Finding the Perpendicular Line: A Friendly Guide to College Algebra

When you're tackling College Algebra, some concepts can feel like they're designed to trip you up – especially lines and slopes. But guess what? Understanding how to find a line perpendicular to another line doesn't have to be one of those tough nuts to crack. So, let’s break it down together!

Imagine you’re given the equation of a line: y = 3x + 2. Looks straightforward, right? This line has a slope of 3, which represents how steep it is. But what if you're tasked to find a line that cuts across this one at a neat 90-degree angle? That’s where the idea of perpendicular lines comes into play.

What Does Perpendicular Mean?

Now, you might be wondering, “What does it mean for lines to be perpendicular?” Picture this: if two lines meet at a right angle, they are considered perpendicular. The nifty thing about the slopes of these lines is their relationship. The slope of a perpendicular line is the negative reciprocal of the original line’s slope.

So, if our existing line has a slope of 3, its negative reciprocal would flip that fraction upside down and switch the sign. Thus, -1/3 becomes the slope of our new line. Pretty neat, huh?

Let's Find Our Line

Now that we have our slope, let’s think about where this new line needs to pass through. We’ve got the point (4, 2) to consider. Here’s the thing: knowing a point allows us to use the point-slope form of a line. The formula looks like this:

y - y₁ = m(x - x₁)

In this formula, (x₁, y₁) is the point we have – in our case, it's (4, 2), and m is the slope we just calculated (-1/3). So, let’s plug those values in, shall we?

y - 2 = -1/3(x - 4)

Now, if we solve this, we start to crystalize the equation of our line. Distributing the slope through gives us:

y - 2 = -1/3x + 4/3

Adding 2 (or 6/3, nice and tidy) to both sides to simplify:

y = -1/3x + 10/3

Checking Out the Multiple-Choice Options

If you recall the options given:

A. y = -3x + 6
B. y = 3x - 6
C. y = -3x - 2
D. y = 3x + 6

We can eliminate options A and B right off the bat, as neither has the -1/3 slope we’re looking for. Option D? That one boasts a positive slope of 3, making it parallel instead of perpendicular. It’s kind of like expecting apples and getting oranges!

That leaves us with option C: y = -3x - 2. While we have the right idea about the slope, let’s tighten this up. The perpendicular slope isn’t -3 but rather -1/3, meaning we need to circle back and double-check our math to keep our work aligned!

Final Touches

At the end (or really, the beginning of your algebra journey), understanding how to determine the slope of a perpendicular line opens up so many tools in your Math toolbox. Whether you're prepping for the CLEP exam or just brushing up on your skills, mastering these lines and slopes will undoubtedly serve you well.

So, if you’re stuck or a bit unsure about lines or any other algebra concepts, remember: take your time, breathe, and break the problem down into digestible pieces. Algebra can sometimes feel like a foreign language, but you’ve got all the tools to become fluent. Good luck and keep practicing! You’ve got this!