Line Equation: Perpendicular To Y=2x+7, Passes Through (-3,5)

Hey math enthusiasts! Ever wondered how to find the equation of a line that's not just any line, but one that's perpendicular to another, and passes through a specific point? It might sound tricky, but trust me, it's totally doable. We're going to break down the process step-by-step, using a classic example. Let's dive into a problem that often pops up in algebra and geometry: finding the equation of a line that passes through the point (-3, 5) and is perpendicular to the line y = 2x + 7. This is a fundamental concept in coordinate geometry, and understanding it will not only help you ace your exams but also give you a solid foundation for more advanced math topics. So, buckle up, and let's get started!

Understanding Perpendicular Lines

Before we jump into the solution, let's quickly recap what it means for lines to be perpendicular. Perpendicular lines are lines that intersect at a right angle (90 degrees). A crucial property of perpendicular lines is their slopes. If one line has a slope of m, a line perpendicular to it will have a slope of -1/m. This is often called the negative reciprocal. Grasping this relationship between slopes is the key to solving problems like the one we're tackling today. Remember, the slope of a line tells us how steep it is and whether it's increasing or decreasing as we move from left to right. A positive slope indicates an increasing line, while a negative slope indicates a decreasing line. The steeper the line, the larger the absolute value of the slope. When two lines are perpendicular, their slopes have a special relationship: they are negative reciprocals of each other. This means that if one line has a slope of, say, 2, the perpendicular line will have a slope of -1/2. This negative reciprocal relationship ensures that the lines intersect at a perfect right angle. So, keep this in mind as we move forward – it's the foundation upon which we'll build our solution. It’s really important to understand this concept, guys, because it's the backbone of solving this type of problem. Think of it as the secret code to unlocking the answer! We use slopes every day, maybe without even realizing it! Imagine you're skiing down a hill – the slope tells you how steep the hill is. Or picture a ramp for a wheelchair; the slope determines how easy or difficult it is to go up the ramp. In math, we use the same idea to describe lines on a graph. The slope is a number that tells us how much the line rises (or falls) for every step we take to the right. It's a measure of the line's steepness and direction. And when we're dealing with perpendicular lines, the relationship between their slopes is crucial. Remember, perpendicular lines intersect at a 90-degree angle, and their slopes are negative reciprocals of each other. This means that if one line has a slope of m, the perpendicular line will have a slope of -1/m. This simple rule is the key to solving problems involving perpendicular lines, so make sure you understand it well!

Identifying the Slope of the Given Line

Our problem involves a line given in slope-intercept form: y = 2x + 7. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. In our case, the given line has a slope of 2. This is the coefficient of the x term. Recognizing this is the first step in finding the equation of the perpendicular line. The slope-intercept form is super handy because it directly reveals the slope and y-intercept of a line. The slope, which is the number multiplied by x, tells us how steep the line is. The y-intercept, which is the constant term, tells us where the line crosses the y-axis. In our case, the equation y = 2x + 7 is already in slope-intercept form, which makes our job easier. We can immediately see that the slope of the given line is 2. This means that for every one unit we move to the right along the line, we move two units up. The line is quite steep, and it's increasing as we move from left to right. But remember, we're not interested in the slope of this line itself. We need to find the slope of a line that's perpendicular to it. And that's where the negative reciprocal comes in! The slope of the line y = 2x + 7 is crucial information because it allows us to determine the slope of any line perpendicular to it. Once we know the slope of the perpendicular line, we can use the point-slope form of a linear equation to find its equation. So, make sure you're comfortable identifying the slope from the slope-intercept form – it's a fundamental skill in algebra and geometry. You'll encounter it again and again, so mastering it now will save you a lot of time and effort in the future. Think of the slope as the DNA of a line – it contains all the essential information about its direction and steepness. And by understanding how to extract this information from an equation, you can unlock a whole world of geometric insights!

Determining the Slope of the Perpendicular Line

Now, we need to find the slope of the line perpendicular to y = 2x + 7. As we discussed, the slope of a perpendicular line is the negative reciprocal of the original line's slope. Since the slope of the given line is 2, the slope of the perpendicular line is -1/2. This is a key step in solving the problem. To find the slope of the perpendicular line, we take the negative reciprocal of the original slope. This means we flip the fraction and change its sign. In our case, the original slope is 2, which can be written as 2/1. Flipping the fraction gives us 1/2, and changing the sign gives us -1/2. So, the slope of the perpendicular line is -1/2. This negative slope tells us that the perpendicular line will be decreasing as we move from left to right. It will slant downwards, in contrast to the original line, which slants upwards. The fact that the slope is 1/2 (in absolute value) also tells us that the perpendicular line will be less steep than the original line. Remember, the larger the absolute value of the slope, the steeper the line. So, a slope of -1/2 indicates a relatively gentle downward slope. This relationship between the slopes of perpendicular lines is a fundamental concept in geometry, and it's essential for solving a wide range of problems. Make sure you understand why the negative reciprocal works – it's not just a random rule! It's based on the geometric properties of right angles and the way slopes are defined. When two lines are perpendicular, their slopes multiply to -1. This is another way to think about the negative reciprocal relationship. So, keep practicing with different examples until you feel confident in your ability to find the slope of a perpendicular line. It's a skill that will serve you well in your mathematical journey. And remember, guys, this is where the magic happens! This little flip and sign change is what allows us to construct a line that's perfectly perpendicular to the original. It's like a mathematical dance move, where we take the original slope and transform it into its opposite, creating a harmonious right angle.

Using the Point-Slope Form

We know the slope of the perpendicular line (-1/2) and a point it passes through (-3, 5). We can use the point-slope form of a linear equation to find the equation of the line. The point-slope form is y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. Plugging in our values, we get y - 5 = -1/2(x - (-3)), which simplifies to y - 5 = -1/2(x + 3). The point-slope form is a powerful tool because it allows us to write the equation of a line as long as we know one point on the line and its slope. It's like having a GPS for lines – you give it a starting point and a direction, and it plots the entire course. In our case, we know that the perpendicular line passes through the point (-3, 5) and has a slope of -1/2. So, we can plug these values into the point-slope form formula: y - y1 = m(x - x1). Here, (x1, y1) is the point (-3, 5), and m is the slope -1/2. Substituting these values, we get: y - 5 = -1/2(x - (-3)). Notice the double negative in the expression (x - (-3)). This is a common pitfall, so be careful to simplify it correctly. Subtracting a negative number is the same as adding a positive number, so (x - (-3)) becomes (x + 3). Now our equation looks like this: y - 5 = -1/2(x + 3). This is the equation of the perpendicular line in point-slope form. It tells us everything we need to know about the line: its slope and a point it passes through. But often, we want to express the equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). To do this, we need to simplify the equation further. However, the point-slope form is perfectly valid and often the most convenient form for solving certain types of problems. So, remember this formula – it's a valuable addition to your mathematical toolkit. And guys, this is where the equation really starts to take shape! We're plugging in the pieces we've gathered – the slope and the point – and the point-slope form is like the mold that holds it all together. It's a beautiful thing to see a line defined by just a few key ingredients!

The Answer and Why It's Correct

The equation we found, y - 5 = -1/2(x + 3), matches option B. This is the correct answer. Let's quickly recap why. We found the negative reciprocal of the original slope to get the slope of the perpendicular line, and then we used the point-slope form with the given point to construct the equation. Boom! We nailed it. The fact that our equation matches one of the given options is a good sign that we're on the right track. But it's always a good idea to double-check our work, especially in math problems. Make sure you've correctly identified the slope of the original line, calculated the negative reciprocal, and plugged the values into the point-slope form. Also, it's helpful to think about the geometric meaning of the equation. Does it make sense that a line with a slope of -1/2 would be perpendicular to a line with a slope of 2? Does it make sense that the line would pass through the point (-3, 5)? Visualizing the lines on a graph can help you confirm your answer. In this case, a line with a negative slope will slant downwards, and the slope of -1/2 is less steep than the original slope of 2. Also, the point (-3, 5) should lie on the line if we plug it into the equation. By checking these things, we can be confident that our answer is correct. And remember, guys, the goal isn't just to get the right answer – it's to understand the process and the concepts behind it. So, take the time to review the steps we've taken and make sure you understand why each one is necessary. This will not only help you solve similar problems in the future but also deepen your overall understanding of mathematics. You've got this!

Conclusion: Mastering Perpendicular Lines

So, there you have it! We've successfully navigated the process of finding the equation of a line perpendicular to another line and passing through a given point. This is a fundamental skill in algebra and geometry, and mastering it will open doors to more advanced concepts. The key takeaways are understanding the negative reciprocal relationship between perpendicular slopes and using the point-slope form to construct the equation. Keep practicing, and you'll become a pro in no time. Remember, math isn't just about memorizing formulas – it's about understanding the underlying principles and applying them creatively. The more you practice, the more natural these concepts will become. And don't be afraid to ask questions! If you're stuck on a particular step, reach out to a teacher, a tutor, or a fellow student. Explaining your thinking to someone else can often help you clarify your own understanding. So, keep exploring, keep questioning, and keep learning! The world of math is full of fascinating ideas and challenges, and you have the potential to unlock its secrets. And hey, guys, you've conquered a tough math problem today! That deserves a pat on the back. You've shown that you can tackle complex concepts and break them down into manageable steps. This is a valuable skill that will serve you well in all areas of life. So, keep challenging yourself, keep pushing your limits, and keep striving for excellence. You're doing great!