Problem 19 - Algebra 1

Question

Without plotting the point, tell whether it is in Quadrant I, Quadrant II, Quadrant III, or Quadrant IV. $$(5,-3)$$

Step-by-Step Solution

Answer

The point (5,-3) is in Quadrant IV.

Step 1: Identify the Coordinates

Identify the coordinates of the point. In the case of (5,-3), the x-coordinate, represented by the first number, is 5, and the y-coordinate, represented by the second number, is -3.

Step 2: Determine the Quadrant

Based on the sign of the coordinates, we can classify the given point into one of the four quadrants. According to our identification, the x-coordinate is positive (+5) and the y-coordinate is negative (-3). This combination of positive x-coordinate and negative y-coordinate corresponds to Quadrant IV.

Key Concepts

Understanding Quadrants in the Coordinate Plane

In coordinate geometry, the coordinate plane is divided into four sections known as quadrants. Each quadrant is defined by the signs of the x and y coordinates.
The quadrants are designated as follows:

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

This system allows us to categorize any point plotted in the plane based on the signs of its coordinates. In our example, the point \((5, -3)\) falls into Quadrant IV because it has a positive x-coordinate and a negative y-coordinate. Quadrants are counted counter-clockwise starting from the top right, which helps to simplify the understanding and classification of points when plotting on the plane.

Basics of the Coordinate System

The coordinate system, often referred to as the Cartesian coordinate system, is a two-dimensional plane created by two perpendicular lines. These lines, known as axes, allow us to specify the location of any point with a pair of numerical values, or coordinates.
This system was introduced by Rene Descartes and has become a foundational tool in geometry and algebra.

• The **horizontal line** is called the x-axis.
• The **vertical line** is called the y-axis.

Where these axes intersect is known as the origin, marked as \((0, 0)\). From this origin, the x-axis extends right and left, while the y-axis extends up and down. Every point is defined relative to this origin with a coordinate pair \((x, y)\). Using this system, one can easily locate points on the plane, determine distances and analyze geometric properties.

Roles of the x-axis and y-axis

The x-axis and y-axis are essential components of the coordinate system. Together, they form the backbone, allowing for precise location plotting in the plane.
The **x-axis** is the horizontal axis. It is used to determine the horizontal position of a point and is methodically labeled with both positive and negative numbers.

• Moving to the right of the origin along the x-axis, coordinates increase positively.
• Moving to the left, coordinates decrease negatively.

On the other hand, the **y-axis** is the vertical axis. It determines the vertical position of points:

• Above the origin, coordinates are positive on the y-axis.
• Below the origin, coordinates are negative.

These axes form the coordinate plane which is crucial for graphing equations and visualizing algebraic principles, providing a structured way to navigate between numerical and geometrical representations.