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.