Problem 33 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The equation is \(y = 6\).

Step 1: Introduction to the Problem

We need to find the equation of a line that contains the point \(5,6\) and is perpendicular to the \(y\) axis. Since it is perpendicular to the \(y\) axis, it must be a horizontal line.

Step 2: Understanding the Slope of Perpendicular Lines

Lines perpendicular to the \(y\) axis have undefined slope; however, lines perpendicular to this have a slope of 0, meaning they are horizontal lines.

Step 3: Finding the Equation of the Line

A horizontal line can be expressed as \(y = k\) where \(k\) is a constant. Since the line passes through the point \(5,6\), we set \(y = 6\). Therefore, the equation is \(y = 6\).

Key Concepts

Perpendicular Lines

In coordinate geometry, perpendicular lines are lines that intersect at right angles, or 90 degrees. This unique intersection creates a special relationship between their slopes. For lines on the Cartesian plane, when one line is vertical (perpendicular to the x-axis) with an undefined slope, any line perpendicular to this would be horizontal. Essentially, the slope of a line perpendicular to the y-axis will instead have a defined slope, specifically a slope of 0, making it a horizontal line.
Understanding perpendicular lines is crucial because their relationship helps in finding unknown slopes or verifying that angles are truly perpendicular. To summarize:

• The slope of one line (vertical) is undefined.
• The perpendicular slope is 0, indicating a horizontal line.

This perpendicular relationship is essential for solving many geometric problems, especially those involving right angles.

Horizontal Line Equation

Horizontal lines play an important role in geometry. The equation of a horizontal line is one of the simplest forms because it represents a constant y-value across all x-values. The general form of a horizontal line is given by:

\[ y = k \]

Here, \(k\) is a constant representing the y-value through which all points of the line pass. This ensures that no matter what value x takes, y remains the same.
Horizontal lines have a few key characteristics:

• They run parallel to the x-axis.
• Their slope is 0 since there is no vertical change as we move along the line.
• They intersect the y-axis at exactly one point, \((0, k)\).

In our exercise example, since the horizontal line passes through the point \((5, 6)\), the equation is simply \(y = 6\). This straightforward nature makes horizontal lines easy to understand and work with in geometry.

Slope of a Line

The slope of a line is a measure of its steepness and direction on a graph. It is represented by the letter \(m\) and calculated as the ratio of the 'rise' (the change in y) over the 'run' (the change in x). Mathematically, slope \(m\) can be expressed as:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are any two distinct points on the line.
Slope provides valuable information about a line:

• A positive slope means the line rises from left to right.
• A negative slope indicates the line falls from left to right.
• A zero slope describes a horizontal line, as there is no vertical change.
• An undefined slope signifies a vertical line, where there is no horizontal change.

In the context of our example, the line perpendicular to the y-axis must be horizontal, thus its slope is 0. This concept is central in understanding how lines relate to one another in terms of orientation and angle.