Identifying Alejandro's Error Perpendicular Line Equation

Introduction

In the realm of coordinate geometry, determining the equation of a line perpendicular to a given line and passing through a specific point is a fundamental concept. This involves understanding the relationship between the slopes of perpendicular lines and applying the point-slope form of a linear equation. In this article, we will delve into a scenario where Alejandro made an error while attempting to find the equation of a line perpendicular to the line $4x - 3y = -8$ and passing through the point $(3, -2)$. We will meticulously analyze Alejandro's steps, pinpoint the error, and provide a comprehensive explanation of the correct procedure.

Problem Statement

The problem at hand requires us to identify the mistake in Alejandro's approach to finding the equation of a line. The line must be perpendicular to the given line, which is $4x - 3y = -8$, and it must pass through the point $(3, -2)$. Alejandro's attempt, presented in a step-by-step manner, allows us to scrutinize each stage of his solution. By carefully examining his work, we can understand where the error occurred and how to rectify it. Understanding the concept of perpendicular lines and their slopes is crucial in solving this problem. Two lines are perpendicular if the product of their slopes is -1. This means that if we know the slope of one line, we can easily find the slope of a line perpendicular to it. Additionally, the point-slope form of a linear equation, $y - y_1 = m(x - x_1)$, is essential for constructing the equation of a line when we know its slope and a point it passes through. This form allows us to directly substitute the known values and derive the equation.

Alejandro's Steps and the Identification of the Error

To accurately identify Alejandro's error, we need to meticulously examine each step he took in his solution. Let's analyze a hypothetical scenario of Alejandro's work:

Upon reviewing these steps, a clear error emerges in Step 2. The slope of the perpendicular line was incorrectly determined. Alejandro seems to have taken the slope of the original line ($ \frac{4}{3} $) and directly used it as the slope of the perpendicular line. However, the fundamental principle of perpendicular lines dictates that their slopes are negative reciprocals of each other. This means that if a line has a slope of $m$, a line perpendicular to it will have a slope of $- \frac{1}{m}$. In this case, the correct slope for the perpendicular line should be the negative reciprocal of $ \frac{4}{3} $, which is $- \frac{3}{4} $. This error in determining the slope propagates through the subsequent steps, leading to an incorrect final equation.

Correcting Alejandro's Error: A Step-by-Step Solution

To rectify Alejandro's mistake and arrive at the correct equation, we need to follow the appropriate steps, paying close attention to the concept of negative reciprocal slopes. Here's a detailed, step-by-step solution:

Step 1: Convert the given equation to slope-intercept form.

The given equation is $4x - 3y = -8$. To convert it to slope-intercept form ($y = mx + b$), we need to isolate $y$:

• Subtract $4x$ from both sides: $-3y = -4x - 8$
• Divide both sides by $-3$: $y = \frac{4}{3}x + \frac{8}{3}$

This form reveals that the slope of the given line is $ \frac{4}{3} $.

Step 2: Determine the slope of the perpendicular line.

As discussed earlier, the slope of a line perpendicular to another line is the negative reciprocal of its slope. Therefore, the slope of the perpendicular line ($m_{\perp}$) is:

• $m_{\perp} = - \frac{1}{\frac{4}{3}} = - \frac{3}{4}$

Step 3: Use the point-slope form to create the equation of the perpendicular line.

The point-slope form of a linear equation is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope. We are given the point $(3, -2)$ and we have calculated the slope of the perpendicular line as $- \frac{3}{4} $. Substituting these values into the point-slope form, we get:

• $y - (-2) = - \frac{3}{4}(x - 3)$

Step 4: Simplify the equation to slope-intercept form.

To simplify the equation, we first distribute the slope and then isolate $y$:

• $y + 2 = - \frac{3}{4}x + \frac{9}{4}$
• Subtract 2 from both sides: $y = - \frac{3}{4}x + \frac{9}{4} - 2$
• Convert 2 to a fraction with a denominator of 4: $y = - \frac{3}{4}x + \frac{9}{4} - \frac{8}{4}$
• Simplify: $y = - \frac{3}{4}x + \frac{1}{4}$

Therefore, the correct equation of the line perpendicular to $4x - 3y = -8$ and passing through the point $(3, -2)$ is $y = - \frac{3}{4}x + \frac{1}{4}$.

Importance of Understanding Perpendicular Slopes

The concept of perpendicular slopes is a cornerstone of coordinate geometry and has far-reaching applications in various fields. In mathematics, it is crucial for solving geometric problems, understanding vector operations, and working with transformations. Understanding perpendicular slopes allows us to accurately determine the relationships between lines and shapes in a coordinate plane. For instance, in computer graphics, perpendicular lines are essential for creating orthogonal projections and rendering 3D objects onto a 2D screen. In physics, the concept is used in analyzing forces and motion, particularly when dealing with components of vectors that are perpendicular to each other. Moreover, in engineering, perpendicularity is a critical factor in structural design, ensuring stability and load distribution. Architects and civil engineers rely on this principle to construct buildings and bridges that can withstand various stresses. For example, the supports of a bridge are often designed to be perpendicular to the road surface to maximize stability. In navigation, perpendicular lines and angles are used in creating maps and determining directions. Sailors and pilots use these concepts to plot courses and avoid collisions. The global positioning system (GPS) also relies on geometric principles, including perpendicularity, to calculate the position of a receiver. Furthermore, understanding perpendicular slopes is vital for problem-solving in calculus, particularly when finding the normal line to a curve at a given point. The normal line is perpendicular to the tangent line, and its equation can be determined using the negative reciprocal of the tangent's slope. This is a fundamental concept in optimization problems and curve analysis.

Common Mistakes and How to Avoid Them

While the process of finding the equation of a perpendicular line is relatively straightforward, several common mistakes can lead to incorrect results. One of the most frequent errors is, as seen in Alejandro's case, failing to take the negative reciprocal of the slope. Students often remember the reciprocal part but forget to change the sign, or vice versa. To avoid this, it is crucial to always remember that perpendicular lines have slopes that are both reciprocals and have opposite signs. Another common mistake is incorrectly applying the point-slope form of the equation. Students may mix up the $x_1$ and $y_1$ values or incorrectly substitute the slope. To prevent this, it is helpful to write the point-slope formula explicitly ($y - y_1 = m(x - x_1)$) before substituting the values. This helps in visualizing the correct placement of each number. Additionally, errors can occur during the algebraic manipulation of the equation. When simplifying the equation from point-slope form to slope-intercept form, mistakes in distribution or combining like terms can lead to an incorrect final equation. To minimize these errors, it is advisable to perform each step carefully and double-check the work. Breaking down the simplification into smaller, manageable steps can also help in identifying and correcting any mistakes. A further pitfall is not converting the given equation to slope-intercept form correctly. If the original equation is not in the form $y = mx + b$, it is essential to rearrange it accurately to identify the correct slope. Mistakes in this initial step can propagate through the entire solution. To ensure accuracy, it is good practice to double-check the algebraic steps involved in isolating $y$. Another common issue arises when students try to memorize formulas without understanding the underlying concepts. This can lead to misapplication of the formulas in different contexts. To overcome this, it is essential to understand the logic behind each step and the relationship between the concepts. For instance, understanding why the negative reciprocal is used for perpendicular slopes will help in recalling the rule correctly. Finally, students sometimes make mistakes due to carelessness, such as copying numbers incorrectly or overlooking negative signs. To avoid such errors, it is crucial to maintain a neat and organized workspace and to take the time to review the solution carefully. Using a pencil and paper can make it easier to correct mistakes as they are identified.

Conclusion

In conclusion, Alejandro's error in determining the equation of the perpendicular line stemmed from incorrectly identifying the slope of the perpendicular line. By failing to take the negative reciprocal of the original line's slope, he arrived at an incorrect equation. Through a step-by-step correction, we demonstrated the accurate method for finding the equation of a perpendicular line. The corrected equation, $y = - \frac{3}{4}x + \frac{1}{4}$, accurately represents the line perpendicular to $4x - 3y = -8$ and passing through the point $(3, -2)$. This exercise underscores the importance of understanding fundamental concepts in coordinate geometry, particularly the relationship between the slopes of perpendicular lines and the correct application of the point-slope form. By avoiding common mistakes and carefully applying the principles of coordinate geometry, one can confidently solve similar problems and achieve accurate results. Furthermore, the broader applications of these concepts in various fields highlight the practical significance of mastering them. From engineering and physics to computer graphics and navigation, the principles of perpendicularity and linear equations play a vital role in numerous real-world applications. Therefore, a thorough understanding of these concepts is not only crucial for academic success but also for practical problem-solving in various professional domains.