Finding Equation Of Perpendicular Line With Same Y-Intercept

Hey guys! Let's dive into a cool math problem today that involves lines, slopes, and y-intercepts. We're going to figure out how to find the equation of a line that's perpendicular to another line and shares the same y-intercept. Sounds like a fun challenge, right? Let's get started!

Understanding the Problem

Before we jump into solving the problem, let's break down what it's asking. We have a given line, and we need to find a new line that meets two specific conditions:

1. Perpendicular: The new line must be perpendicular to the given line. Remember, perpendicular lines intersect at a 90-degree angle, forming a perfect right angle.
2. Same y-intercept: The new line must cross the y-axis at the same point as the given line. The y-intercept is the value of y when x is 0.

Our given line is y = (1/5)x + 1. This equation is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. So, from our given equation, we can easily see that the slope is 1/5 and the y-intercept is 1. Our mission is to find the equation of a new line that’s perpendicular to this one and also has a y-intercept of 1. Let’s explore the concept of slopes and y-intercepts a bit more to make sure we’re all on the same page. The slope of a line is a measure of its steepness and direction. It tells us how much the line rises or falls for every unit of horizontal change. A positive slope means the line goes upwards from left to right, while a negative slope means the line goes downwards. The steeper the line, the larger the absolute value of the slope. The slope is often referred to as “rise over run,” where rise is the vertical change and run is the horizontal change. The y-intercept, on the other hand, is the point where the line intersects the y-axis. This is the point where the x-coordinate is zero. In the slope-intercept form (y = mx + b), the y-intercept is represented by the constant term, b. So, if we have an equation like y = 2x + 3, the y-intercept is 3, meaning the line crosses the y-axis at the point (0, 3). Understanding these basic concepts is crucial for tackling problems involving linear equations. Now that we have a solid grasp of slopes and y-intercepts, let’s dive deeper into the relationship between the slopes of perpendicular lines, which is a key piece of the puzzle for solving our main problem. Knowing how slopes change when lines are perpendicular will give us the tools we need to find the equation of the line we’re looking for. So, let’s get to it!

Slopes of Perpendicular Lines

This is a super important concept for solving our problem. Perpendicular lines have slopes that are negative reciprocals of each other. What does that mean? Well, if one line has a slope of m, a line perpendicular to it will have a slope of -1/m. So, to find the slope of a perpendicular line, you flip the fraction (take the reciprocal) and change the sign.

For example:

• If a line has a slope of 2, a perpendicular line will have a slope of -1/2.
• If a line has a slope of -3/4, a perpendicular line will have a slope of 4/3.

Let's apply this to our given line, y = (1/5)x + 1. The slope of this line is 1/5. To find the slope of a line perpendicular to it, we take the negative reciprocal of 1/5. First, we flip the fraction to get 5/1, which is just 5. Then, we change the sign to get -5. So, the slope of our perpendicular line will be -5. Understanding the relationship between slopes of perpendicular lines is crucial in many areas of geometry and linear algebra. The negative reciprocal concept allows us to easily determine if two lines are perpendicular and to find the equation of a line perpendicular to a given line. This is a fundamental concept used in various applications, from computer graphics and engineering to physics and navigation. For instance, in computer graphics, ensuring lines are perpendicular is vital for creating accurate and visually appealing images. In engineering, perpendicularity is often a key factor in structural design, ensuring stability and load distribution. In physics, understanding perpendicular forces and vectors is essential for analyzing motion and equilibrium. The negative reciprocal relationship is also used in navigation systems, where the calculation of perpendicular paths is necessary for determining optimal routes and avoiding obstacles. By mastering this concept, you’re not just solving mathematical problems; you’re also gaining insights into how mathematical principles are applied in real-world scenarios. So, next time you see two lines intersecting at a right angle, remember the negative reciprocal rule and appreciate the mathematical harmony at play. Now that we have the slope of the perpendicular line, which is -5, and we know the y-intercept should be the same as the given line, which is 1, we are just one step away from writing the equation of the perpendicular line. Let’s put it all together and finalize our solution!

Using the Slope-Intercept Form

Okay, we're in the home stretch! We know the slope of our perpendicular line (-5) and the y-intercept (1). We can use the slope-intercept form of a line, y = mx + b, to write the equation. Remember, m is the slope and b is the y-intercept.

Plugging in our values, we get:

y = -5x + 1

And that's it! This is the equation of the line that is perpendicular to y = (1/5)x + 1 and has the same y-intercept. Let's just recap quickly what we did to arrive at this answer. We started by understanding the problem, recognizing that we needed to find a line perpendicular to the given line and sharing the same y-intercept. We then delved into the concept of perpendicular lines and their slopes, learning that the slopes are negative reciprocals of each other. We applied this knowledge to find the slope of our perpendicular line, which turned out to be -5. Next, we identified the y-intercept of the given line, which is 1, and understood that our new line needed to have the same y-intercept. Finally, we used the slope-intercept form of a line, y = mx + b, to combine the slope and y-intercept we found, resulting in the equation y = -5x + 1. This step-by-step approach is key to solving many mathematical problems. By breaking down the problem into smaller, manageable parts, we can tackle even complex questions with confidence. Remember to always review the problem, identify the relevant concepts, apply the necessary formulas, and finally, double-check your answer to ensure it makes sense. Now, let’s discuss some common mistakes people make when dealing with problems like this and how you can avoid them. Being aware of these pitfalls can further solidify your understanding and improve your problem-solving skills. So, let’s move on to the next section and explore these common errors.

Common Mistakes to Avoid

When working with perpendicular lines and y-intercepts, there are a few common mistakes people often make. Let's talk about them so you can steer clear!

1. Forgetting the negative: When finding the slope of a perpendicular line, it's easy to remember to take the reciprocal but forget to change the sign. Always remember to make the slope negative (if it was positive) or positive (if it was negative).
2. Incorrectly calculating the reciprocal: Make sure you flip the fraction correctly. For example, the reciprocal of 2 (which can be written as 2/1) is 1/2, not 2.
3. Confusing slope and y-intercept: The slope is the number multiplied by x, and the y-intercept is the constant term. Don't mix them up!
4. Not using the slope-intercept form correctly: When plugging in the slope and y-intercept, make sure you put them in the correct places in the equation y = mx + b.

Avoiding these common mistakes is crucial for getting the correct answer. One of the best ways to avoid these mistakes is to practice regularly and develop a solid understanding of the underlying concepts. The more you work with these types of problems, the more comfortable you’ll become with identifying the slope and y-intercept, finding the negative reciprocal, and applying the slope-intercept form. Another helpful strategy is to double-check your work. After you’ve found the equation of the line, take a moment to review your steps and ensure that each calculation is correct. You can also try graphing the original line and the perpendicular line you found to visually confirm that they are indeed perpendicular and share the same y-intercept. This visual check can often help you catch errors that you might have missed in your calculations. Additionally, consider working through similar problems with varying slopes and y-intercepts. This will help you build a deeper understanding of the concepts and become more adept at applying them in different scenarios. Don’t hesitate to seek help or clarification if you’re struggling with a particular step or concept. Talking through the problem with a classmate, teacher, or online forum can often provide valuable insights and help you identify areas where you need more practice. Remember, making mistakes is a natural part of the learning process. The key is to learn from your mistakes and use them as opportunities to improve your understanding and skills. By being aware of these common pitfalls and actively working to avoid them, you’ll be well on your way to mastering the concepts of perpendicular lines and y-intercepts. Now that we’ve covered the common mistakes, let’s recap the main points and see how we can apply these concepts to other problems.

Wrapping Up

Great job, guys! We've successfully found the equation of a line perpendicular to a given line with the same y-intercept. Just to recap, here's what we did:

1. Identified the slope and y-intercept of the given line.
2. Found the negative reciprocal of the slope to get the slope of the perpendicular line.
3. Used the slope-intercept form (y = mx + b) to write the equation of the new line.

Remember, the key to solving these types of problems is understanding the relationship between the slopes of perpendicular lines and how to use the slope-intercept form. These concepts are foundational in algebra and geometry, and mastering them will open doors to more advanced topics. To solidify your understanding, try working through additional problems with different slopes and y-intercepts. Challenge yourself to find equations of lines that are parallel or perpendicular to each other, and experiment with different forms of linear equations, such as the point-slope form and the standard form. Understanding how to convert between these forms will give you greater flexibility in solving problems and a deeper appreciation for the versatility of linear equations. Furthermore, consider exploring real-world applications of these concepts. Linear equations are used extensively in fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations can describe the motion of objects under constant velocity. In engineering, they can be used to model the behavior of circuits and structures. In economics, they can represent supply and demand curves. And in computer science, they are used in various algorithms and data structures. By connecting these mathematical concepts to real-world scenarios, you’ll not only enhance your understanding but also appreciate the practical significance of what you’re learning. So, keep practicing, keep exploring, and keep challenging yourself. The more you engage with these concepts, the more confident and proficient you’ll become in mathematics. And remember, if you ever encounter a problem you’re not sure how to solve, don’t hesitate to break it down into smaller steps, review the fundamental principles, and seek help when needed. With perseverance and a solid understanding of the core concepts, you can tackle any mathematical challenge that comes your way. Keep up the great work, and happy problem-solving!

I hope this explanation helps you guys understand how to find the equation of a perpendicular line with the same y-intercept. Keep practicing, and you'll become a pro in no time!