Problem 63 - College Algebra 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) do not represent the same point because they are located in different positions on the coordinate plane and hence they belong to different quadrants.

Step 1: Understanding the Coordinates

Let's first understand that a coordinate, represented by an ordered pair (x, y), shows a specific location on a coordinate plane. The x-value (first number) describes the horizontal position and the y-value (second number) describes the vertical position. If we swap these two numbers, we are essentially swapping the vertical and horizontal positions which results in a different point on the plane.

Step 2: Applying the concept

The ordered pair (5,-2) represents a point on the plane that is 5 units to the right along the x-axis (because it's positive) and 2 units below the x-axis (since it's negative). On the other hand, the ordered pair (-2,5) is 2 units to the left of origin (because it's negative) along the x-axis and 5 units above the x-axis (because it's positive).

Step 3: Concluding the difference

Consequently, the points (5,-2) and (-2,5) are located in different quadrants and thus they cannot be the same.