Problem 57 - Intermediate Algebra

Question

Use the given conditions to determine in which quadrant of a rectangular coordinate system each point \((x, y)\) is located. \(x>0\) and \(y<0\)

Step-by-Step Solution

Answer

The point (x, y) is located in Quadrant IV.

Step 1: Understanding Quadrants

A rectangular coordinate system is divided into four quadrants. Quadrant I is where both coordinates are positive. Quadrant II is where the x-coordinate is negative and the y-coordinate is positive. Quadrant III is where both coordinates are negative. Quadrant IV is where the x-coordinate is positive, and the y-coordinate is negative.

Step 2: Locate Point Based on Given Conditions

From the given conditions, we know that the x-coordinate () and the y-coordinate () < 0. According to the quadrant definitions: - A positive x-coordinate occurs in Quadrants I and IV. - A negative y-coordinate occurs in Quadrants III and IV.

Step 3: Determine The Intersection of Conditions

Since the only quadrant where both conditions are met is Quadrant IV (where x > 0 and y < 0), the point  (x, y) based on the given conditions is located in Quadrant IV.

Key Concepts

Rectangular Coordinate System

Imagine a large sheet of graph paper. A rectangular coordinate system, also known as the Cartesian plane, is very similar to this setup. It has two lines, called axes, that intersect at a point known as the origin. These axes divide the plane into four sections called quadrants. The horizontal line is called the x-axis, and the vertical line is called the y-axis.

Each quadrant in the coordinate system has a distinct combination of positive and negative values for x and y coordinates:

• Quadrant I: Both x and y are positive.
• Quadrant II: x is negative, and y is positive.
• Quadrant III: Both x and y are negative.
• Quadrant IV: x is positive, and y is negative.

Understanding this structure is crucial for locating points within the system.

Coordinate Points

In a rectangular coordinate system, each position is described with a pair of numbers, also known as coordinate points. These points are written as \((x, y)\) where x represents the horizontal position, and y represents the vertical position.

Let's dive into how these coordinates work:

• The x-coordinate indicates how far to move left or right from the origin. Positive values move right, and negative values move left.
• The y-coordinate indicates how far to move up or down from the origin. Positive values move up, and negative values move down.

This dual-number system allows us to pinpoint any specific location on the plane with accuracy and can be used to determine which quadrant a point is located in by checking the signs of these coordinates.

Graphical Representation

Visualizing points on a graph within the coordinate system is known as graphical representation. It is an effective way to understand mathematical concepts visually. When you see a point plotted, with the specific x and y values, you can immediately find its location relative to the four quadrants.

For example, if you were to plot a point with coordinates \((x>0, y<0)\), you would start at the origin, move to the right on the x-axis, and then move downward on the y-axis because y is negative. This action lands you in Quadrant IV.

Graphical representation helps you:

• Quickly recognize where a point belongs in the coordinate plane.
• Identify patterns and relationships between different points or sets of points.
• Simplify complex algebraic concepts by converting them into a visual form.

It is a powerful tool in both education and practical applications, providing a clear picture of abstract numerical relationships.