Equation Of Perpendicular Line: A Step-by-Step Guide

by Sebastian Müller 53 views

Hey guys! Today, we're diving into a super common problem in algebra: finding the equation of a line that's perpendicular to another line and passes through a specific point. It might sound a bit intimidating at first, but trust me, we'll break it down into simple, manageable steps. So, let's get started!

Understanding Perpendicular Lines and Their Slopes

Before we jump into the problem, let's quickly review what it means for lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle (90 degrees). The key to working with perpendicular lines lies in their slopes. The slopes of perpendicular lines have a special relationship: they are negative reciprocals of each other. What does that mean? Well, if one line has a slope of, say, m, then a line perpendicular to it will have a slope of -1/m. This negative reciprocal relationship is crucial for solving problems like the one we're tackling today. Think of it as flipping the fraction and changing the sign! For example, if a line has a slope of 2/3, a perpendicular line will have a slope of -3/2. Similarly, if a line has a slope of -4, a perpendicular line will have a slope of 1/4. Understanding this negative reciprocal relationship is the foundation for finding the equation of a perpendicular line.

Why is this important? Because the slope tells us the steepness and direction of a line. A line with a positive slope goes uphill from left to right, while a line with a negative slope goes downhill. The steeper the line, the larger the absolute value of its slope. When lines are perpendicular, their slopes have this negative reciprocal relationship, ensuring they intersect at a perfect right angle. This concept is not only fundamental in algebra but also has applications in geometry, trigonometry, and even real-world scenarios like architecture and engineering. So, grasping this relationship will definitely help you ace your math courses and beyond! Now that we've refreshed our understanding of perpendicular lines and their slopes, let's move on to the specific problem we're trying to solve.

The Problem: Finding the Perpendicular Line

Okay, so here's the challenge: We need to find the equation of a line that is perpendicular to the line y = -3/4x + 1 and passes through the point (9, 12). Let's break this down piece by piece. First, we have a given line, y = -3/4x + 1. Notice that this equation is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. In our case, the slope of the given line is -3/4. This is super important because, as we just discussed, we need the negative reciprocal of this slope to find the slope of the perpendicular line. Remember, we flip the fraction and change the sign. So, the negative reciprocal of -3/4 is 4/3. This means the line we're trying to find will have a slope of 4/3. Great! We've got the slope. Now, we also know that our line needs to pass through the point (9, 12). This gives us a specific location on the coordinate plane that our line must go through. We have the slope, we have a point – now we just need to put it all together to find the equation of the line.

To recap, we're looking for a line with a slope of 4/3 that goes through the point (9, 12). We've identified the slope of the perpendicular line by taking the negative reciprocal of the original line's slope. We also have a specific point that our line must pass through. The next step is to use this information to find the equation of the line. There are a couple of ways we can do this, and we'll explore the most common and efficient method: using the point-slope form of a linear equation. This form is particularly handy when you have a point and a slope, which is exactly what we have in this situation. So, let's dive into the point-slope form and see how it helps us solve this problem!

Using the Point-Slope Form

The point-slope form of a linear equation is a really useful tool, especially when you know a point on the line and the slope of the line. The formula looks like this: y - y1 = m(x - x1), where m is the slope, and (x1, y1) is the given point. This formula might seem a bit intimidating at first, but it's actually quite straightforward to use. All you need to do is plug in the values you know and then simplify the equation. In our case, we know that the slope of our perpendicular line is 4/3, and it passes through the point (9, 12). So, we can plug these values into the point-slope form: y - 12 = 4/3(x - 9). See? We've just substituted the values into the formula. Now, the next step is to simplify this equation and get it into slope-intercept form (y = mx + b), which is often the preferred way to express a linear equation.

To simplify, we first distribute the 4/3 on the right side of the equation: y - 12 = 4/3x - 12. Notice that 4/3 multiplied by 9 equals 12. Now, we want to isolate y on the left side, so we add 12 to both sides of the equation: y = 4/3x - 12 + 12. This simplifies to y = 4/3x. And there you have it! We've found the equation of the line in slope-intercept form. The line perpendicular to y = -3/4x + 1 and passing through the point (9, 12) is y = 4/3x. Pretty cool, right? By using the point-slope form, we were able to easily plug in the known values and simplify the equation to find our answer. This method is a powerful tool in your algebra arsenal, so make sure you're comfortable using it. Now, let's double-check our work to make sure we've got it right!

Double-Checking Our Work

It's always a good idea to double-check your work, especially in math. So, let's make sure that the equation we found, y = 4/3x, is indeed correct. There are a couple of ways we can do this. First, we can check if the slope is correct. We know that the slope of the original line was -3/4, and we found that the slope of the perpendicular line is 4/3. These are indeed negative reciprocals of each other, so the slopes are correct. Awesome! Next, we need to make sure that the line y = 4/3x actually passes through the point (9, 12). To do this, we can plug in the x-coordinate (9) into our equation and see if we get the y-coordinate (12). Let's try it: y = 4/3 * 9. This simplifies to y = 12. Bingo! When we plug in x = 9, we get y = 12, which means the point (9, 12) lies on the line y = 4/3x. So, we've verified that our equation has the correct slope and passes through the given point. This confirms that our answer is correct!

Double-checking your work is a crucial habit to develop in mathematics. It helps you catch any mistakes you might have made and ensures that you're confident in your solutions. In this case, we checked both the slope and the point to make sure our equation was accurate. This process not only gives you peace of mind but also deepens your understanding of the concepts involved. By verifying your answers, you're essentially reinforcing your knowledge and building a stronger foundation for future math problems. Now that we've successfully found the equation of the perpendicular line and double-checked our work, let's summarize the key steps we took to solve this problem.

Summarizing the Steps

Okay, let's quickly recap the steps we took to find the equation of the line that is perpendicular to y = -3/4x + 1 and contains the point (9, 12). First, we identified the slope of the given line, which was -3/4. Then, we found the negative reciprocal of this slope, which is 4/3. This is the slope of our perpendicular line. Next, we used the point-slope form of a linear equation, y - y1 = m(x - x1), to plug in the slope (4/3) and the point (9, 12). This gave us the equation y - 12 = 4/3(x - 9). We then simplified this equation to get it into slope-intercept form, y = mx + b. After distributing and isolating y, we arrived at the equation y = 4/3x. Finally, we double-checked our work by verifying that the slope was correct and that the point (9, 12) lies on the line. We plugged in x = 9 and confirmed that y = 12, which validated our answer.

By breaking the problem down into these steps, we made it much easier to solve. This step-by-step approach is a valuable strategy for tackling any math problem. It allows you to focus on one aspect at a time, making the overall process less overwhelming. Remember, understanding the concepts is key. Knowing the relationship between the slopes of perpendicular lines and being comfortable with the point-slope form are essential for solving these types of problems. Practice makes perfect, so try working through similar examples to solidify your understanding. With a little practice, you'll be finding equations of perpendicular lines like a pro! And that's a wrap, guys! We've successfully navigated this problem and learned some valuable skills along the way. Keep practicing, and you'll conquer any math challenge that comes your way!

Practice Problems

Want to really nail down this concept? Try these practice problems! They're designed to help you solidify your understanding of finding equations of perpendicular lines. Work through them step-by-step, and don't forget to double-check your answers. The more you practice, the more confident you'll become in your ability to solve these types of problems. Remember, math is like any other skill – the more you use it, the better you get. So, grab a pencil and paper, and let's get started!

  1. Find the equation of the line that is perpendicular to y = 2x - 3 and passes through the point (4, 1).
  2. Find the equation of the line that is perpendicular to y = -1/2x + 5 and passes through the point (-2, 3).
  3. Find the equation of the line that is perpendicular to y = 5/3x + 1 and passes through the point (0, -2).
  4. Find the equation of the line that is perpendicular to y = -4x - 2 and passes through the point (1, 6).
  5. Find the equation of the line that is perpendicular to y = 1/4x + 7 and passes through the point (-8, 0).

These problems cover a range of scenarios, so working through them will give you a well-rounded understanding of the topic. Pay close attention to the slopes and the given points, and remember the negative reciprocal relationship. Use the point-slope form to your advantage, and always double-check your work. If you get stuck, review the steps we discussed earlier in this guide. And most importantly, don't be afraid to ask for help if you need it. Learning math is a journey, and it's okay to stumble along the way. The key is to keep practicing and keep asking questions. So, go ahead and tackle these practice problems – you've got this!

Conclusion

Alright, guys, we've reached the end of our guide on finding the equation of a perpendicular line! We've covered a lot of ground, from understanding the relationship between perpendicular lines and their slopes to using the point-slope form and double-checking our work. We've also tackled some practice problems to solidify your understanding. Hopefully, you now feel more confident in your ability to solve these types of problems. Remember, the key to success in math is understanding the underlying concepts and practicing regularly. Don't be afraid to make mistakes – they're a natural part of the learning process. The important thing is to learn from your mistakes and keep moving forward.

Finding the equation of a perpendicular line is a fundamental skill in algebra, and it's a building block for more advanced topics in mathematics. So, mastering this concept will definitely serve you well in your future math studies. Keep practicing, keep asking questions, and most importantly, keep believing in yourself. You've got the tools and the knowledge to succeed! And that's it for today's guide. I hope you found it helpful and informative. Until next time, happy problem-solving!