Problem 63 - Precalculus Essentials

Question

Explain why (5,-2) and (-2,5) do not represent the same point.

Step-by-Step Solution

Answer

The points (5, -2) and (-2, 5) don't represent the same point because they have different positions relative to the origin. (5, -2) is 5 units to the right and 2 units below the origin, while (-2, 5) is 2 units to the left and 5 units above the origin.

Step 1: Understand Cartesian coordinate system

In a Cartesian coordinate system, each point is identified by a pair of coordinates, represented as (x, y). The x-coordinate shows the position of the point along the horizontal axis and the y-coordinate shows the position along the vertical axis.

Step 2: Compare the coordinates

The first given pair of coordinates is (5, -2). This means the point is 5 units to the right and 2 units below the origin (0, 0).\n\nThe second pair of coordinates is (-2, 5). This means the point is 2 units to the left and 5 units above the origin. Because their relative positions to the origin are different, they can't be the same point.

Key Concepts

Coordinates

In the world of mathematics, when we discuss the position of a point in a plane, we talk about coordinates. Coordinates are represented as an ordered pair \((x, y)\), where:

• The first number, \(x\), is known as the x-coordinate and deals with the horizontal placement.
• The second number, \(y\), is the y-coordinate, which manages the vertical placement.

Together, these two numbers tell us exactly where to find a point in a two-dimensional space.
Since it’s all about positioning, the order of these numbers matters greatly. For instance, the coordinates \((5, -2)\) are different from \((-2, 5)\) as they point to completely different places on the plane. It's crucial to always keep the order correct to ensure you find the intended location.

x-axis

The x-axis is a fundamental part of the Cartesian coordinate system. It runs horizontally across the plane and is used to measure how far "left or right" a point is from the origin.

• Points on the x-axis have their y-coordinate as 0.
• You move right for positive x-coordinates and left for negative x-coordinates.

For example, the point (5, -2) has an x-coordinate of 5, which places it 5 units to the right of the origin. On the other hand, the point (-2, 5) has an x-coordinate of -2, meaning it is 2 units to the left of the origin.
Understanding the x-axis helps you visualize horizontal displacement in the coordinate system.

y-axis

In contrast to the x-axis, the y-axis runs vertically along the Cartesian plane. The y-axis is essential in determining how far "up or down" a point is located from the origin.

• Points on the y-axis have their x-coordinate as 0.
• A positive y-coordinate indicates a position above the origin, while a negative value places the point below.

Consider the points (5, -2) and (-2, 5): - *(5, -2)* with a y-coordinate of -2 is 2 units below the origin. - *(-2, 5)* with a y-coordinate of 5 is 5 units above the origin.
This vertical navigation, dictated by the y-coordinate, helps in understanding where points are situated in terms of height.

Origin

The origin is the starting point in the Cartesian coordinate system, denoted as (0, 0). It is the point where both the x-axis and y-axis intersect.

• The coordinates of the origin are always (0, 0).
• It provides a reference point for locating all other points on the plane.

When you see a coordinate pair like (5, -2) or (-2, 5), each number tells you how far from the origin you need to move to find the point. The origin is central to understanding the Cartesian plane since it serves as the baseline for measuring any given position.
Whenever you're plotting or identifying points, remembering that everything starts from the origin is key to mastering coordinate geometry.