Problem 66 - Prealgebra

Question

Name the quadrant in which \(P(-5,-6)\) is located.

Step-by-Step Solution

Answer

Point P(-5, -6) is located in the third quadrant.

Step 1: Understand the Coordinate System

In the Cartesian coordinate system, there are four quadrants. The first quadrant is for coordinates (+, +), the second quadrant for (-, +), the third quadrant for (-, -), and the fourth quadrant for (+, -).

Step 2: Identify the Coordinates

Point P has coordinates (-5, -6). The x-coordinate is -5 and the y-coordinate is -6.

Step 3: Determine the Quadrant

Since both coordinates of point P are negative, the point lies in the third quadrant, which is represented as (-, -).

Key Concepts

quadrants

In the Cartesian coordinate system, the entire coordinate plane is divided into four distinct sections known as quadrants.
These quadrants are numbered counterclockwise, beginning with the top-right quadrant.
Here's a quick breakdown:

• **First Quadrant (I)**: Both x and y coordinates are positive ( (+,+))
• **Second Quadrant (II)**: x is negative, but y is positive ( (-,+))
• **Third Quadrant (III)**: Both coordinates are negative ( (-,-))
• **Fourth Quadrant (IV)**: x is positive and y is negative ( (+,-))

Every point on the coordinate plane falls into one of these quadrants, depending on the signs of its x and y coordinates. For example, with point P(-5, -6), both coordinates are negative. Hence, it lies in the third quadrant (III). Understanding quadrants helps in analyzing the position and movements of points in mathematical and practical applications.

coordinate system

A coordinate system is a method for describing the position of points in space with numerical coordinates.
An essential component of geometry and engineering, coordinate systems provide a way to precisely locate and plot points.
The most widely used system is the **Cartesian coordinate system**, developed by René Descartes. In the Cartesian coordinate system:

• There are two axes—**x-axis** (horizontal) and **y-axis** (vertical)
• The point where the x-axis and y-axis intersect is called the **origin** (0,0)

The coordinate system allows for a simple two-dimensional representation of points using ordered pairs (x, y). Each value in this pair indicates a distance from the respective axis. This system provides a framework for graphing equations and analyzing geometric figures, forming the basis of much of analytic geometry.

coordinate plane

The coordinate plane is a two-dimensional surface defined by the intersection of the x-axis and y-axis.
It provides a space for plotting points, drawing lines, and depicting geometric shapes in a visually organized manner.
This plane is sometimes called the **xy-plane**. Key characteristics of the coordinate plane include:

• The plane is divided by the two axes into four quadrants
• Each point on the plane is represented by an ordered pair (x, y)

Points are plotted by moving horizontally along the x-axis and vertically along the y-axis, featuring an organized method of placement.
Understanding how to use and navigate the coordinate plane is fundamental for solving problems in mathematics, physics, and engineering. It connects abstract mathematical concepts with tangible visual representations.