Finding Perpendicular Lines An In-Depth Guide
This is a common question in algebra and coordinate geometry, and understanding the principles behind perpendicular lines is crucial to solving it. In this article, we will discuss how to determine the equation of a line perpendicular to a given line. To successfully solve this question and others like it, we must grasp the concept of slopes and how they relate to perpendicularity. Let's explore the principles behind determining equations of perpendicular lines, focusing on how slopes dictate this relationship. We'll begin by dissecting the given equation and then systematically evaluate the provided options. Remember, the key to mastering coordinate geometry lies in understanding the underlying concepts and applying them methodically. This exploration will empower you to confidently navigate similar problems in the future.
Decoding the Original Equation
To identify a line perpendicular to the given equation, we must first determine the slope of the original line. The given equation is in standard form: $3x - 8y = -16$. To find the slope, we need to convert this equation into slope-intercept form, which is $y = mx + b$, where m represents the slope and b represents the y-intercept. Let’s begin by isolating the y term:
- Subtract 3x from both sides: $-8y = -3x - 16$
- Divide both sides by -8: $y = \frac{-3x}{-8} + \frac{-16}{-8}$
- Simplify: $y = \frac{3}{8}x + 2$
Now that the equation is in slope-intercept form, we can easily identify the slope. The slope of the given line is $\frac{3}{8}$. Understanding this slope is essential for determining the slope of any line perpendicular to it. The slope of a perpendicular line is the negative reciprocal of the original line's slope. This means we flip the fraction and change its sign. So, if the original slope is $\frac{3}{8}$, the perpendicular slope will be $-\frac{8}{3}$. This principle is a cornerstone of coordinate geometry and is vital for solving problems involving perpendicular lines. In the following sections, we will analyze the provided options, comparing their slopes to this calculated perpendicular slope, ensuring we select the equation that accurately represents a line perpendicular to the original.
The Negative Reciprocal: Key to Perpendicularity
The fundamental concept behind perpendicular lines is the negative reciprocal of their slopes. When two lines are perpendicular, the product of their slopes is -1. This relationship allows us to quickly determine if two lines are perpendicular by examining their slopes. To find the slope of a line perpendicular to a given line, you simply take the negative reciprocal of the given line's slope. This means you flip the fraction (reciprocal) and change its sign (negative). For example, if a line has a slope of 2 (which can be written as $\frac{2}{1}$), the slope of a line perpendicular to it would be $-\frac{1}{2}$. Similarly, if a line has a slope of $-\frac{4}{5}$, the perpendicular slope would be $\frac{5}{4}$. Understanding this relationship is crucial for solving problems involving perpendicular lines in coordinate geometry. It allows us to directly link the slope of one line to the slope of any line perpendicular to it. In our problem, we identified the original slope as $\frac{3}{8}$, and therefore, the perpendicular slope must be $-\frac{8}{3}$. This understanding will now guide us as we analyze the answer choices, ensuring we select the equation that accurately reflects this perpendicular relationship. Let's move forward by examining each option in light of this crucial concept.
Analyzing the Answer Choices
Now that we know the slope of a line perpendicular to the given line ($\frac{3}{8}$) must be $-\frac{8}{3}$, we can examine each answer choice to see which one has this slope. Each answer choice is in slope-intercept form ($y = mx + b$), which makes it easy to identify the slope (the coefficient of x). Let's analyze each option:
- **A. $y = \frac8}{3}x + 8${3}$. This is the reciprocal of the original slope but not the negative reciprocal. Therefore, this option is incorrect.
- **B. $y = -\frac3}{8}x + 1${8}$. This is the negative of the original slope but not the negative reciprocal. Therefore, this option is incorrect.
- **C. $y = \frac3}{8}x - 4${8}$. This is the same as the original slope, indicating a parallel line, not a perpendicular one. Therefore, this option is incorrect.
- **D. $y = -\frac8}{3}x - 2${3}$. This is the negative reciprocal of the original slope. Therefore, this option is the correct answer.
By systematically analyzing each option and comparing its slope to the required perpendicular slope, we can confidently identify the correct answer. This methodical approach is crucial for solving similar problems in coordinate geometry, ensuring accuracy and a clear understanding of the underlying principles. In the following section, we'll solidify our understanding by summarizing the steps and highlighting key takeaways.
Solution and Conclusion
After converting the original equation $3x - 8y = -16$ to slope-intercept form, we found its slope to be $\frac{3}{8}$. Then, we determined that the slope of a line perpendicular to this line must be the negative reciprocal, which is $-\frac{8}{3}$. By examining the answer choices, we identified that option D. $y = -\frac{8}{3}x - 2$ has the correct slope. Therefore, the equation that represents a line perpendicular to the given line is D. $y = -\frac{8}{3}x - 2$. Understanding the relationship between slopes of perpendicular lines is essential for solving problems like this. Remember, perpendicular lines have slopes that are negative reciprocals of each other. By converting equations to slope-intercept form, identifying slopes, and applying the negative reciprocal concept, you can confidently solve similar problems. This approach not only leads to the correct answer but also reinforces the fundamental principles of coordinate geometry.
In summary, to find the equation of a line perpendicular to a given line:
- Convert the given equation to slope-intercept form ($y = mx + b$) to identify its slope.
- Calculate the negative reciprocal of the slope.
- Examine the answer choices and select the equation with the matching perpendicular slope.
By following these steps and understanding the underlying principles, you can confidently tackle any problem involving perpendicular lines.
