Chapter 3 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 33Section Section 11

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Reading Graphs; Linear Equations in Two Variables

Interpret graphs.Write a solution as an ordered pair.Decide whether a given ordered pair is a solution of a given equation.Complete ordered pairs for a given equation.Complete a table of values.Plot ordered pairs.

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44

33

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66

55

3.13.13.13.1


Objective 11

Interpret graphs.

Slide 3.1 - 3


Interpret graphs.

Recall that a bar graph is used to show comparisons. It consists of a series of bars (or simulations of bars) arranged either vertically or horizontally. In a bar graph, values from two categories are paired with each other.

Slide 3.1 - 4

A line graph is used to show changes or trends in data over time. To form a line graph, we connect a series of points representing data with line segments.


EXAMPLE 1

Compare sales of motor scooters in 1999 and 2001.

Solution:

Sales were about 25 thousand in 1999 and about 50 thousand in 2001, so sales doubled over those years.

Interpreting a Bar Graph

Slide 3.1 - 5


Estimate the number of households that purchased a real tree in 2004.

About how much did the number of households purchasing real trees increase from 2002 to 2004?

EXAMPLE 2

Solution: about 27 million

Interpreting a Line Graph

Slide 3.1 - 6

Solution: about 5 million


Interpret Graphs. (cont’d)

A linear equation in two variables is an equation that can be written in the form

where A, B, and C are real numbers and A and B are not both 0.

Some examples of linear equations in two variables in this form, called standard form, are

and

Slide 3.1 - 7

Other linear equations in two variables, such as

and are not written in standard form, but could be. We discuss the forms of linear equations in more detail in Section 3.4.

,Ax B Cy

3 ,4 9x y ,0x y 8.2x y

4 5y x 3 ,7 2x y

Linear equations in two variables


Objective 22

Write a solution as an ordered pair.

Slide 3.1 - 8


A solution of a linear equation in two variables requires two numbers, one for each variable. For example, a true statement results when we replace x with 2 and y with 13 in the equation

since

Write a solution as an ordered pair.

Slide 3.1 - 9

4 5,y x 13 4 52 . The pair of numbers x = 2 and y = 13 gives a solution of the

equation The phrase “x = 2 and y = 13” is abbreviated 4 5.y x

2,13

x-value y-value

The x-value is always given first. A pair of numbers such as (2,13) is called an ordered pair, since the order in which the numbers are written is important.

The ordered pairs (2,13) and (13,2) are not the same. The second pair indicates that x = 13 and y = 2. For ordered pairs to be equal, their x-coordinates must be equal and their y-coordinates must be equal.


Objective 33

Slide 3.1 - 10

Decide whether a given ordered pair is a solution of a given equation.


Decide whether a given ordered pair is a solution of a given equation.

We substitute the x- and y-values of an ordered pair into a linear equation in two variables to see whether the ordered pair is a solution. An ordered pair that is a solution of an equation is said to satisify the equation.

Slide 3.1 - 11

When listing ordered pairs, be sure to always list the x-value first.

Infinite numbers of ordered pairs can satisfy a linear equation in two variables.


Decide whether each ordered pair is a solution of the equation

EXAMPLE 3

Solution:

4,20

5 2 20.x y

Deciding Whether Ordered Pairs Are Solutions of an Equation

Slide 3.1 - 12

2, 5 225 2 05

25 4 2 200

10 10 20 0 20

20 40 20 20 20

No

Yes


Objective 44

Slide 3.1 - 13

Complete ordered pairs for a given equation.


EXAMPLE 4

Solution:

2 9.y x

4 9y 22 9y

5y

8x

2, 2, 5

7 9 92 9x 16

2 2

2x

Completing Ordered Pairs

Slide 3.1 - 14

Complete each ordered pair for the equation

,7 8,7


Objective 55

Complete a table of values.

Slide 3.1 - 15


EXAMPLE 5

Complete the table of values for the equation Then write the results as ordered

pairs. Solution:

2 3 12.x y

2 0 3 12y

Completing Tables of Value

Slide 3.1 - 16

3 12

3 3

y

4y

4

26

3

2

2 3 120x 2 2

2 2

1x

6x

2 3 3 12y

3

3 6

3

y

2y

63 12 66 y 132 3 2x

2 3

2 2

x

3

2x

99 12 92x

0,4

3, 2 32

,3

6,0


Objective 66

Plot ordered pairs.

Slide 3.1 - 17


Plot ordered pairs.

Slide 3.1 - 18

Every linear in two variables equation has an infinite number of ordered pairs as solutions. Each choice of a number for one variable leads to a particular real number for the other variable.

To graph these solutions, represented as ordered pairs (x,y), we need two number lines, one for each variable. The two number lines are drawn as shown below. The horizontal number line is called the x-axis and the vertical line is called the y-axis. Together, these axes form a rectangular coordinate system, also called the Cartesian coordinate system.


The coordinate system is divided into four regions, called quadrants. These quadrants are numbered counterclockwise, starting with the one in the top right quadrant.

Slide 3.1 - 19

Plot ordered pairs. (cont’d)

Points on the axes themselves are not in any quadrant.

The point at which the x-axis and y-axis meet is called the origin, labeled 0 on the previous diagram. This is the point corresponding to (0, 0).

The x-axis and y-axis determine a plane— a flat surface illustrated by a sheet of paper. By referring to the two axes, we can associate every point in the plane with an ordered pair. The numbers in the ordered pair are called the coordinates of the point.

In a plane, both numbers in the ordered pair are needed to locate a point. The ordered pair is a name for the point.


For example, locate the point associated with the ordered pair (2,3) by starting at the origin.

Slide 3.1 - 20


Since the x-coordinate is 2, go 2 units to the right along the x-axis.

Since the y-coordinate is 3, turn and go up 3 units on a line parallel to the y-axis.

The point (2,3) is plotted in the figure to the right. From now on the point with x-coordinate 2 and y-coordinate 3 will be referred to as point (2,3).


Plot the given points in a coordinate system:

EXAMPLE 6

3,5 ,

Plotting Ordered Pairs

Slide 3.1 - 21

2,6 , 4,0 , 5, 2 , 5, 2 , 0, 6 .


EXAMPLE 7

3.563 7007.7.y x

Completing Ordered Pairs to Estimate the Number of Twin Births

Slide 3.1 - 22

Complete the table of ordered pairs for the equation, , where x = year and y = number

of twin births in thousands. Round answers to the nearest whole number. Interpret the results for 2002.

20013.563 7007.7y 7.129.6 7007.7y 122y

19993.563 7007.7y 7122.4 7007.7y 115y

20023.563 7007.7y 7133.1 7007.7y 125y

115 122 125

Solution:

There were about 125 thousand twin births in the U.S. in 2002.


The ordered pairs of twin births in the U.S. for 1998, 2000, and 2003 are graphed to the right. This graph of ordered pairs of data is called a scatter diagram. Notice how the how the axes are labeled: x represents the year, and y represents the number of twin births in thousands.


Slide 3.1 - 23

A scatter diagram enables us to tell whether two quantities are related to each other. These plotted points could be connected to form a straight line, so the variables x (years) and y (number of births have a linear relationship.


Think of ordered pairs as representing an input value x and an output value y. If we input x into the equation, the output is y. We encounter many examples of this type of relationship every day.

The cost to fill a tank with gasoline depends on how many gallons are needed; the number of gallons is the input, and the cost is the output

The distance traveled depends on the traveling time; input a time and the output is a distance.

The growth of a plant depends on the amount of sun it gets; the input is the amount of sun, and the output is growth.


Slide 3.1 - 24

Chapter 3 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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Transcript of Chapter 3 Section 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.