Finding The Perpendicular Line Equation In Slope-Intercept Form A Step-by-Step Guide

Hey everyone! Today, we're diving into a cool math problem that involves finding the equation of a line. But not just any line – a line that's perpendicular to another line and passes through a specific point. Sounds like a fun challenge, right? Let's break it down step by step.

Understanding the Problem

Okay, so the main goal here is to figure out the equation of a line in what we call slope-intercept form. Remember that? It's the classic y = mx + b format, where m represents the slope of the line and b is the y-intercept (where the line crosses the y-axis). Now, there are two key pieces of information we need to consider to solve this problem:

1. We're given a line: y - 4 = -2/3(x - 6). This is our starting point. We need to understand its slope to find the slope of the perpendicular line.
2. We're given a point: (-2, -2). This point must lie on the new line we're trying to find. This is crucial for determining the exact equation.

So, our mission is to manipulate these pieces of information, do some mathematical maneuvering, and arrive at the equation of the perpendicular line in the beautiful y = mx + b form. Let's get started!

Step 1: Finding the Slope of the Given Line

The very first thing we need to do is figure out the slope of the line we've been given: y - 4 = -2/3(x - 6). Now, this equation looks a little different from our standard y = mx + b form, doesn't it? It's currently in what's called point-slope form. But don't worry, we can easily transform it!

The point-slope form of a linear equation looks like this: y - y1 = m(x - x1), where m is the slope, and (x1, y1) is a point on the line. Comparing this to our given equation, y - 4 = -2/3(x - 6), we can immediately see something important: the slope of this line is -2/3. See how the -2/3 is sitting in the m position? Awesome!

But to be absolutely sure and to make things clearer for the next steps, let's convert this equation to slope-intercept form (y = mx + b). To do this, we simply need to distribute the -2/3 and then isolate y:

y - 4 = -2/3(x - 6) y - 4 = -2/3 * x + (-2/3) * (-6) y - 4 = -2/3x + 4 y = -2/3x + 4 + 4 y = -2/3x + 8

Ta-da! We've got it in slope-intercept form. We can clearly see that the slope (m) of the given line is indeed -2/3. This is a crucial piece of the puzzle. Remember this slope, because we're about to use it to find the slope of the perpendicular line. Understanding how to convert between point-slope and slope-intercept form is a fundamental skill in algebra, guys. Make sure you're comfortable with this!

Step 2: Finding the Slope of the Perpendicular Line

Now comes the cool part – figuring out the slope of a line that's perpendicular to our given line. Remember, perpendicular lines intersect at a 90-degree angle, a perfect right angle. This geometric relationship gives us a very special connection between their slopes.

Here's the key: The slopes of perpendicular lines are negative reciprocals of each other. What does that mean in plain English? Well, it means two things:

1. Flip the fraction: If your original slope is a fraction (like -2/3), flip it upside down (to get -3/2).
2. Change the sign: If your original slope is negative, make it positive. If it's positive, make it negative.

So, let's apply this to our situation. The slope of our given line is -2/3. To find the slope of the perpendicular line, we:

1. Flip it: -2/3 becomes -3/2
2. Change the sign: -3/2 becomes 3/2

Therefore, the slope of the line perpendicular to y = -2/3x + 8 is 3/2. We've found our new slope! This is a huge step forward. Remember this negative reciprocal relationship; it's a cornerstone of coordinate geometry. Knowing how slopes of perpendicular lines relate is super important, so keep this in your mathematical toolkit.

Step 3: Using Point-Slope Form with the New Slope and Point

Alright, we've got the slope of our perpendicular line (which is 3/2), and we know it passes through the point (-2, -2). Now, we're going to use this information to construct the equation of our line. And guess what? We're going back to point-slope form! Remember, it's y - y1 = m(x - x1). This form is super handy when you have a slope (m) and a point (x1, y1). Guys, this is where all the pieces start coming together!

Let's plug in the values we know:

• m = 3/2 (the slope of our perpendicular line)
• (x1, y1) = (-2, -2) (the point the line passes through)

So, our equation in point-slope form looks like this:

y - (-2) = 3/2(x - (-2))

Notice the double negatives? We need to simplify those. y - (-2) becomes y + 2, and x - (-2) becomes x + 2. So, our equation now looks like this:

y + 2 = 3/2(x + 2)

We've successfully used the point-slope form! But remember, our ultimate goal is to get the equation in slope-intercept form (y = mx + b). So, we've got one more step to go.

Step 4: Converting to Slope-Intercept Form

We're in the home stretch now! We have the equation of our perpendicular line in point-slope form: y + 2 = 3/2(x + 2). To get it into the familiar slope-intercept form (y = mx + b), we just need to do a little bit of algebraic manipulation.

First, we'll distribute the 3/2 on the right side of the equation:

y + 2 = 3/2 * x + 3/2 * 2 y + 2 = 3/2x + 3

Now, we need to isolate y. To do this, we subtract 2 from both sides of the equation:

y + 2 - 2 = 3/2x + 3 - 2 y = 3/2x + 1

And there we have it! We've successfully converted our equation to slope-intercept form! The equation of the line perpendicular to y - 4 = -2/3(x - 6) and passing through the point (-2, -2) is:

y = 3/2x + 1

Isn't it satisfying when all the steps come together and you get the final answer? This process of converting between different forms of linear equations is a key skill in algebra and geometry.

Conclusion

Wow, we did it! We successfully navigated the world of perpendicular lines and slope-intercept form. We started with a given line, found the slope of a perpendicular line, and then used a point and our new slope to construct the equation of the perpendicular line in y = mx + b format.

This problem really highlights the power of understanding the relationships between slopes of lines and the different forms of linear equations. We used point-slope form as a stepping stone to get to slope-intercept form, and that's a common and very useful technique.

So, next time you encounter a problem like this, remember the steps we took:

1. Find the slope of the given line.
2. Calculate the slope of the perpendicular line (remember the negative reciprocal!).
3. Use point-slope form with the new slope and the given point.
4. Convert to slope-intercept form.

With practice, these steps will become second nature, and you'll be solving these kinds of problems like a pro. Keep up the great work, guys, and remember to always break down complex problems into smaller, manageable steps. You've got this! Understanding the concepts deeply is more important than just memorizing formulas, so make sure you grasp the 'why' behind the 'how'.

Introduction Unlocking the Secrets of Linear Equations

In the vast realm of mathematics, linear equations hold a position of fundamental importance, serving as the building blocks for more advanced concepts and practical applications. Among the various forms of linear equations, the slope-intercept form reigns supreme for its clarity and ease of use. Guys, understanding this form is crucial for tackling a wide array of problems! We're not just talking about abstract math here; linear equations pop up in physics, economics, computer science – you name it. They help us model real-world relationships, predict trends, and make informed decisions. Today, we're zeroing in on a specific scenario: finding the equation of a line that's perpendicular to another line and passes through a given point. This might sound a bit complex at first, but fear not! We'll break it down step by step, using a friendly and conversational approach to make sure you grasp every concept. The key is to build a solid foundation, so let's get started! So, what exactly are we setting out to achieve? Our main goal is to become fluent in the language of linear equations, specifically mastering the art of finding the equation of a line that meets two key criteria: it must be perpendicular to a given line, meaning it intersects the original line at a right angle, and it must pass through a specific point in the coordinate plane. To do this, we'll dive deep into the underlying principles, explore the connections between different forms of linear equations, and work through a detailed example to solidify your understanding. Think of it like learning a new language – the more you practice, the more fluent you'll become! By the end of this comprehensive guide, you'll not only be able to solve this type of problem with confidence but also gain a deeper appreciation for the elegance and power of linear equations. So, let's put on our mathematical hats and embark on this exciting journey together! Remember, the journey of a thousand miles begins with a single step. And in our case, that first step is understanding the slope-intercept form itself. Let's unravel its mysteries! Guys, you've got this – let's learn this together!

Delving into Slope-Intercept Form The Cornerstone of Linear Equations

At the heart of our quest lies the venerable slope-intercept form, a mathematical expression that elegantly captures the essence of a straight line. This form, represented as y = mx + b, holds the key to unlocking many secrets of linear equations. It's like the Rosetta Stone for understanding lines! In this equation, y and x are our trusty variables, representing the coordinates of any point that graces the line. The magic, however, lies in the coefficients m and b. The coefficient m boldly proclaims the slope of the line, a numerical measure of its steepness and direction. Think of it as the line's incline – how much it rises or falls for every step you take to the right! A positive slope signals an upward climb as you move from left to right, while a negative slope indicates a descent. The larger the absolute value of m, the steeper the incline or decline. Now, let's turn our attention to b, the y-intercept. This unassuming constant reveals the precise point where our line intersects the y-axis, that vertical boundary in our coordinate plane. It's like the line's home base on the y-axis! When x is zero, y gracefully settles at the value of b, marking the line's passage through this crucial intersection. Understanding these two key components, the slope m and the y-intercept b, empowers us to not only visualize a line but also to craft its equation. It's like having the blueprint for building a straight line! We can extract these values from a graph, calculate them from two points on the line, or even manipulate an existing equation to unveil them. Mastering this form is paramount, as it streamlines numerous mathematical tasks, from graphing lines to solving systems of equations. Guys, think of slope-intercept form as your secret weapon for tackling linear equations! It provides a clear and concise way to represent lines, making it easier to analyze their behavior and relationships. So, embrace the power of y = mx + b – it will serve you well in your mathematical adventures! But what happens when lines decide to intersect at a perfect right angle? That's where the concept of perpendicularity comes into play. Let's explore this fascinating relationship between lines!

Unveiling Perpendicular Lines The Dance of Slopes

When two lines pirouette across the coordinate plane and meet at a perfect 90-degree angle, we declare them perpendicular. This geometric harmony translates into a special relationship between their slopes, a relationship we must understand to solve our problem. It's like a secret handshake between lines that form right angles! The key concept here is the negative reciprocal. If one line boasts a slope of m, its perpendicular partner flaunts a slope of -1/m. This means we perform two operations to transform the original slope: we flip it (take the reciprocal) and change its sign (from positive to negative or vice versa). Think of it as a mathematical makeover for the slope! For instance, if a line gracefully ascends with a slope of 2, its perpendicular counterpart descends sharply with a slope of -1/2. If another line takes a gentler slope of 1/3, its perpendicular friend plunges steeply with a slope of -3. This negative reciprocal relationship guarantees that the lines will intersect at a right angle, forming those perfect corners we call perpendicular. It's a beautiful example of how mathematics reflects geometric harmony! Understanding this principle empowers us to find the slope of a line perpendicular to any given line. Simply take the negative reciprocal of the original slope, and you've unlocked the slope of its perpendicular partner. It's like having a mathematical decoder ring for finding perpendicular slopes! But how do we use this knowledge to actually find the equation of a perpendicular line? That's where the point-slope form enters our story. Guys, get ready to add another powerful tool to your mathematical arsenal!

Point-Slope Form A Versatile Equation for Lines

While slope-intercept form reigns supreme for its clarity, the point-slope form shines in its versatility. This equation, expressed as y - y1 = m(x - x1), proves invaluable when we know a line's slope (m) and a point (x1, y1) it gracefully passes through. It's like a tailor-made equation, crafted to fit a specific point and slope! Unlike slope-intercept form, which explicitly reveals the y-intercept, point-slope form keeps it hidden, focusing instead on the relationship between any point (x, y) on the line and the known point (x1, y1). Think of it as a flexible equation that adapts to different situations! The term (y - y1) represents the vertical change, while (x - x1) captures the horizontal change. Their ratio, of course, is none other than the slope m, the very essence of the line's inclination. This form allows us to construct the equation of a line even without knowing its y-intercept, as long as we have a point and the slope. It's a lifesaver when the y-intercept is elusive or difficult to calculate! To wield this form effectively, simply plug in the known slope m and the coordinates of the point (x1, y1), then simplify the equation. You can leave the equation in point-slope form, or you can manipulate it to unveil the familiar slope-intercept form, depending on your needs. It's like having a transformer equation, ready to adapt to different forms! In our quest to find the equation of a perpendicular line, the point-slope form will serve as a crucial bridge, connecting the perpendicular slope we calculated with the given point the line must traverse. Guys, this is where the magic truly happens – let's see how it all comes together!

Solving the Puzzle Finding the Perpendicular Line Equation

Now, let's bring all our mathematical tools together and tackle the problem head-on: finding the equation, in slope-intercept form, of the line that is perpendicular to the line y - 4 = -2/3(x - 6) and passes through the point (-2, -2). It's like assembling a complex machine, where each piece plays a vital role! First, we must decipher the slope of the given line. Notice that the equation is presented in point-slope form, a clever disguise that we must penetrate. By carefully comparing it to the general form y - y1 = m(x - x1), we can discern the slope m lurking within: it's -2/3. Think of it as spotting a hidden clue in a mathematical mystery! Armed with this knowledge, we can summon the concept of negative reciprocals to find the slope of our perpendicular line. Flip -2/3 and change its sign, and we proudly obtain 3/2, the slope of our target line. It's like unlocking the secret code for perpendicularity! Next, we embrace the point-slope form, our versatile equation that thrives on slopes and points. We know the perpendicular slope is 3/2, and the line must elegantly pass through the point (-2, -2). Plugging these values into y - y1 = m(x - x1), we obtain y - (-2) = 3/2(x - (-2)), which simplifies to y + 2 = 3/2(x + 2). It's like crafting a custom-fit equation for our line! But our quest demands slope-intercept form, the gold standard for clarity. So, we must transform our point-slope equation. First, we distribute the 3/2 on the right side, yielding y + 2 = 3/2x + 3. Then, we gracefully subtract 2 from both sides, isolating y and revealing our final answer: y = 3/2x + 1. It's like unveiling the masterpiece hidden within the equation! This equation, in its elegant slope-intercept form, proudly represents the line that is perpendicular to the given line and passes through the specified point. We have successfully navigated the mathematical maze and emerged victorious. Guys, give yourselves a pat on the back – you've earned it!

Conclusion Mastering the Art of Perpendicular Lines

In this comprehensive journey, we've not only conquered a specific problem but also gained a deeper understanding of the fundamental concepts underlying linear equations. We've explored the elegant slope-intercept form, the versatile point-slope form, and the crucial relationship between slopes of perpendicular lines. It's like building a strong foundation for future mathematical adventures! By dissecting the problem step-by-step, we've learned to identify key information, apply relevant principles, and transform equations with confidence. Think of it as developing a mathematical problem-solving toolkit! We now possess the skills to find the equation of a line perpendicular to another line and passing through a given point, a valuable ability in various mathematical and real-world contexts. But more importantly, we've cultivated a deeper appreciation for the beauty and interconnectedness of mathematical concepts. It's like discovering the hidden harmonies within the world of numbers! Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying logic, making connections, and developing a problem-solving mindset. Guys, embrace the challenge, explore the patterns, and never stop learning! So, go forth and conquer linear equations, knowing that you now possess the knowledge and skills to tackle them with grace and confidence. And remember, the journey of mathematical discovery is a lifelong adventure – enjoy the ride! Keep practicing, keep exploring, and keep the mathematical spirit alive! Now, let's put your newfound knowledge to the test. Can you apply these principles to solve other similar problems? Challenge yourself, and watch your mathematical prowess grow!