3.1READING GRAPHS; LINEAR EQUATIONS IN TWO VARIABLES

Interpret graphs.

bar graph

line graph

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

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

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

Other linear equations in two variables, such as

and

,Ax B Cy

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

4 5y x 3 ,7 2x y

Linear equations in two variables

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.

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.

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

2, 5 225 2 05

25 4 2 200

10 10 20 0 20

20 40 20 20 20

No

Yes

Decide if the ordered pair is a solution to the equation

2x + 3y = 12

a. (3,2)

b. (-2,-7)

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

Complete each ordered pair for the equation

,7 8,7

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 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

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

X Y

0

0

3

-2

€

4x + 6y = 24

Plot ordered pairs.Every linear in two variables equation has an infinite number

of ordered pairs as solutions.

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.

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. This is the point corresponding to (0, 0).

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

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

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

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

What About Negative?

Plot the given points in a coordinate system:

3,5 , 2,6 , 4,0 , 5, 2 , 5, 2 , 0, 6 .

3.563 7007.7.y x 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.

This is what we call a “Scatter Diagram.” Whenever you make a diagram be sure to label the x and y axis as well as the Title.

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.

Homework:

• 2.8: 3-87 EOO• 3.1: 1-75 ODD