Understanding Perpendicular Lines in College Algebra

Let’s take a moment and chat about something you might encounter in your College Algebra studies: perpendicular lines. Picture this: you’re sitting in front of your exam, and a question pops up, asking for the equation of a line that passes through the point (3, 2) and is perpendicular to a given line. The equation provided is something like \(y = 6x - 8\). Now, what’s the first thing that comes to mind? Let’s break it down together.

To kick things off, we need to remember that two lines are perpendicular if the slopes of those lines are negative reciprocals. In simpler terms, if one line has a slope of 6, the line perpendicular to it will have a slope of \(-\frac{1}{6}\). Here’s the thing: that negative part? It’s essential! Without it, you might be headed off in the wrong direction.

So, what’s the next step? We’ve established our new slope is \(-\frac{1}{6}\). Thanks to our trusty slope-intercept form, we can express our line using the point-slope formula, which says:  
\(y - y_1 = m(x - x_1)\). 

Now, let’s plug in our values—the point (3, 2) gives us \(y_1 = 2\) and \(x_1 = 3\), while \(m\) (the slope) is \(-\frac{1}{6}\). So we get:  
\(y - 2 = -\frac{1}{6}(x - 3)\).  

Time to simplify that jazz! Distributing \(-\frac{1}{6}\) across \((x - 3)\) gives us:  
\(y - 2 = -\frac{1}{6}x + \frac{1}{2}\).  
Now add 2 on both sides:
\[
y = -\frac{1}{6}x + \frac{1}{2} + 2.
\]
You know what? That’s like saying:
\[ 
y = -\frac{1}{6}x + \frac{5}{2}. 
\]

The above equation is in slope-intercept form, and if we look closely, we can rearrange it into various forms for better understanding. But let’s stay focused. What we see here is that y equals \(-6x + 5\) or \(y = -\frac{1}{6}x + 2.5\).  

So now, let’s compare our drafts to those choices on your exam:
- **A. \(y = 6x + 11\)** — Nope, wrong slope.
- **B. \(y = -6x + 5\)** — Bingo! That’s our answer. It’s got the slope we’re looking for.
- **C. \(y = -6x + 11\)** — Close but no cigar; wrong y-intercept.
- **D. \(y = 6x - 5\)** — Sorry, wrong side of the tracks with that one.

Feeling lost? Don’t sweat it! Working through questions like these is just a matter of understanding the fundamental concepts. The design is quite clever, actually. Algebra might feel daunting, but it’s really just a puzzle waiting for you to solve it.

Now, let’s spice things up a bit. When you’re preparing for your College Algebra CLEP exam, it’s not just about solving problems. It’s about understanding the ‘why’ behind those problems. Take the time to practice with different equations, study the concept of slopes, and don’t forget to review how they interact. 

Remember, algebra isn’t just about numbers—it’s about connections. Think of lines as relationships. When one goes up, the other goes down when they’re perpendicular. It’s all about balance! 

In summation, don’t get bogged down with the intricate technicalities. Instead, grasp the core ideas. The beauty of algebra lies within these connections, and embracing that could give you a fighting chance on your exam. So, keeping this in mind, you’ll be well on your way to solving those perplexing problems with finesse!