Problem 55 - Understanding Elementary Algebra with Geometry

Question

Why is the point \((x, y)\) called an ordered pair?

Step-by-Step Solution

Answer

The point (x, y) is called an ordered pair because the order of the coordinates uniquely identifies a specific location in a coordinate system.

Step 1: Understanding the Components

Consider the point (x, y). It consists of two numbers, x and y. The first number in this pair is called the x-coordinate, and the second number is called the y-coordinate.

Step 2: Order Matters

In an ordered pair, the sequence in which the coordinates are listed is crucial. The x-coordinate always comes first, and the y-coordinate comes second. This order must be maintained to accurately represent a specific point in a coordinate system.

Step 3: Unique Representation

The specific order of the coordinates ensures that each ordered pair uniquely identifies a point on a graph. Changing the order will lead to a different location. For example, (2, 3) is different from (3, 2).

Key Concepts

Ordered Pairs

An ordered pair, such as (x, y), is a fundamental concept in mathematics. It's called an 'ordered pair' because the order in which the numbers appear is important.
The first number in the pair is known as the x-coordinate, and the second is the y-coordinate.
This structure allows us to exactly pinpoint a location on a graph.

It's significant to remember that (x, y) is different from (y, x).
For instance, (2, 3) will be a different point on the graph from (3, 2).
Thus, the name 'ordered pair' emphasizes the necessity of sequence.
Changing the sequence of x and y will locate a different point in the coordinate system.

x-coordinate

The x-coordinate is the first number in an ordered pair (x, y).
It shows the horizontal position of the point on a graph.

To locate the x-coordinate, you start from the origin point (0,0) and move horizontally:

• Move right if the x-coordinate is positive.
• Move left if the x-coordinate is negative.

For example, in the pair (3, 2), the x-coordinate is 3.
This means you move 3 units to the right from the origin.

Getting comfortable with x-coordinates helps in better understanding and plotting points on a graph efficiently.

y-coordinate

The y-coordinate is the second number in an ordered pair (x, y).
It indicates the vertical position of the point on a graph.

To find the y-coordinate, after determining the x-coordinate, you move vertically:

• Move up if the y-coordinate is positive.
• Move down if the y-coordinate is negative.

For instance, in the pair (3, 2), the y-coordinate is 2.
This shows you need to move 2 units up from the x-position.

By understanding how to locate y-coordinates, you can accurately plot the vertical aspects of points on a coordinate system.