Problem 19 - Precalculus

Question

Write an equation of: (a) a vertical line passing through the given point; (b) a horizontal line passing through the given point. $$(-3,4)$$

Step-by-Step Solution

Answer

Vertical line: \(x = -3\); Horizontal line: \(y = 4\).

Step 1: Identify Characteristics of a Vertical Line

A vertical line passes through a given x-coordinate and stays constant across all y-values. In this case, the line will pass through the x-coordinate of the point \((-3,4)\).

Step 2: Formulate Equation for the Vertical Line

Since the x-coordinate remains constant, the equation for the vertical line is\(x = -3\).

Step 3: Identify Characteristics of a Horizontal Line

A horizontal line passes through a given y-coordinate and remains constant across all x-values. In this case, the line will pass through the y-coordinate of the point \((-3,4)\).

Step 4: Formulate Equation for the Horizontal Line

Since the y-coordinate remains constant, the equation for the horizontal line is\(y = 4\).

Key Concepts

Vertical Lines

A vertical line in coordinate geometry is a line that moves up and down parallel to the y-axis. It doesn't slant or slope in any direction, making it have an undefined slope. The most important concept about vertical lines is that they have a constant x-value, regardless of what the y-value is. This means that every point along a vertical line can have different y-coordinates, but the x-coordinate will always remain the same. For example, consider a line that passes through the point

• (-3, 4)

This line is vertical, so its equation is very simple:

• x = -3

This equation tells us that all points on this line have an x-coordinate of -3. It doesn't matter what the y-coordinate is; it could be 4, 0, 100, or any other number. The x-coordinate will always remain -3.

Horizontal Lines

Horizontal lines are different from vertical lines because they move left to right, parallel to the x-axis. In coordinate geometry, horizontal lines have a 0 slope, meaning they neither move up nor move down. They maintain a constant y-value across all x-values, making every point on a horizontal line share the same y-coordinate. For a line passing through the point

• (-3, 4)

This horizontal line will have the equation:

• y = 4

This equation indicates that no matter what the x-coordinate is, the y-coordinate will always be 4. So whether x is -3, 0, or 5, the y will always remain constant at 4.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is the study of geometry using a coordinate system. This approach helps to describe geometric figures and relationships between them using numbers and equations. The coordinate system consists of two perpendicular number lines, known as axes.

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

In coordinate geometry, points are defined by their positions along these axes, known as coordinates. A point is often denoted as

• (x, y)

The x-coordinate tells you where the point is along the x-axis, and the y-coordinate reveals its position along the y-axis. By using equations, coordinate geometry allows us to understand and describe lines, such as vertical and horizontal lines, circles, and other geometric shapes with precision. This makes it a powerful tool for solving complex problems and visually representing mathematical concepts.