Problem 48 - Prealgebra

Question

OPEN ENDED Give an example of an ordered pair, and identify the \(x\) - and \(y\) -coordinate.

Step-by-Step Solution

Answer

Example ordered pair: (3, 5); x-coordinate: 3, y-coordinate: 5.

Step 1: Define an Ordered Pair

An ordered pair is a pair of elements written in a specific order, usually written in the form \(x, y\). It represents a point on the Cartesian coordinate system, where \x\ is the horizontal coordinate and \y\ is the vertical coordinate. Let's use the ordered pair \(3, 5\) as an example.

Step 2: Identify the x-coordinate

The first number in the ordered pair is the \(x\)-coordinate. For the ordered pair \(3, 5\), the \(x\)-coordinate is \x = 3\. This represents the position on the horizontal axis.

Step 3: Identify the y-coordinate

The second number in the ordered pair is the \(y\)-coordinate. For the ordered pair \(3, 5\), the \(y\)-coordinate is \y = 5\. This represents the position on the vertical axis.

Key Concepts

Understanding the Cartesian Coordinate System

The Cartesian coordinate system is a two-dimensional plane created by two perpendicular number lines, usually called the x-axis and y-axis. These axes divide the plane into four quadrants, which help us locate points using ordered pairs. The x-axis runs horizontally, while the y-axis runs vertically. Together, they form a grid-like structure that allows us to map mathematical relationships and data visually.

The origin, or the point where the x-axis and y-axis intersect, is designated as \(0, 0\) and is the central reference point in this system. By moving right or left along the x-axis and up or down the y-axis, we can plot any point using an ordered pair \(x, y\). This method provides a simple yet powerful framework for graphing equations and analyzing mathematical concepts.

The Role of the x-coordinate

The x-coordinate in an ordered pair \(x, y\) represents the point's location on the horizontal axis of the Cartesian coordinate system. The x-axis is akin to the spine of the graph, providing horizontal alignment for points.

For instance, in the ordered pair \(3, 5\), the x-coordinate is 3. This means the point is located 3 units to the right of the origin along the horizontal axis. If the x-coordinate were negative, it would indicate a position to the left of the origin. Understanding the x-coordinate is crucial for determining a point's position in a horizontal direction. It allows us to navigate side-to-side on the graphical plane.

The Importance of the y-coordinate

The y-coordinate in an ordered pair provides the vertical position of a point on the Cartesian coordinate system. It indicates how far up or down a point is from the x-axis, the horizontal line that runs through the origin.

Taking the previous example of the ordered pair \(3, 5\), the y-coordinate is 5. This tells us that the point is located 5 units above the x-axis, giving it vertical height on the graph. A negative value for the y-coordinate would mean the point is below the x-axis. Being aware of the y-coordinate's role helps us make sense of vertical movement within the coordinate plane, essential for understanding how data points relate to each other in a graphical context.