Unraveling the Equation of a Line: A Step-by-Step Guide

When it comes to mastering College Algebra, one fundamental concept that often takes center stage is understanding linear equations. But fear not! We’re here to break it down, step by step.

Let’s imagine you’ve been tasked with finding the equation of a line that passes through the point (2, -3) and has a slope of -2. Sounds daunting? Don’t worry, by the end, you’ll be cruising through these problems like a pro!

Let’s Break It Down: What’s the Point-Slope Form?

You might be scratching your head, thinking, “What’s this point-slope form you’re talking about?” Well, here’s the thing: the point-slope form is a handy way to write the equation of a line when you know a point on that line and its slope. The formula is:

[ y - y1 = m(x - x1) ]

Here, (m) represents the slope, while ((x1, y1)) is the point through which the line passes. So, in our case, our point is ((2, -3)) and our slope is (-2).

Plugging in the Numbers

Alright, let’s plug those values into our formula:

[ y - (-3) = -2(x - 2) ]

Doesn’t look too intimidating, does it? Simplifying this gives us:

[ y + 3 = -2(x - 2) ]

Now, let’s distribute that (-2) to make things a tad clearer.

[ y + 3 = -2x + 4 ]

Rearranging to Standard Form

Next up, we want to express it in a form that’s often seen in textbooks, known as standard form. To do that, we simply rearrange the equation:

[ y = -2x + 4 - 3 ]

Which leads to:

[ y = -2x - 3 ]

And voila! We’ve found our equation of the line.

Evaluating Our Options

Now that we have our final equation, let’s take a look at the options presented:

• A. y = 2x - 3
• B. y = (-2/2)x - 3
• C. y = 2x + 3
• D. y = -2x - 3

So, which one is correct? Well, option D – (y = -2x - 3) – is spot on!

Now, why are the others incorrect? Well:

• Option A changes the slope and the constant term. Yikes!
• Option B tries to muster up some division that just doesn’t fit, giving us (-1x - 3) (spoiler: not what we need).
• Option C does have the same slope but flips the sign on that constant, so it’s destined for the wrong side of the equation, too.

Why This Matters

Understanding how to derive the equation of a line isn’t just a College Algebra requirement – it’s a skill that pops up in various topics throughout mathematics. Whether you're grappling with calculus, statistics, or even real-world applications, a firm grasp on linear equations lays the groundwork for success.

So, as you prep for that College Algebra CLEP exam, remember this method. You might find it popping up more often than you think!

Wrapping It Up

Mastering linear equations like this will not only give your confidence a boost but also pave the way for more complex math problems down the line. Remember, it’s all about breaking it down into manageable pieces, much like we did here. Repetition is key, so don’t forget to practice similar problems as you study. Happy learning!