#### 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 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.

#### 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:

xy-graph

The horizontal axis, called the x-axis, represents values of

*x*, and the vertical axis, called the y-axis, represents values of

*y*. From now on, the word "graph" will refer to the

*xy*-graph, unless specified otherwise.

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