Problem 97 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

Points (5,-2) and (-2,5) are not the same because they are located in different quadrants in the Cartesian coordinate system.

Step 1: Understanding Coordinate System

In the Cartesian coordinate system, a point in a two-dimensional plane is defined by an ordered pair of numbers (a,b), where 'a' refers to the x-coordinate (position along the x-axis) and 'b' refers to the y-coordinate (position along the y-axis). The first number corresponds to the x-coordinate and the second number to the y-coordinate.

Step 2: Evaluate Coordinate Points

The points given in the exercise are (5,-2) and (-2,5). The first point (5,-2) can be interpreted as being 5 units to the right on the x-axis and 2 units downward on the y-axis from the origin (0,0). The other point (-2,5), on the other hand, is 2 units to the left on the x-axis and 5 units upward on the y-axis.

Step 3: Contrasting the Two Points

Upon comparison, it is clear that the two points do not represent the same position on two-dimensional space. The first point lies in the fourth quadrant, whereas the second one lies in the second quadrant. Thus, they are not the same points.