Problem 39 - Intermediate Algebra

Question

Contains the point \((5,6)\) and is perpendicular to the \(y\) axis

Step-by-Step Solution

Answer

The line equation is \( y = 6 \).

Step 1: Understanding the Problem

We need to find the equation of a line that contains the point (5,6) and is perpendicular to the y-axis. Recall that perpendicular to the y-axis means the line will be a horizontal line.

Step 2: Identifying Line Characteristics

A line that is perpendicular to the y-axis is horizontal. Horizontal lines have a constant y-value and can be expressed with the equation \( y = c \), where \( c \) is a constant.

Step 3: Applying the Point to Find the Line Equation

Since the line passes through the point \((5,6)\), the y-value of the line is 6. Therefore, substitute \( y = 6 \) in the equation of the line.

Step 4: Writing the Line Equation

The equation of the line that passes through (5, 6) and is perpendicular to the y-axis is \( y = 6 \).

Key Concepts

Perpendicular Lines

Perpendicular lines are lines that intersect each other at a perfect right angle, which is 90 degrees. In coordinate geometry, this often means one line may have a negative reciprocal slope of another. However, when dealing with horizontal and vertical lines, their relationship is a bit different. A line perpendicular to the y-axis is not a vertical line, but rather a horizontal one. This is because a horizontal line runs parallel to the x-axis and does not slant upwards or downwards, keeping a consistent y-value at every point along it. The important thing to remember is direction:

• Horizontal lines are perpendicular to the y-axis.
• Vertical lines are perpendicular to the x-axis.

By understanding this, you can quickly identity that any line perpendicular to the y-axis is horizontal, balancing out how opposites attract in geometry!

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves graphing equations on a plane. The coordinate plane consists of two number lines: the x-axis, which is horizontal, and the y-axis, vertical. Together, these create a grid that can be used to plot points, lines, and shapes. Each point on this plane is expressed as a pair of numbers \(x,y\), representing its exact location.
Coordinate geometry lets us analyze the relationships between points and lines. For example:

• To find a line parallel to an axis, notice if both coordinates stay constant or vary.
• The slope determines the line's direction and steepness.

In our example, the line is perpendicular to the y-axis, meaning we primarily focus on manipulating the y-coordinate, while the x may vary. This gives great power in determining relationships simply from coordinates!

Equation of a Line

The equation of a line tells us mathematically how a line is positioned and angled within a coordinate plane. The most common form for a line's equation is the slope-intercept form: \(y = mx + b\), where \m\ is the slope and \b\ is the y-intercept. However, in the case of horizontal lines, the slope \(m\) is 0, because there is no vertical change as we move along the line. This simplifies the equation to \(y = c\), where \c\ is a constant y-value, indicating the uniform height of the line.

• For vertical lines, the equation is \(x = a\), where \a\ is a constant x-value.
• Horizontal line's equation depends solely on maintaining a constant y-coordinate.

When given a point like \(5,6\), and asked for a perpendicular line to the y-axis, the equation simply adopts the y-coordinate from the point, resulting in \(y = 6\). This showcases how straightforward it can be to pinpoint horizontal lines when perpendicularity to an axis is involved!