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Graphing Equations

Ordered Pairs

Terms

Problems

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:

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

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.

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