Perpendicular Line Equation: Find The Solution!

Hey Plastik Magazine readers! Today, we're diving into a fun math problem involving perpendicular lines. This might sound intimidating, but don't worry, we'll break it down step by step so it's super easy to understand. Our mission is to figure out which equations represent a line that's perpendicular to the line 5x - 2y = -6 and also passes through the point (5, -4). We've got five options to choose from, and the challenge is to select the three correct ones. Ready to put on your math hats and get started?

Understanding Perpendicular Lines

Before we jump into solving the problem, let's quickly recap what it means for 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 you have two perpendicular lines, the product of their slopes is always -1. In simpler terms, if one line has a slope of m, the slope of a line perpendicular to it will be -1/m. This is often referred to as the negative reciprocal. Grasping this relationship is crucial for tackling problems like the one we have today, so make sure you've got this concept down solid, guys. We will use this later to solve the question, so keep this in mind.

Finding the Slope of the Given Line

So, let's start with the line given to us: 5x - 2y = -6. To figure out the slope of this line, we need to rearrange the equation into slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. Let's do the algebra: First, we'll subtract 5x from both sides of the equation, which gives us -2y = -5x - 6. Next, we'll divide both sides by -2 to isolate y. This results in y = (5/2)x + 3. Now we can clearly see that the slope of the given line is 5/2. Remember that slope is super important when we talk about lines that are perpendicular to each other. It's like the foundation for figuring out how these lines interact on a graph, so getting this part right is key to solving the whole problem.

Determining the Perpendicular Slope

Now that we know the slope of the given line is 5/2, we can find the slope of any line perpendicular to it. Remember the negative reciprocal thing we talked about earlier? To find the perpendicular slope, we flip the fraction and change the sign. So, the negative reciprocal of 5/2 is -2/5. This means any line with a slope of -2/5 will be perpendicular to the line 5x - 2y = -6. This is a crucial step, guys, because it narrows down our options considerably. Knowing the perpendicular slope helps us quickly identify which equations could possibly represent the line we're looking for. We're one step closer to cracking this math puzzle!

Analyzing the Options

Okay, now comes the fun part – let's dive into the options and see which ones fit the bill! We're looking for equations that have a slope of -2/5 and pass through the point (5, -4). We will analyze each choice one by one, making sure to check both conditions. Remember, the goal is to select three correct options, so we need to be thorough and precise in our analysis. Let’s get started and see what we find!

Option A: y = (-2/5)x - 2

Let's start with Option A: y = (-2/5)x - 2. This equation is already in slope-intercept form (y = mx + b), which makes it super easy to identify the slope. We can see that the slope (m) is -2/5. This matches the perpendicular slope we calculated earlier, so this equation checks the first box. Now, we need to see if this line passes through the point (5, -4). To do this, we'll substitute x = 5 and y = -4 into the equation and see if it holds true. So, we get -4 = (-2/5)(5) - 2. Simplifying the right side, we have -4 = -2 - 2, which simplifies further to -4 = -4. Since the equation holds true, Option A is a winner! It has the correct slope and passes through the given point. One down, two more to go!

Option B: 2x + 5y = -10

Next up is Option B: 2x + 5y = -10. This equation isn't in slope-intercept form yet, so we'll need to rearrange it to find the slope. Let's isolate y. First, subtract 2x from both sides: 5y = -2x - 10. Then, divide both sides by 5: y = (-2/5)x - 2. Aha! We see that the slope is -2/5, which matches our perpendicular slope. Now, let's check if the point (5, -4) lies on this line. Substitute x = 5 and y = -4 into the original equation: 2(5) + 5(-4) = -10. This simplifies to 10 - 20 = -10, which is indeed true. So, Option B also represents the line we're looking for. We're on a roll, guys!

Option C: 2x - 5y = -10

Now let's tackle Option C: 2x - 5y = -10. Just like with Option B, we need to rearrange this equation into slope-intercept form to easily identify the slope. Let's subtract 2x from both sides: -5y = -2x - 10. Now, divide both sides by -5: y = (2/5)x + 2. Oops! This time, the slope is 2/5, not -2/5. This means Option C is not perpendicular to the given line, so we can eliminate it right away. It's essential to catch these differences early on to avoid any confusion. Remember, we're looking for that negative reciprocal slope, so this one doesn't fit the bill.

Option D: y + 4 = (-2/5)(x - 5)

Let's move on to Option D: y + 4 = (-2/5)(x - 5). This equation is in point-slope form, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. From this form, we can directly see that the slope is -2/5, which is exactly what we need. The point-slope form also conveniently shows us a point on the line, which in this case appears to be (5, -4). To confirm, we can see that the equation directly corresponds to the point-slope form with the given point. So, Option D checks out too! We've found our third equation. It's always great when an equation is presented in a way that makes it easy to spot the key information, isn't it?

Option E: y - 4 = (5/2)(x + 5)

Finally, let's examine Option E: y - 4 = (5/2)(x + 5). This equation is also in point-slope form. We can immediately see that the slope is 5/2. But wait a minute! This is the original slope, not the perpendicular slope. This means Option E is parallel to the given line, not perpendicular. So, we can eliminate this option. It's crucial to pay close attention to those slopes, guys. They tell us so much about the relationship between lines!

Conclusion: The Correct Equations

Alright, guys, we've done it! We've successfully navigated through the options and identified the three equations that represent a line perpendicular to 5x - 2y = -6 and pass through the point (5, -4). The correct options are:

• A. y = (-2/5)x - 2
• B. 2x + 5y = -10
• D. y + 4 = (-2/5)(x - 5)

We tackled this problem by first understanding the concept of perpendicular lines and their slopes. Then, we found the slope of the given line and calculated the perpendicular slope. Finally, we analyzed each option, checking both the slope and whether the line passed through the given point. Remember, practice makes perfect, so keep flexing those math muscles! Until next time, keep exploring the fascinating world of mathematics!