Finding The Equation Of A Perpendicular Line
In mathematics, understanding the relationship between lines, especially perpendicular lines, is crucial. This article will guide you through the process of finding the equation of a line that is perpendicular to a given line and passes through a specific point. We will focus on expressing the final equation in slope-intercept form, which is a fundamental concept in linear algebra. This article aims to clarify the steps involved, making it easy to understand and apply in various mathematical contexts.
Understanding Slope-Intercept Form
Before we dive into the problem, let's briefly discuss the slope-intercept form. The slope-intercept form of a linear equation is expressed as y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). This form is particularly useful because it directly reveals two key characteristics of the line: its steepness (slope) and its position on the coordinate plane (y-intercept). Mastering the slope-intercept form is fundamental for understanding and manipulating linear equations in various mathematical applications. When you're dealing with linear equations, recognizing and utilizing the slope-intercept form is a crucial skill for solving problems and understanding the behavior of lines. Being able to quickly identify the slope and y-intercept from an equation in this form allows for efficient graphing and analysis of linear relationships. This form is not just a mathematical tool but a foundational concept that bridges algebra and geometry, making it an essential part of any mathematics curriculum. Understanding slope-intercept form is crucial not only for academic success but also for real-world applications such as interpreting data trends and making predictions based on linear models. Whether you're a student learning about linear equations for the first time or a professional using mathematical models in your work, a solid grasp of slope-intercept form is indispensable.
Perpendicular Lines and Their Slopes
The key concept in this problem is the relationship between the slopes of perpendicular lines. Two lines are perpendicular if they intersect at a right angle (90 degrees). The slopes of perpendicular lines have a unique relationship: they are negative reciprocals of each other. This means that if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. For instance, if a line has a slope of 2, a perpendicular line will have a slope of -1/2. This inverse relationship is critical for solving problems involving perpendicular lines. Understanding this relationship allows us to quickly determine the slope of a perpendicular line when we know the slope of the original line. Recognizing that perpendicular lines form right angles is fundamental to geometry and has far-reaching implications in various fields, including engineering, architecture, and computer graphics. The negative reciprocal relationship is not just a mathematical curiosity but a practical tool for ensuring accuracy and stability in designs and calculations. Whether you are calculating the angle of a roof truss or designing a bridge, the principle of perpendicular lines and their slopes is essential for structural integrity and functionality. The application of this concept extends beyond theoretical mathematics, making it a valuable skill in a wide range of professions.
Problem Setup: Finding the Perpendicular Line
Now, let's address the specific problem. We are given the line y = -1/2x - 5 and the point (2, 7). Our goal is to find the equation of the line that is perpendicular to the given line and passes through the specified point. The first step is to identify the slope of the given line. In the equation y = -1/2x - 5, the slope is -1/2. To find the slope of the perpendicular line, we take the negative reciprocal of -1/2, which is 2. Therefore, the perpendicular line will have a slope of 2. This is a crucial step as it sets the foundation for the rest of the solution. Recognizing the slope of the original line is the first key to unlocking the equation of the perpendicular line. Once you have the slope of the perpendicular line, you can move forward to use the point-slope form or the slope-intercept form to find the full equation. This initial step highlights the importance of understanding the relationship between the slopes of perpendicular lines, as it directly influences the subsequent calculations and the final solution. Accurate identification of the original slope and its negative reciprocal is paramount for solving the problem correctly. The ability to quickly determine the slope of a perpendicular line is a valuable skill in various mathematical contexts.
Using the Point-Slope Form
Now that we know the slope of the perpendicular line is 2, and it passes through the point (2, 7), we can use the point-slope form to find the equation. The point-slope form is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is the given point. Plugging in the values, we get y - 7 = 2(x - 2). This equation represents the line in point-slope form. While point-slope form is a valid representation of a line, we often need to convert it to slope-intercept form for various applications. Using the point-slope form provides a direct way to construct the equation of a line when you have a point and the slope. This is particularly useful when dealing with geometric problems where specific points and slopes are known. The point-slope form is not just a formula but a powerful tool for building equations from limited information. Understanding this form allows you to efficiently solve problems involving lines and their relationships. The ability to seamlessly switch between point-slope form and slope-intercept form is a hallmark of strong algebraic skills. The point-slope form is a bridge between geometric concepts and algebraic expressions, making it an indispensable tool in mathematics.
Converting to Slope-Intercept Form
To convert the equation from point-slope form to slope-intercept form (y = mx + b), we need to simplify the equation and isolate y. Starting with y - 7 = 2(x - 2), we first distribute the 2 on the right side: y - 7 = 2x - 4. Next, we add 7 to both sides to isolate y: y = 2x - 4 + 7. Simplifying, we get y = 2x + 3. This is the equation of the line in slope-intercept form. The slope of this line is 2, and the y-intercept is 3. Transforming equations into slope-intercept form is a fundamental skill in algebra, enabling us to easily visualize and analyze linear functions. The process of converting from point-slope form to slope-intercept form involves algebraic manipulation that reinforces understanding of equation solving. This conversion is not just about getting the equation in a specific format but about gaining deeper insight into the characteristics of the line. Being able to manipulate equations into different forms enhances problem-solving abilities and provides a more comprehensive understanding of linear relationships. The slope-intercept form is the ultimate goal in many linear equation problems, as it provides a clear and concise representation of the line's properties.
Final Answer and Verification
The final answer is y = 2x + 3. This equation represents the line that is perpendicular to y = -1/2x - 5 and passes through the point (2, 7). To verify our answer, we can check if the slope is the negative reciprocal of the original line's slope and if the point (2, 7) satisfies the equation. The slope of the original line is -1/2, and the slope of our line is 2, which confirms they are negative reciprocals. Next, we substitute x = 2 into our equation: y = 2(2) + 3 = 4 + 3 = 7. This confirms that the point (2, 7) lies on the line. Verification is a crucial step in mathematics to ensure the accuracy of the solution. Checking the slope and the point provides a robust confirmation that the equation is correct. The final answer, y = 2x + 3, represents a complete and verified solution to the problem. This process of finding the equation of a perpendicular line showcases the interconnectedness of algebraic concepts and geometric principles. The ability to solve such problems is a testament to a strong understanding of linear equations and their properties.
Conclusion
In conclusion, finding the equation of a line perpendicular to a given line and passing through a specific point involves several key steps: identifying the slope of the given line, finding the negative reciprocal to determine the slope of the perpendicular line, using the point-slope form to create an initial equation, and converting to slope-intercept form for the final answer. This process demonstrates the importance of understanding the relationships between slopes of perpendicular lines and the various forms of linear equations. The final solution, y = 2x + 3, is a testament to the power of algebraic manipulation and the elegance of mathematical problem-solving. Mastering these skills is essential for success in mathematics and related fields. The concepts discussed here are fundamental building blocks for more advanced topics in algebra and geometry. Continuous practice and application of these principles will solidify understanding and enhance problem-solving abilities. The journey from the initial problem setup to the final verified solution highlights the logical and systematic nature of mathematics, making it a rewarding and intellectually stimulating endeavor.
