3.1 Linear Equations in Two Variables; The Rectangular Coordinate System

Objective 1

Interpret graphs.

Slide 3.1-3

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.

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.

Slide 3.1-4

Interpret graphs.

Refer to the line graph below.Estimate the average price of a gallon of gasoline in 2001.

About how much did theaverage price of a gallonof gasoline decreasefrom 2001 to 2002?

Solution:

about $1.45

about $0.10

Slide 3.1-5

Interpreting a Line GraphCLASSROOM EXAMPLE 1

Linear Equation in Two VariablesA 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.

,Ax B Cy

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

and3 ,4 9x y ,0x y 8.2x y Linear equations in two variables

Slide 3.1-6

Interpret graphs. (cont’d)

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.

4 5y x 3 ,7 2x y

Objective 2

Write a solution as an ordered pair.

Slide 3.1-7

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

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

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. 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.

Slide 3.1-8

Write a solution as an ordered pair.

Objective 3

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

Slide 3.1-9

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.

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.

Slide 3.1-10

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

Decide whether each ordered pair is a solution of the equation.

Solution:

4,20

5 2 20x y

2, 5 225 2 05

25 4 2 200

10 10 20 0 20

20 40 20 20 20

No

Yes

Slide 3.1-11

Deciding Whether Ordered Pairs Are Solutions of an EquationCLASSROOM EXAMPLE 2

Objective 4

Complete ordered pairs for a given equation.

Slide 3.1-12

Solution:

2 9y x

4 9y 22 9y

5y

8x

2, 2, 5

7 9 92 9x 162 2

2x

Complete each ordered pair for the equation.

,7 8,7

Slide 3.1-13

Completing Ordered PairsCLASSROOM EXAMPLE 3

Objective 5

Complete a table of values.

Slide 3.1-14

Ordered pairs are often displayed in a table of values. Although we usually write tables of values vertically, they may be written horizontally.

Slide 3.1-15

Complete a table of values.

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

Solution:

2 3 12x y 2 0 3 12y

3 123 3y

4y

4

26

32

2 3 120x 2 22 2

1x

6x

2 3 3 12y

33 6

3y

2y

63 12 66 y 132 3 2x

2 32 2x

32

x

99 12 92x

0, 4

3, 2 32,3

6,0

Slide 3.1-16

Completing Tables of ValuesCLASSROOM EXAMPLE 4

Objective 6

Plot ordered pairs.

Slide 3.1-17

Every linear in two variables equation has an infinite number of ordered pairs (x, y) 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 thex-axis and the vertical line is calledthe y-axis. Together, these axisform a rectangular coordinate system, also called the Cartesian coordinate system.

Slide 3.1-18

Plot ordered pairs.

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, 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.

Slide 3.1-19

Plot ordered pairs. (cont’d)

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.

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).

Slide 3.1-20

Plot ordered pairs. (cont’d)

Plot the given points in a coordinate system:

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

Slide 3.1-21

Plotting Order PairsCLASSROOM EXAMPLE 5

3.049 5979,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 2005.

2003 5.049 5979y 6113.2 5979y 134.2y

2003 2.049 5979y 6104.1 5979y 125.1y 125 134

Solution: There were about 134 thousand twin births in the U.S. in 2005.

Slide 3.1-22

Completing Ordered Pairs to Estimate the Number of Twin BirthsCLASSROOM EXAMPLE 6

The ordered pairs of twin births in the U.S. for 2001 until 2006 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.

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.

Slide 3.1-23

Plot ordered pairs. (cont’d)

Do not assume that this scatter diagram or resulting equation would provide reliable data for other years, since the data for those years may not follow the same pattern.

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, andthe cost is the output

The distance traveled depends on the traveling time; inputa 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

Plot ordered pairs. (cont’d)