Intermediate Algebra (11th edition)

SECTION 3.1 The Rectangular Coordinate System 137

The position of any point in this plane is determined by referring to the horizontal

number line, or x-axis,and the vertical number line, or y-axis.The x-axis and the

y-axis make up a rectangular(or Cartesian,for Descartes) coordinate system.

In an ordered pair, the first component indicates position relative to the x-axis, and

the second component indicates position relative to the y-axis. For example, to locate,

or plot,the point on the graph that corresponds to the ordered pair we move

three units from 0 to the right along the x-axis and then two units up parallel to the

y-axis. See FIGURE 3. The numbers in an ordered pair are called the coordinatesof the

corresponding point.

We can apply this method of locating ordered pairs to the line graph in FIGURE 1.

We move along the horizontal axis to a year and then up parallel to the vertical axis

to approximate spending for that year. Thus, the ordered pair indicates

that in 2006 personal spending on medical care was about $1800 billion.

1 2006, 1800 2

x 1 3, 2 2 ,

y

Quadrant III Quadrant IV

Quadrant II Quadrant I

(–5, 6)

(–5, 0)

(0, 0)

(0, –6)

(4, –1)

(3, 2)

(– 4, –5)

FIGURE 3

CAUTION The parentheses used to represent an ordered pair are also used to

represent an open interval (Section 1.1). The context of the discussion tells whether

ordered pairs or open intervals are being represented.

The four regions of the graph, shown in FIGURE 3, are called quadrants I, II, III,

and IV,reading counterclockwise from the upper right quadrant. The points on the

x-axis and y-axis do not belong to any quadrant.

OBJECTIVE 3 Find ordered pairs that satisfy a given equation.Each solu-

tion of an equation with two variables, such as

includes two numbers, one for each variable. To keep track of which number goes

with which variable, we write the solutions as ordered pairs. (If x and y are used as

the variables, the x-value is given first.)For example, we can show that is a

solution of by substitution.

Let

Multiply.

✓ True

Because the ordered pair makes the equation true, it is a solution. On the

other hand, is nota solution of the equation

Let

Multiply.

False

To find ordered pairs that satisfy an equation, select a number for one of the vari-

ables, substitute it into the equation for that variable, and solve for the other variable.

Since any real number could be selected for one variable and would lead to a

real number for the other variable, linear equations in two variables have an

infinite number of solutions.

13 = 6

10 + 3  6

2152 + 3112  6 x=5,y=1.

2 x+ 3 y= 6

1 5, 1 2 2 x+ 3 y=6.

1 6, - 22

6 = 6

12 - 6  6

2162 + 31 - 22  6 x=6, y=- 2.

2 x+ 3 y= 6

2 x+ 3 y= 6

1 6, - 22

2 x+ 3 y= 6,

Use parentheses

to avoid errors.