Problem 32 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(21-32,\) indicate which quadrant contains the given point. If a point lies on one of the coordinate axes, indicate which one. $$(-586,0)$$

Step-by-Step Solution

Answer

The point \((-586, 0)\) lies on the x-axis.

Step 1: Understand the Cartesian Coordinate System

The Cartesian coordinate system consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point \((-586, 0)\) contains coordinates in the form \( (x, y) \) where \(x = -586 \) and \( y = 0 \).

Step 2: Determine the Placement on the Axes

Because the y-coordinate is \(0\), the point \((-586, 0)\) lies on the x-axis. A point lies on the x-axis if its y-coordinate is zero, regardless of the value of the x-coordinate.

Key Concepts

Quadrants

In the Cartesian coordinate system, the plane is divided into four regions known as the quadrants. These quadrants are crucial in identifying the position of a point in the plane.
Each quadrant is defined by the signs of the coordinates of the points that lie within it:

• Quadrant I: The x and y coordinates are both positive \((+,+)\)
• Quadrant II: The x coordinate is negative, and the y coordinate is positive \((-,+)\)
• Quadrant III: Both x and y coordinates are negative \((-,-)\)
• Quadrant IV: The x coordinate is positive, and the y coordinate is negative \((+,-)\)

Understanding which quadrant a point is in helps to quickly identify its relative position in the plane.

Coordinate Axes

The Cartesian coordinate system is defined by two perpendicular axes that intersect at their zero points. These are known as the coordinate axes.
The two coordinate axes are:

• The x-axis, which runs horizontally
• The y-axis, which runs vertically

The point at which these axes intersect is called the origin, denoted by \( (0,0)\).
Points can lie in any of the four quadrants created by these axes or directly on the axes themselves. Points lying on the axes have either their x or y coordinate set to zero:

• If the x coordinate is zero, the point lies on the y-axis
• If the y coordinate is zero, the point lies on the x-axis

This system of axes helps in precisely locating points on the plane.

x-axis

The x-axis is the horizontal axis in the Cartesian coordinate system. It plays a crucial role in defining the position of points in the plane.
Here are some key points about the x-axis:

• Points on the x-axis have their y-coordinate set to zero
• The coordinates of points on the x-axis follow the format \( (x,0) \)
• If a point lies on the x-axis, it does not belong to any of the quadrants

For example, the point \( (-586,0) \) lies on the x-axis because its y-coordinate is zero, regardless of the x-coordinate value.
This understanding of the x-axis is fundamental when plotting or identifying points in the Cartesian coordinate system.