4-1

C H A P T E R F O U R

Systems of Linear Equations and Inequalities

C H A P T E R F O U R C O N T E N T S

4.1 Solving Systems of Linear Equations by Graphing4.2 Solving Systems of Linear Equations

by Substitution4.3 Solving Systems of Linear Equations

by Elimination4.4 Applications of Linear Systems4.5 System of Linear Inequalities4.6 Systems of Linear Equations in Three Variables

4.1 Solving Systems of Linear Equations by GraphingTHINGS TO KNOWBefore working through this section, be sure you are familiar VIDEO ANIMATION INTERACTIVEwith the following concepts:

1. Determine If an Ordered Pair Is a Solution to an Equation (Section 3.2, Objective 1)

2. Graph a Line Using the Slope and a Point (Section 3.3, Objective 3)

3. Use the Slope-Intercept Form to Graph a Linear Equation (Section 3.4, Objective 2)

4. Determine the Relationship between Two Lines (Section 3.4, Objective 6)

OBJECTIVES

1 Determine If an Ordered Pair Is a Solution to a System of Linear Equations in Two Variables

2 Determine the Number of Solutions to a System Without Graphing

3 Solve Systems of Linear Equations by Graphing

You Try It

You Try It

You Try It

You Try It

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OBJECTIVE 1 DETERMINE IF AN ORDERED PAIR IS A SOLUTION TO A SYSTEM OF LINEAR EQUATIONS IN TWO VARIABLES

Figure 1 shows that the approximate revenue for digital music is projected to increasesteadily through 2014, while revenue for physical music (CDs and vinyl) is projected todecrease. The -coordinate of the point where the two lines intersect gives the year whenrevenues are expected to be equal for both music types, and the -coordinate of the inter-section point gives the expected revenue when the values are equal for both music types.

yx

4-2 Chapter 4 Systems of Linear Equations and Inequalities

(eText Screens 4.1-1–4.1-17)

2

1

4

3

6

5

12 13 14 15111098

Years since 2000

Digital and Physical Recorded MusicRevenue, 2008–2014

Rec

orde

d M

usic

Rev

enue

($

billi

ons) y � �0.56x � 9.5

y � 0.29x � 0.7

DigitalPhysical

(12, 2.78)

Source: Forrester Research Internet Music Forecast, 12/09

Figure 1

Figure 1 illustrates the graph of a system of linear equations in two variables

Definition System of Linear Equations in Two Variables

A system of linear equations in two variables is a collection of two or more linearequations in two variables considered together.

Note: Although a system of linear equations in two variables can contain more than twoequations, we will only consider cases involving two equations in this text.

The system graphed in Figure 1 is given by the equations .

The following examples also represent systems of linear equations in two variables.

Within a given system of linear equations in two variables, the same two variables will beused. For example, the variables and are used in both equations of the first system,while the variables and are used in the second system.

From Section 3.2 we know that an ordered pair is a solution to an equation in two variablesif a true statement results when substituting the values for the variables. Each ordered pairsolution is a point on the graph of the equation. The intersection point in Figure 1, ,lies on the graph of both equations, so it is a solution to both equations. Substituting for and for in either equation results in a true statement (view the checks). Theordered pair is an example of a solution to a system of linear equations.112, 2.782

y2.78x12

112, 2.782

bayx

b10p + 3q = 28

q = -5d 1

3 a -

2

5 b = 6

-

1

6 a +

4

3 b =

3

7

b5x - y = -9

3x + 4y = 15

b y = 0.29x - 0.7

y = -0.56x + 9.5

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