Perpendicular Line Equation: Step-by-Step Guide
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 a bit intimidating at first, but trust me, we'll break it down into easy-to-follow steps. We'll use an example question to make things super clear. Let's get started!
Understanding Perpendicular Lines
Before we jump into the calculations, let's quickly review what it means for two lines to be perpendicular. 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 with slope m1 and line 2 with slope m2, they are perpendicular if and only if the product of their slopes is -1. Mathematically, this is expressed as m1 * m2* = -1. In simpler terms, the slope of a line perpendicular to another is the negative reciprocal of the original line's slope. So, if a line has a slope of 2, a line perpendicular to it will have a slope of -1/2. This understanding is crucial for solving our problem.
Now, let's think about why this negative reciprocal relationship exists. Imagine a line with a positive slope, going upwards as you move from left to right. A perpendicular line needs to go downwards to form a right angle, hence the negative sign. The reciprocal part comes from the fact that the steepness of the perpendicular line is inversely related to the steepness of the original line. A steeper line will have a less steep perpendicular line, and vice versa. Keep this concept in mind as we move forward, as it's the foundation for finding the equation of perpendicular lines. We'll use this principle extensively when we tackle our example question.
Furthermore, visualizing perpendicular lines on a graph can be incredibly helpful. Try sketching a few lines and their perpendicular counterparts. You'll notice how the right angle is always formed at the intersection, and how the slopes are indeed negative reciprocals of each other. This visual understanding can make the algebraic manipulations we'll do later much more intuitive. So, make sure you're comfortable with the concept of perpendicularity before moving on. It's a fundamental idea in geometry and is used in various real-world applications, from architecture to navigation. Now that we've got a solid grasp on what perpendicular lines are, let's move on to the next step: finding the slope of our given line.
Finding the Slope of the Given Line
The first step in finding the equation of a line perpendicular to a given line is to determine the slope of the given line. This is because, as we discussed, the slope of the perpendicular line is directly related to the slope of the original line. To find the slope, we typically need to rewrite the equation of the line in slope-intercept form. The slope-intercept form of a linear equation is y = mx + b, where m represents the slope and b represents the y-intercept. So, our goal is to manipulate the given equation to isolate y on one side.
Let's consider our example equation: 3x + 5y = -9. To get this into slope-intercept form, we need to isolate y. First, we subtract 3x from both sides of the equation: 5y = -3x - 9. Next, we divide both sides by 5 to solve for y: y = (-3/5)x - 9/5. Now, we can clearly see that the slope of the given line is -3/5. This is a crucial piece of information because we'll use it to find the slope of the perpendicular line.
Remember, the coefficient of x in the slope-intercept form is always the slope of the line. It tells us how much y changes for every unit change in x. A negative slope, like -3/5, indicates that the line is decreasing as we move from left to right. Now that we have the slope of the given line, we can easily find the slope of the line perpendicular to it. This is where the negative reciprocal concept comes into play. So, let's move on to the next section and see how we can use this information to find the perpendicular slope. Make sure you're comfortable with the process of converting equations to slope-intercept form, as it's a fundamental skill in algebra and is used in many different contexts.
Calculating the Perpendicular Slope
Now that we've found the slope of our given line (which was -3/5), the next step is to calculate the slope of the line perpendicular to it. Remember our rule? The slope of a perpendicular line is the negative reciprocal of the original line's slope. This means we need to flip the fraction and change its sign. So, if our original slope is -3/5, we flip it to get -5/3, and then change the sign to get 5/3. Therefore, the slope of the line perpendicular to 3x + 5y = -9 is 5/3.
Let's break this down a bit further. Taking the reciprocal means swapping the numerator and the denominator. So, -3/5 becomes -5/3. Then, we take the negative of this, which means multiplying by -1. So, -(-5/3) becomes 5/3. It's like a double negative situation, where the two negatives cancel each other out. This process ensures that our new slope represents a line that is indeed perpendicular to the original. A common mistake is to only take the reciprocal or only change the sign, but it's crucial to do both to get the correct perpendicular slope.
Why is this negative reciprocal relationship so important? It's because it guarantees that the lines intersect at a right angle. If we were to graph both lines, we would see that they form a perfect 90-degree angle at their point of intersection. This perpendicularity is used in many real-world applications, from constructing buildings to designing roads. So, understanding how to calculate the perpendicular slope is a valuable skill. Now that we have the slope of the perpendicular line, we're one step closer to finding its equation. We know the slope, and we also know a point that the line passes through. Let's see how we can use this information to write the equation of the line.
Using the Point-Slope Form
Okay, so we've got the slope of the perpendicular line (5/3), and we know it passes through the point (3, 0). Now, we need to use this information to find the equation of the line. The most convenient form for this is the point-slope form. The point-slope form of a linear equation is y - y1 = m(x - x1), where m is the slope, and (x1, y1) is a point on the line. This form is super handy when you have a slope and a point, which is exactly our situation.
Let's plug in our values. We have m = 5/3, x1 = 3, and y1 = 0. So, our equation becomes y - 0 = (5/3)(x - 3). Simplifying this, we get y = (5/3)(x - 3). This is a perfectly valid equation for the line, but we can go a step further and rewrite it in slope-intercept form (y = mx + b) or standard form (Ax + By = C), depending on what the question asks for or what form you prefer.
Let's convert it to slope-intercept form first. We distribute the 5/3: y = (5/3)x - (5/3)3. Simplifying further, we get y = (5/3)x - 5. Now we have the equation in slope-intercept form, which clearly shows the slope (5/3) and the y-intercept (-5). This form is great for visualizing the line on a graph. Alternatively, we can convert to standard form. To do this, we want to eliminate the fraction and get the x and y terms on the same side. Multiply both sides of y = (5/3)x - 5 by 3: 3y* = 5x - 15. Then, subtract 5x from both sides: -5x + 3y = -15. Finally, multiply both sides by -1 to make the coefficient of x positive: 5x - 3y = 15. This is the equation in standard form.
The point-slope form is a powerful tool, and mastering it will make solving these types of problems much easier. Remember, the key is to identify the slope and a point on the line, and then plug those values into the formula. From there, you can easily manipulate the equation into other forms if needed. Now, let's take a look at the final answer and make sure we've answered the question completely.
The Final Answer
Alright, guys, we've done all the hard work! We found the slope of the given line, calculated the perpendicular slope, and used the point-slope form to find the equation of the perpendicular line. Now, let's recap and present our final answer clearly.
We started with the equation 3x + 5y = -9. We found that the slope of this line is -3/5. Then, we calculated the slope of the perpendicular line, which is the negative reciprocal of -3/5, giving us 5/3. We were also given the point (3, 0) that the perpendicular line passes through. Using the point-slope form, we found the equation y - 0 = (5/3)(x - 3). We then simplified this equation into slope-intercept form: y = (5/3)x - 5. And finally, we converted it to standard form: 5x - 3y = 15.
So, the equation of the line that is perpendicular to the given line 3x + 5y = -9 and passes through the point (3, 0) is 5x - 3y = 15. This is our final answer! You can double-check your work by graphing both lines and making sure they intersect at a right angle and that the perpendicular line passes through the point (3, 0). This is a good practice to ensure you've done everything correctly.
Remember, the key to solving these types of problems is to break them down into smaller, manageable steps. First, find the slope of the given line. Second, calculate the perpendicular slope. Third, use the point-slope form to find the equation. And finally, simplify the equation into the desired form. With practice, you'll become a pro at finding equations of perpendicular lines! So, keep practicing, and don't hesitate to review these steps whenever you need a refresher. You've got this!
Practice Problems
To really nail down this concept, let's try a few practice problems. The best way to learn math is by doing it, so grab a pencil and paper and let's work through these together!
- Find the equation of the line perpendicular to 2x - y = 4 and passing through the point (1, 2).
- What is the equation of the line perpendicular to y = -3x + 5 and passing through the point (-2, 0)?
- Determine the equation of the line perpendicular to x + y = 7 and passing through the point (0, -1).
Try solving these problems using the steps we discussed earlier. Remember to find the slope of the given line, calculate the perpendicular slope, use the point-slope form, and then simplify the equation. Don't be afraid to make mistakes – that's how we learn! If you get stuck, go back and review the previous sections or ask for help. The important thing is to keep practicing and building your understanding.
These practice problems will help you solidify your understanding of finding perpendicular lines. You'll start to see patterns and become more comfortable with the process. And remember, math is like any other skill – the more you practice, the better you'll get. So, keep at it, and you'll be solving these problems like a pro in no time!
