Beginning Algebra, 11th Edition

SECTION 3.1 Linear Equations in Two Variables; The Rectangular Coordinate System^177

Actual Data Source:U.S. Department
of Energy.

The line graph in FIGURE 1relates years to average prices for a gallon of gasoline.

We can also represent these two related quantities using a table of data, as shown in

the margin. In table form, we can see more precise data rather than estimating it.

Trends in the data are easier to see from the graph, which gives a “picture” of the data.

We can extend these ideas to the subject of this chapter, linear equations in two

variables. A linear equation in two variables, one for each of the quantities being re-

lated, can be used to represent the data in the table or graph. The graph of a linear

equation in two variables is a line.

Average Price

Year (in dollars per gallon)

2001 1.46

2002 1.36

2003 1.59

2004 1.88

2005 2.30

2006 2.59

2007 2.80

2008 3.25

Linear Equation in Two Variables

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

where A, B, and Care real numbers and Aand Bare not both 0.

AxByC,

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

form,are

3 x+ 4 y=9, x-y=0, and x+ 2 y=-8.

⎧⎨⎩

Linear equations

in two variables

NOTE Other linear equations in two variables, such as

and

are not written in standard form, but could be algebraically rewritten in this form. We

discuss the forms of linear equations in more detail in Section 3.4.

y= 4 x+ 5 3 x= 7 - 2 y,

OBJECTIVE 2 Write a solution as an ordered pair.Recall from Section 1.3

that a solutionof an equation is a number that makes the equation true when it

replaces the variable. For example, the linear equation in onevariable

has solution 7 , since replacing xwith 7 gives a true statement.

A solution of a linear equation in twovariables requires twonumbers, one for

each variable.For example, a true statement results when we replace xwith 2 and y

with 13 in the equation since

Let and.

The pair of numbers and gives a solution of the equation

The phrase “ and ” is abbreviated

x-value y-value

Ordered pair

with the x-value, 2, and the y-value, 13, given as a pair of numbers written inside

parentheses. The x-value is always given first.A pair of numbers such as is

called an ordered pair.

1 2, 13 2

12 , 132

x= 2 y= 13

x= 2 y= 13 y= 4 x+ 5.

13 = 4122 + 5. x= 2 y= 13

y= 4 x+ 5,

x- 2 = 5