Problem 1 - Algebra and Trigonometry

Question

The point that is 3 units to the right of the y-axis and 5 units below the x-axis has coordinates (____,____).

Step-by-Step Solution

Answer

The point has coordinates (3, -5).

Step 1: Identify the Point's Location

We are told the point is 3 units to the right of the y-axis. Points to the right of the y-axis have positive x-values. Thus, the x-coordinate of the point is 3.

Step 2: Determine Vertical Position

The point is also 5 units below the x-axis. Points below the x-axis have negative y-values. Thus, the y-coordinate of the point is -5.

Step 3: Construct the Coordinate Pair

Now we can combine our findings into a coordinate pair. The point is located at (3, -5).

Key Concepts

Understanding the x-axis

In geometry, the x-axis is a crucial part of the coordinate plane, which allows us to locate points precisely. Imagine it as the horizontal line that stretches across the plane from left to right. The x-axis serves as the baseline for all horizontal measurements.

Here’s what you need to know about the x-axis:

• The x-axis is always horizontal.
• It is marked by incremental units, which can represent positive or negative numbers.
• Points positioned on the right of the y-axis along the x-axis have positive x-coordinates.
• Points positioned on the left of the y-axis on the x-axis have negative x-values.

When we say a point is a certain number of units to the right of the y-axis, we are referring to its positive x-coordinate. For instance, if a point is 3 units to the right, it means its x-coordinate is 3.

Exploring the y-axis

The y-axis is another essential line on the coordinate plane, running vertically from top to bottom. It acts as the reference line for vertical measurements.

Here's how the y-axis is characterized:

• The y-axis is always vertical.
• It also uses incremental units that can be positive or negative values.
• Points above the x-axis on the y-axis have positive y-coordinates.
• Points below the x-axis have negative y-values.

In our exercise, the point lies 5 units below the x-axis. This means its y-coordinate should be -5, indicating that it is negative due to its position beneath the x-axis.

Grasping the Coordinate Plane

The coordinate plane is a two-dimensional system where each point can be identified by an ordered pair of numbers, usually written as \((x, y)\). This system helps in visualizing and solving geometric problems.

Key features of the coordinate plane:

• It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
• The point where the two axes intersect is called the origin, marked by coordinates \((0, 0)\).
• Each point on the plane is determined by an x-coordinate (horizontal position) and a y-coordinate (vertical position).

In the given exercise, we found the point to be 3 units to the right of the y-axis and 5 units below the x-axis. By placing these findings together, we locate the point as \((3, -5)\) on the coordinate plane, where the first number represents its x-coordinate and the second, its y-coordinate.