Perpendicular Line Equation: A Step-by-Step Solution

Hey guys! Today, we're diving into a classic math problem: finding the equation of a line that's perpendicular to a given line and passes through a specific point. This might sound intimidating, but don't worry, we'll break it down into easy-to-follow steps. We'll use the example of finding a line perpendicular to 3x + 5y = -9 and passing through the point (3, 0). Let's get started!

Understanding Perpendicular Lines

Before we jump into the calculations, let's make sure we're all on the same page about perpendicular lines. Perpendicular lines are lines that intersect at a right angle (90 degrees). The key concept here is the relationship between their slopes. If we have two lines, line 1 and line 2, and their slopes are m₁ and m₂ , then the lines are perpendicular if and only if:

m₁ * m₂ = -1

In other words, the slopes of perpendicular lines are negative reciprocals of each other. This means you flip the fraction and change the sign. For example, if a line has a slope of 2/3, a line perpendicular to it will have a slope of -3/2. This negative reciprocal relationship is absolutely crucial for solving these types of problems. It's the foundation upon which we'll build our solution, so make sure you've got this concept down pat. We'll be using it extensively throughout the process, so a solid understanding of negative reciprocals will make everything much smoother. Think of it like the cornerstone of a building – without it, the rest of the structure won't stand properly. So, take a moment to really internalize this idea. It's not just about memorizing a formula; it's about understanding the geometric relationship between lines that meet at right angles. Once you grasp this concept, the rest of the process will fall into place much more easily. So, let's move forward with confidence, knowing that we've got a solid understanding of the fundamental principle at play.

Step 1: Find the Slope of the Given Line

Okay, let's apply this to our problem. Our given line is 3x + 5y = -9. To find its slope, we need to rewrite the equation in slope-intercept form, which is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept. So, let's isolate 'y' in our equation. First, we subtract 3x from both sides:

5y = -3x - 9

Then, we divide both sides by 5:

y = (-3/5)x - 9/5

Now, we can clearly see that the slope of the given line is -3/5. This is our m₁ . We've successfully extracted the slope from the given equation by putting it into a familiar form. This process of rearranging the equation is a fundamental skill in algebra, and it's essential for many types of problems, not just this one. So, mastering this technique will serve you well in your mathematical journey. Remember, the key is to isolate 'y' on one side of the equation, and then the coefficient of 'x' will be your slope. It's like peeling back the layers of an onion – we're taking the equation and revealing its underlying structure. In this case, the structure we're interested in is the slope, which is the key to finding the perpendicular line. Now that we have the slope of the given line, we're one step closer to our final answer. We've successfully identified a crucial piece of the puzzle, and we're ready to move on to the next step. So, let's keep up the momentum and see what's next!

Step 2: Determine the Slope of the Perpendicular Line

Now that we know the slope of the given line is -3/5, we can find the slope of the line perpendicular to it. Remember, the slopes of perpendicular lines are negative reciprocals. So, we flip the fraction and change the sign. The negative reciprocal of -3/5 is 5/3. This will be our m₂ , the slope of the perpendicular line. We've just performed a crucial step in solving our problem, and it all hinges on understanding the relationship between perpendicular lines and their slopes. The concept of negative reciprocals might seem a bit abstract at first, but it's a powerful tool that allows us to connect geometry and algebra. It's like having a secret code that unlocks the relationship between these lines. By flipping the fraction and changing the sign, we're essentially finding the slope that will create a right angle when the lines intersect. This is a beautiful example of how mathematical concepts are interconnected and how understanding one concept can lead to understanding others. So, let's take a moment to appreciate the elegance of this relationship before we move on. We've successfully found the slope of the perpendicular line, and now we're ready to use this information to find the equation of the line. So, let's keep going and see how we can put all the pieces together.

Step 3: Use the Point-Slope Form

We now know the slope of the perpendicular line (5/3) and a point it passes through (3, 0). This is where the point-slope form of a linear equation comes in handy. The point-slope form is:

y - y₁ = m(x - x₁)

where m is the slope and (x₁, y₁) is a point on the line. Plugging in our values, we get:

y - 0 = (5/3)(x - 3)

This simplifies to:

y = (5/3)(x - 3)

We've successfully used the point-slope form to create an equation that represents our perpendicular line. This form is particularly useful when you have a point and a slope, as it allows you to directly construct the equation without having to solve for the y-intercept first. It's like having a shortcut that bypasses some of the usual steps. The point-slope form is a versatile tool that can be applied in various situations, so it's definitely worth mastering. It's one of those mathematical concepts that, once you understand it, you'll find yourself using it again and again. Now that we have the equation in point-slope form, we can further simplify it to get it into slope-intercept form, which is often the preferred way to express linear equations. So, let's move on to the next step and see how we can transform our equation into a more familiar format.

Step 4: Convert to Slope-Intercept Form (Optional)

While we have a valid equation, it's often helpful to convert it to slope-intercept form (y = mx + b) for clarity and comparison. Let's distribute the 5/3:

y = (5/3)x - (5/3) * 3

This simplifies to:

y = (5/3)x - 5

Now we have the equation of the perpendicular line in slope-intercept form. We can clearly see the slope (5/3) and the y-intercept (-5). Converting to slope-intercept form gives us a clear picture of the line's characteristics, making it easier to visualize and compare with other lines. It's like taking a raw sketch and turning it into a polished drawing – we're refining the equation to make it more presentable and informative. While the point-slope form is useful for constructing the equation, the slope-intercept form is often preferred for analysis and interpretation. It allows us to quickly identify the slope and y-intercept, which are key features of a linear equation. So, by converting to slope-intercept form, we've essentially made our equation more accessible and user-friendly. This step is optional, but it's a good practice to get into, as it often leads to a deeper understanding of the equation and its properties. Now that we have our equation in slope-intercept form, we're almost at the finish line. Let's take one final step to ensure we've answered the question completely.

Step 5: Convert to Standard Form

To match the answer options, let's convert the equation to standard form, which is Ax + By = C, where A, B, and C are integers, and A is usually positive.
Starting with our slope-intercept form:

y = (5/3)x - 5

Subtract (5/3)x from both sides:

-(5/3)x + y = -5

Multiply both sides by 3 to eliminate the fraction:

-5x + 3y = -15

Multiply both sides by -1 to make A positive:

5x - 3y = 15

This matches one of our answer choices! We've successfully transformed our equation into standard form, which is a common way to represent linear equations. Standard form has its own set of advantages, such as making it easier to find intercepts and compare with other equations in standard form. It's like having another tool in our toolbox that we can use depending on the situation. Converting to standard form is often a matter of preference or the requirements of the problem, but it's a valuable skill to have. It demonstrates a thorough understanding of linear equations and the different ways they can be represented. By going through this final step, we've ensured that our answer is not only correct but also presented in a format that matches the given options. This attention to detail is crucial in mathematics and can make the difference between getting the right answer and missing it. So, let's celebrate our accomplishment and recognize the importance of mastering these different forms of linear equations. We've successfully navigated the process of finding a perpendicular line, and we've learned a lot along the way.

Conclusion

So, the equation of the line that is perpendicular to 3x + 5y = -9 and passes through the point (3, 0) is 5x - 3y = 15. We did it! By breaking down the problem into smaller, manageable steps, we were able to find the solution. Remember, the key is to understand the relationship between the slopes of perpendicular lines and how to use the point-slope form. You've now got the tools to tackle similar problems with confidence. Keep practicing, and you'll become a pro at finding perpendicular lines! We've covered a lot of ground in this guide, from understanding the concept of perpendicular lines to manipulating equations into different forms. We've seen how each step builds upon the previous one, and how a solid understanding of the fundamentals is essential for success. The process of finding a perpendicular line is not just about getting the right answer; it's about developing a deeper understanding of linear equations and their properties. So, take pride in your accomplishment and remember that the more you practice, the more comfortable and confident you'll become. Mathematics is a journey, and each problem you solve is a step forward. So, keep exploring, keep learning, and keep pushing your boundaries. You've got this! And remember, if you ever get stuck, don't hesitate to break the problem down into smaller steps and focus on the fundamentals. You'll be surprised at how much you can achieve with a little bit of patience and perseverance. So, go forth and conquer, and remember to have fun along the way! Until next time, keep those lines perpendicular and those slopes reciprocal!

Repair Input Keyword

What is the equation of the line perpendicular to the line 3x + 5y = -9 and passing through the point (3, 0)?