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4.1 Solving Systems of Linear Equations Graphically and Numerically 4.2 Solving Systems of Linear Equations by Substitution 4.3 Solving Systems of Linear Equations by Elimination 4.4 Systems of Linear Inequalities

Systems of Linear Equations in Two Variables

We can do anything we want to do if we stick to it long enough. —HELEN KELLER

mericans have been moving toward a more mobile lifestyle. In recent years, the percentage of U.S. households relying solely on mobile phone service has increased, while the percentage of households relying solely on landline phone service has decreased. The following figure shows that linear equations can be used to model the percentage P of households relying on each type of phone service during year x. What does the point of intersection represent? Telephone Service

P

40

A

Percent of Households

Landline-only Households

30 20 10 0 ’05 ’06 ’07 ’08 ’09 ’10

Wireless-only Households

x

Year

ISBN 1-256-49082-2

Source: National Center for Health Statistics.

This graph illustrates a system of linear equations. If we use the graph to estimate both the year and the percentage at the intersection point, we are solving the system of linear equations graphically. In this chapter we discuss graphical, numerical, and symbolic methods for solving systems of linear equations. 245

Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.

246

CHAPTER 4 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES

4.1

Solving Systems of Linear Equations Graphically and Numerically Basic Concepts ● Solutions to Systems of Equations

A LOOK INTO MATH N

In business, linear equations are sometimes used to model supply and demand for a product. For example, if the price of a gourmet coffee drink is too high, the demand for the drink will decrease because consumers are interested in saving money. Similarly, if the price of the coffee drink is too low, supply will decrease because suppliers are interested in making money. To find an appropriate price for the coffee drink, a system of linear equations can be solved. In this section, we will solve systems of linear equations graphically and numerically.

y

Basic Concepts

In Chapter 3 we showed that the graph of y = mx + b is a line with slope m and y-intercept b, as illustrated in Figure 4.1. Each point on this line represents a solution to the equation y = mx + b. Because there are infinitely many points on a line, there are infinitely many solutions to this equation. However, many applications require that we find one particular solution to a linear equation. One way to find such a solution is to graph a second line in the same xy-plane and determine the point of intersection (if one exists). N REAL-WORLD CONNECTION Consider the following application of a line. If renting a moving truck for one day costs $25 plus $0.50 per mile driven, then the equation C = 0.5x + 25 represents the cost C in dollars of driving the rental truck x miles. The graph of this line is shown in Figure 4.2(a) for x Ú 0.

x y = mx + b b

Figure 4.1

NEW VOCABULARY

n Intersection-of-graphs method n System of linear equations in two variables n Solution to a system n Inconsistent system n Consistent system with independent equations n Consistent system with dependent equations

Truck Rental Cost

C

125 100

Rental Cost of $75

C

125 100

Driving 100 miles costs $75. C = 75 (100, 75) C = 0.5x + 25

Cost (dollars)

Cost (dollars)

75 50

75 50 25

C = 0.5x + 25

25 0 25 50 75 100 125

x

0

25

50

75

100 125

x

Distance (miles) (a)

Distance (miles) (b)

STUDY TIP

Try to find a consistent time and place to study your notes and do your homework. When the time comes to study for an exam, do so at your usual study time in your usual place rather than “pulling an...