The Cartesian Coordinate System
Suppose we know the sum of two numbers is 6.
We can represent this situation with the equation x + y = 6.
There infinitely many possibilities for x and y. For example:
1 and 5 1 + 5 = 6 
4 and 10 4 + 10 = 6 
0 and 6 0 + 6 = 6 
We can visualize this relationship between x and y using two number lines,
one for x and the other for y.
First, draw a horizontal number line.
This is usually called the xaxis and is labeled with the variable x.
Next, draw a vertical number line perpendicular to the xaxis.
The two number lines should intersect at their zeros.
The vertical number line is usually called the yaxis and is labeled with the
variable y.
The point of intersection, the zero of each number line, is called the origin.
To make it easier to locate a point, we draw a rectangular grid as a
background.
The axes and the grid define a flat surface called the xyplane.
The number lines and grid form a rectangular coordinate system. We
typically use x and y for the variables, so a rectangular coordinate system
is often called an xycoordinate system. The French mathematician Rene
Descartes (15961650) is credited with developing this type of coordinate
system, so it is also referred to as the Cartesian coordinate system.
The x and yaxes divide the plane into four regions called quadrants. We
label these with Roman numerals I, II, III, and IV in a counterclockwise
direction beginning in the upper right.
Quadrant
I
II
III
IV 
Sign of x
positive
negative
negative
positive 
Sign of y positive
positive
negative
negative 
A point on an axis does not lie in a quadrant.
Example 1
State the quadrant in which each labeled point lies.
Solution
a. Quadrant II. Point A lies in the upper left quadrant.
b. Quadrant IV. Point B lies in the lower right quadrant.
c. None. Points on either axis do not lie in a quadrant.
d. Quadrant III. Point D lies in the lower left quadrant.
e. Quadrant I. Point E lies in the upper right quadrant.
