Ordered pairs
In mathematics, an ordered pair is a set of two numbers usually written in the form (a, b). The order of the two numbers is important—(a, b) is different from (b, a) unless a equals b.
Ordered pairs are commonly used to specify a location on a map or coordinate plane. Sometimes, a map will use a letter to represent one of the numbers in the ordered pair.
In the above map of Central America, coordinates are used to specify the positions of a few countries: Mexico (B, 1), Belize (B, 3), Guatemala (C, 2), Honduras (C, 3), and Nicaragua (D, 4). On the coordinate plane to the right, points A and B are specified using the ordered pairs (3, -1) and (2, -3), respectively.
Cartesian coordinate plane
The use of ordered pairs is most often seen in the Cartesian coordinate plane. An ordered pair, (x, y), is a set of numbers that tells us the coordinates of a point in the coordinate plane. The first number, called the x-coordinate, corresponds to a position on the x-axis, and the second number, called the y-coordinate, corresponds to a position on the y-axis.
Finding a point in the coordinate plane
To determine the position of a given an ordered pair, (x, y), start from the origin (0, 0), then count x squares along the x-axis to determine the horizontal position of a point. To determine vertical position, count y squares along the y-axis, as shown in the figure below.
Ordered pairs as solutions
The solution to a two-variable equation or graph of an equation can also be written as an ordered pair.
Below are three of the unlimited many solutions to the equation y = -x + 3:
(-1, 4), (0, 3), and (2, 1).
We can verify this by testing whether or not:
1. the ordered pair satisfies the equation (makes the equation a true statement when the coordinates of the ordered pair are substituted into the equation).
Substituting the ordered pairs:
| (-1, 4) | (0, 3) | 2, 1) |
|---|---|---|
| 4 = -(-1) + 3 4 = 1 + 3 4 = 4 |
3 = -(0) + 3 3 = 3 |
1 = -(2) + 3 1 = -2 + 3 1 = 1 |
2. the ordered pairs are points on the graph of the equation.
Since the three points are on the graph of y = -x + 3, shown above, the ordered pairs are solutions to the equation.
Ordered pairs as vectors
In physics, a vector can be represented as an ordered pair, <x, y>, where the first number is called the vector's x-component and the second number is called the vector's y-component. The ordered pair indicates the position of a point on the coordinate plane through which a vector, starting at the origin, passes.
Vectors <2, 4> and <2, -3> are graphed in the coordinate plane above.