the objective function?
Graph showing a shaded region for systems of inequalities with corner points labeled.

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1
Identify the corner points of the shaded region from the graph. The corner points are (0, 0), (0, 13), (4, 5), and (5, 0).
Substitute each corner point into the objective function z = 2x + 3y to calculate the value of z at each point. For example, for the point (0, 0), substitute x = 0 and y = 0 into the equation.
Repeat the substitution for the other corner points: (0, 13), (4, 5), and (5, 0). For each point, calculate the value of z = 2x + 3y.
Compare the calculated values of z for all the corner points. The maximum value of z corresponds to the maximum value of the objective function, and the minimum value of z corresponds to the minimum value of the objective function.
State the corner point where the maximum value occurs and the corner point where the minimum value occurs, along with their respective z values.

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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 in the form of maximizing or minimizing a value. In this case, the objective function is z = 2x + 3y, which needs to be evaluated at specific points to find its maximum and minimum values within a defined region.
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Corner Points

Corner points, or vertices, of a feasible region are the points where the boundary lines intersect. In linear programming, the maximum and minimum values of the objective function occur at these corner points. The graph shows the corner points (0,0), (0,13), (4,5), and (5,0), which are essential for evaluating the objective function.
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Feasible Region

The feasible region is the area on a graph that satisfies all constraints of a linear programming problem. It is typically bounded by the lines representing the constraints. The shaded region in the graph indicates the feasible region where the objective function will be evaluated to determine its maximum and minimum values.
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Related Practice
Textbook Question

Solve each system for x and y, expressing either value in terms of a or b, if necessary. Assume that a ≠ 0 and b ≠ 0. For the linear function f(x) = mx + b, f(−2) = 11 and ƒ(3) = -9. Find m and b.

Textbook Question

Exercises 57–59 will help you prepare for the material covered in the next section. Subtract: 3x−4−2x+2\(\frac{3}{x - 4}\) - \(\frac{2}{x + 2}\)

Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x2+y2<16, y≥2x

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Textbook Question

In Exercises 27–62, graph the solution set of each system of inequalities or indicate that the system has no solution. x2+y2≤1, y−x2>0

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Textbook Question

In Exercises 57–59, graph the region determined by the constraints. Then find the maximum value of the given objective function, subject to the constraints. This is a piecewise function. Refer to the textbook.

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Textbook Question

Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.