Perpendicular Line Equation: Step-by-Step Solution

Introduction

In the realm of mathematics, specifically within the study of linear equations, understanding the relationship between lines and their perpendicularity is crucial. This article will serve as a comprehensive guide to determine the equation of a line that satisfies two key conditions: perpendicularity to a given line and passage through a specific point. We will delve into the fundamental concepts of slope, the slope-intercept form of a line, and the relationship between slopes of perpendicular lines. By mastering these concepts, you'll be well-equipped to tackle a variety of problems involving perpendicular lines. Let's dive in, guys!

This exploration into perpendicular lines is essential not only for academic pursuits but also for practical applications in fields like engineering, physics, and computer graphics. The ability to accurately define and manipulate linear equations is a fundamental skill that underpins many advanced mathematical and scientific concepts. So, let’s get started and unlock the secrets of perpendicular lines together!

To make this journey as smooth as possible, we will break down the problem into manageable steps. First, we'll revisit the concept of slope and its significance in defining the steepness and direction of a line. Then, we'll explore the slope-intercept form, a powerful tool for representing linear equations. Finally, we'll tie these concepts together to understand the unique relationship between the slopes of perpendicular lines. With a solid foundation in these core ideas, we'll be ready to tackle the specific problem at hand: finding the equation of a line perpendicular to y=5 and passing through the point (-7,-5). So, buckle up, math enthusiasts, and let's embark on this exciting mathematical adventure!

Understanding the Basics: Slope and Linear Equations

Before we tackle the specific problem, let's quickly review some foundational concepts. The slope of a line, often denoted by the letter m, quantifies its steepness and direction. It is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run). Mathematically, if we have two points on a line, (x1, y1) and (x2, y2), the slope is given by the formula:

m = (y2 - y1) / (x2 - x1)

A positive slope indicates a line that rises as you move from left to right, while a negative slope indicates a line that falls. A slope of zero represents a horizontal line, and an undefined slope (division by zero) represents a vertical line. Understanding the slope is key to visualizing and working with linear equations.

Now, let's talk about the slope-intercept form of a linear equation. This form is a convenient way to represent a line and is given by:

y = mx + b

where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). This form is particularly useful because it directly reveals the slope and y-intercept, making it easy to graph the line and analyze its properties. By simply looking at the equation, you can immediately determine the steepness and the point where the line intersects the vertical axis. This is a powerful tool in your mathematical arsenal!

For example, the equation y = 2x + 3 represents a line with a slope of 2 and a y-intercept of 3. This means the line rises 2 units for every 1 unit it moves horizontally, and it crosses the y-axis at the point (0, 3). The slope-intercept form provides a clear and concise way to express linear relationships and is essential for understanding the behavior of lines. So, remember this form well, as it will be our trusty companion in solving problems involving linear equations.

Perpendicular Lines: The Key Relationship

The cornerstone of our problem lies in understanding the relationship between the slopes of perpendicular lines. Two lines are considered perpendicular if they intersect at a right angle (90 degrees). The crucial property that connects perpendicular lines is their slopes. If a line has a slope of m1, and another line is perpendicular to it with a slope of m2, then the product of their slopes is -1. Mathematically, this is expressed as:

m1 * m2 = -1

This means that the slope of a line perpendicular to a given line is the negative reciprocal of the given line's slope. To find the negative reciprocal, you flip the fraction and change the sign. For instance, if a line has a slope of 2/3, a line perpendicular to it will have a slope of -3/2. This relationship is fundamental to solving problems involving perpendicularity and forms the basis for our approach.

Consider a horizontal line, which has a slope of 0. The negative reciprocal of 0 is undefined, which corresponds to a vertical line. This makes intuitive sense, as horizontal and vertical lines are always perpendicular. Similarly, if a line has an undefined slope (a vertical line), any line perpendicular to it must be horizontal, with a slope of 0. This reciprocal relationship is not just a mathematical curiosity; it’s a geometric necessity. The slopes must be negative reciprocals to ensure the lines meet at a perfect 90-degree angle. This elegant connection between slope and perpendicularity is what allows us to solve our original problem.

Understanding this negative reciprocal relationship is crucial for finding the equation of a line perpendicular to another. It gives us a direct way to calculate the slope of the perpendicular line, which is the first step in determining its equation. So, keep this principle firmly in mind as we move forward to solve our specific problem. The negative reciprocal is your secret weapon in the world of perpendicular lines!

Solving the Problem: Finding the Equation

Now, let's apply our knowledge to solve the given problem. We need to find the equation of a line that is perpendicular to y = 5 and passes through the point (-7, -5). The first step is to determine the slope of the given line, y = 5. This equation represents a horizontal line, as the y-value is constant regardless of the x-value. Horizontal lines have a slope of 0.

Since we need to find a line perpendicular to y = 5, we need to find the negative reciprocal of 0. As we discussed earlier, the negative reciprocal of 0 is undefined, which corresponds to a vertical line. Vertical lines have the equation x = c, where c is a constant. This means our perpendicular line will be a vertical line.

Now, we know the equation of our perpendicular line will be in the form x = c. To find the specific value of c, we use the fact that the line passes through the point (-7, -5). In the coordinate pair (-7, -5), the x-coordinate is -7. Since our line is vertical and must pass through this point, the equation of the line is simply x = -7. This is because all points on this vertical line will have an x-coordinate of -7, regardless of their y-coordinate.

Therefore, the equation of the line that is perpendicular to y = 5 and passes through the point (-7, -5) is x = -7. This solution elegantly combines our understanding of slopes, perpendicularity, and linear equations. We first identified the slope of the given line, then used the negative reciprocal relationship to determine the slope of the perpendicular line. Finally, we used the given point to pin down the specific equation of the perpendicular line. This step-by-step approach demonstrates the power of applying fundamental mathematical principles to solve complex problems. So, there you have it – we’ve successfully navigated the world of perpendicular lines and found our solution!

Conclusion

In this article, we have successfully navigated the process of finding the equation of a line perpendicular to a given line and passing through a specific point. We started by reviewing the fundamental concepts of slope and the slope-intercept form of a linear equation. We then delved into the crucial relationship between the slopes of perpendicular lines, understanding that they are negative reciprocals of each other. This understanding formed the basis for solving our problem.

We methodically tackled the problem by first identifying the slope of the given line (y = 5), recognizing it as a horizontal line with a slope of 0. We then used the negative reciprocal relationship to deduce that the perpendicular line must be vertical, with an undefined slope. Knowing that vertical lines have the form x = c, we used the given point (-7, -5) to determine the specific equation of the perpendicular line, which is x = -7.

This exercise highlights the importance of a strong foundation in mathematical principles. By understanding concepts like slope, the slope-intercept form, and the relationship between perpendicular lines, we can confidently approach and solve a wide range of problems. The ability to break down a problem into smaller, manageable steps, as we did in this article, is a valuable skill in mathematics and beyond. So, remember these steps and the underlying concepts, and you'll be well-equipped to tackle any challenge involving linear equations and perpendicular lines.

Moreover, the concepts explored in this article have far-reaching applications in various fields. From engineering and physics to computer graphics and data analysis, the ability to work with linear equations and geometric relationships is essential. By mastering these fundamentals, you are not just learning math; you are equipping yourself with powerful tools for problem-solving and critical thinking in a variety of contexts. So, keep practicing, keep exploring, and keep building your mathematical foundation. The world of mathematics is full of exciting discoveries, and you are well on your way to making your own!