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)
.

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
3*x* + 2*y* - 4
?

3*x* + 2*y* - 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
.

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:

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:

Graphs of Three Points

Often, the
*xy*
-graph is drawn with the grid removed, and each interval labeled:

xy-graph

The point (0, 0) -- at the center of the graph -- is called the origin.