Problem 30 - 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. $$(8,47)$$

Step-by-Step Solution

Answer

The point \(8,47\) is in the first quadrant.

Step 1: Understand the Coordinate Plane

The coordinate plane is divided into four quadrants. The first quadrant is where both the x-coordinate and y-coordinate are positive. The second quadrant is where the x-coordinate is negative and the y-coordinate is positive. The third quadrant is where both coordinates are negative. The fourth quadrant is where the x-coordinate is positive and the y-coordinate is negative. Points on coordinate axes don't belong to any quadrant.

Step 2: Analyze the Given Point

Examine the given point \(8,47\). Notice that the x-coordinate is 8 (positive) and the y-coordinate is 47 (positive).

Step 3: Identify the Quadrant

Since both coordinates are positive, the point \(8,47\) lies in the first quadrant.

Key Concepts

Coordinate Plane

The coordinate plane is a two-dimensional space created by two perpendicular lines called the **x-axis** and **y-axis**. Where these two axes intersect is called the origin, denoted by \(0,0\). Every point on this plane can be described using a pair of numbers, called coordinates. The x-coordinate tells us how far to move left or right from the origin, and the y-coordinate tells us how far to move up or down. This system helps us locate points and understand their positions in space.

Quadrants

The coordinate plane is divided into four sections known as quadrants. These quadrants are numbered counterclockwise from the upper right:

• **First Quadrant (I)**: Both x and y coordinates are positive \( (x, y) \). For instance, the point \(8, 47\) resides here because both 8 and 47 are positive.
• **Second Quadrant (II)**: The x-coordinate is negative, while the y-coordinate is positive \( (-x, y) \).
• **Third Quadrant (III)**: Both x and y coordinates are negative \( (-x, -y) \).
• **Fourth Quadrant (IV)**: The x-coordinate is positive and the y-coordinate is negative \( (x, -y) \).

Knowing the signs of the coordinates helps us quickly identify which quadrant a particular point falls into.

Coordinate Axes

The x-axis and y-axis play fundamental roles in the coordinate plane. They not only help in dividing the plane into quadrants but also have their own significance:

• If a point lies directly on the **x-axis**, its y-coordinate is zero, which means it only has an x-value \( (x, 0) \).
• If a point lies directly on the **y-axis**, its x-coordinate is zero, indicating it only has a y-value \( (0, y) \).

Such points do not belong to any of the four quadrants. Understanding the axes is essential for interpreting the positions of points accurately.

Positive Coordinates

Positive coordinates occur when points are located in either the **First Quadrant** or the **Second Quadrant** (for y-coordinate only). These coordinates are essential when identifying a point in the coordinate plane:

• If both x and y coordinates are positive, the point will be in the **First Quadrant**.
• If the x-coordinate is negative but the y-coordinate is positive, it will be in the **Second Quadrant**.

For instance, the point \(8, 47\) has both coordinates positive, placing it squarely in the **First Quadrant**. Points with positive coordinates help us understand various spatial relations and measurements in the plane.