Question

A line has the equation \(y-4 x=5\). What is the slope of a line that is perpendicular to this line? (A) \(-4\) (B) \(-\frac{1}{4}\) (C) \(\frac{4}{3}\) (D) 4

Short answer

The slope of the perpendicular line is \(-\frac{1}{4}\), option (B).

Step-by-step solution

Rearrange the Equation

The given line equation is \( y - 4x = 5 \). We need to rearrange it into the slope-intercept form \( y = mx + b \), where \( m \) is the slope. To do so, add \( 4x \) to both sides to isolate \( y \):\[y = 4x + 5\]The slope \( m \) of this line is \( 4 \).

Understand the Perpendicular Slope Concept

The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. The slope of our original line is \( 4 \). Therefore, the negative reciprocal will be calculated by flipping the fraction of \( 4 \) (treated as \( \frac{4}{1} \)) and changing the sign.

Calculate the Perpendicular Slope

To find the negative reciprocal of \( 4 \), first write \( 4 \) as \( \frac{4}{1} \). The reciprocal is \( \frac{1}{4} \), and by making it negative, we have \( -\frac{1}{4} \). So, the slope of the line that is perpendicular to our given line is \(-\frac{1}{4} \).

Key concepts

Slope-Intercept Form

The slope-intercept form of a line provides a simple way to express the equation of a line. It is written as \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. This form is especially useful because it allows for straightforward identification of the slope and y-intercept values directly from the equation.
When given an equation not in this form, like \( y - 4x = 5 \), the first step is to rearrange it to look like \( y = mx + b \). We do this by isolating \( y \) on one side. Add \( 4x \) to both sides to obtain \( y = 4x + 5 \). Now, the equation is in slope-intercept form, and you can easily identify the slope as \( 4 \).
You can quickly identify how steep a line is by looking at the slope \( m \). If the slope is positive, the line ascends as it goes from left to right, and if it's negative, the line descends.

Negative Reciprocal

The concept of a negative reciprocal plays a critical role in understanding perpendicular lines. If two lines are perpendicular, their slopes are negative reciprocals of each other. To find this negative reciprocal, you take the original slope, turn it upside down (this is the reciprocal), and change its sign.
For instance, if the slope of the line is \( 4 \), it can be expressed as \( \frac{4}{1} \). The reciprocal of \( \frac{4}{1} \) is \( \frac{1}{4} \).
Next, you need to switch the sign, turning it into \(-\frac{1}{4}\). This resulting value is the slope of any line that is perpendicular to your original line. It's essential to understand these mathematical operations to find perpendicular slopes accurately:

• Reciprocal: Flip the fraction.
• Negative: Change the sign.

Equation of a Line

The equation of a line is a mathematical statement that describes all points along the line. The most straightforward form is the slope-intercept form \( y = mx + b \), but there are also other forms like the point-slope or standard form. However, slope-intercept is particularly helpful for quickly identifying the line’s slope and y-intercept.
Knowing the equation helps not only in graphing but also in solving various geometrical problems, like finding parallel and perpendicular lines. For example, if you're given an equation \( y - 4x = 5 \), rearranging it into \( y = 4x + 5 \) quickly shows you the line's slope \( 4 \).
With this, you can determine properties or tasks related to the line, such as:

• **Graphing**: Easily plot points using the slope and intercept.
• **Finding Parallel Lines**: Use the same slope \( m \).
• **Finding Perpendicular Lines**: Calculate a negative reciprocal slope.

Understanding how to manipulate and read the equation of a line aids in deeper comprehension of linear algebra concepts.