Graphing Equations
Ordered Pairs
Ordered Pairs
An ordered pair is a pair of numbers in a specific order. For example, (1, 2) and (- 4, 12) are ordered pairs. The order of the two numbers is important: (1, 2) is not equivalent to (2, 1) -- (1, 2)≠(2, 1) .
Using Ordered Pairs to Represent Variables
Ordered pairs are often used to represent two variables. When we write (x, y) = (7, - 2) , we mean x = 7 and y = - 2 . The number which corresponds to the value of x is called the x-coordinate and the number which corresponds to the value of y is called the y-coordinate.
Example 1. If
(x, y) = (- 1, 4)
, what is the value of
3x + 2y - 4
?
3x + 2y - 4 = 3(- 1) + 2(4) - 4 = - 3 + 8 - 4 = 1
Example 2. Which of the following ordered pairs
(x, y)
are solutions to the equation
- 6 = 1
?
{(4, 1),(5, 2),(- 3, 1),(- 3, -1),(1, 4)}
(x, y) = (4, 1)
:
-6 = 
- 6 = 7 - 6 = 1
. Solution.
(x, y) = (5, 2)
:
-6 = 
-6 = 
-6 = - 
≠1
. Not a solution.
(x, y) = (- 3, 1)
:
-6 = 
-6 = - 7 - 6 = - 13≠1
. Not a solution.
(x, y) = (- 3, - 1)
:
-6 = 
- 6 = 7 - 6 = 1
. Solution.
(x, y) = (1, 4)
:
-6 = 
-6 = 
-6 = - 
≠1
. Not a solution.
Thus,
{(4, 1),(- 3, -1)}
are solutions to
- 6 = 1
.
Graphing Ordered Pairs
We have graphed values on the number line in pre- algebra and in earlier chapters of algebra. However, we can only graph points of one variable on the number line; thus, we need a 2-dimensional (2 variable) way of representing points -- the xy-graph:
To graph a point on the
xy
-graph, first find the
x
-coordinate on the
x
-axis. Then move up on the graph the number of spaces which is equal to the
y
-coordinate (or move down if the
y
-coordinate is negative). For example, to graph
(2, 3)
, find
2
on the
x
-axis. Then move up
3
spaces. To graph
(- 2, 1)
, find
-2
on the
x
-axis, then move up
1
space. To graph
(1.5, - 1)
, find
1.5
on the
x
-axis, then move down
1
space:
Often, the xy -graph is drawn with the grid removed, and each interval labeled:
The point (0, 0) -- at the center of the graph -- is called the origin.
