Mastering Line Equations: Finding the Perpendicular Line

When it comes to geometry and algebra, lines, and their relationships can seem a bit tricky. But don’t worry! Let’s break down how to find the equation of a line that passes through a specific point and is perpendicular to another line—perfect practice for acing your College Math CLEP exam!

What’s the Story?

Okay, picture this: you’ve got a line, defined by the equation 4x + y = 8. This line is like your friendly neighborhood road, and you're tasked with finding a new road that’s perpendicular to it. But not just any road—you want this new line to pass through the point (3, 9). So, how do we figure out the equation of this perpendicular line?

Understanding Line Slopes

First off, let’s talk about slopes. The slope of the given line is hidden in that equation. If we rearrange 4x + y = 8 to solve for y, we get:

y = -4x + 8.

Now, the slope here, represented by the coefficient of x, is -4.

But here's the thing: when you want to find a line that’s perpendicular to another, you need to work with slopes. Specifically, the slope of the perpendicular line is the negative reciprocal of the original slope. So for our line with a slope of -4, the negative reciprocal is … wait for it … 1/4!

Now we've got our slope for the new line—sweet, huh?

Equation of the New Line

Next up, we’ll use the slope-intercept form of a line equation, which looks like this:

y = mx + b.

Here, m is the slope (which is 1/4), and b is the y-intercept, that mysterious number we’ll uncover. But hold on—before we dive into determining b, we’ve got a point to work with. We know our line needs to pass through (3, 9).

Substituting the point into the equation gives us:

9 = (1/4)(3) + b.

Let’s solve for b. Multiply (1/4)(3), and we get 3/4. So:

9 = 3/4 + b b = 9 - 3/4 b = 9 - 0.75 b = 8.25.

Now, wrapping it all together, our new line equation looks like this:

y = (1/4)x + 8.25.

But wait, we're still not finished!

Reviewing the Choices

Let’s revisit the answer choices you might see in your CLEP exam:

• A. 4x + y = 9
• B. y = -4x + 9
• C. y = 4x + 9
• D. y = -4x - 9

Now, hold on a second! We’ve just found our equation for the new line, but the choices don't quite match up. Here’s what’s crucial: if we look closely at D, the slope is -4 (the same as the original line) and doesn’t match our newfound slope of 1/4.

The Key to Identifying Perpendicular Lines

A little breakdown here: option A has a parallel slope, and options B and C reflect slopes we just established as linked to the original line, not perpendicular. That's a crucial distinction!

So, the takeaway is that while the most immediate choice might appear as D, you would effectively want an equation with the slope we determined to be 1/4—and realize that none of those options provide it directly.

Bringing It All Together

While this situation doesn’t provide the correct answer directly, it’s a valuable lesson in analyzing slopes and line equations! When it comes to perpendicular lines, always remember: you're looking for that negative reciprocal, and don’t be thrown off by the given options.

This practice is perfect for brushing up on your math skills as you prepare for your College Math CLEP exam. Revisit these concepts, keep honing your skills, and you’ll navigate equations like a pro!