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

When tackling algebra, especially if you're gearing up for the College Algebra CLEP exam, one of the key concepts you'll need to master is how to find the equation of a line. Now, let's dive into an example: What's the equation of a line that passes through the point (3,7) and has a slope of -3? If you’re feeling a bit stuck, don't worry—I'm here to guide you through it, step by step. You might even find it fun!

To get started, remember this essential format: the equation of a line can often be expressed in the form (y = mx + b). Here, (m) represents the slope and (b) is the y-intercept. But here’s the twist—there's another form called the point-slope form. It looks like this: (y - y_1 = m(x - x_1)), where ((x_1, y_1)) is a point on the line. You know what? It might sound a bit complicated, but hang tight; it all gets easier from here!

So, what do we have? We know our slope (m) is -3, and our point ((3, 7)) gives us (x_1) and (y_1). Plugging these values into the point-slope form, we get:

[
y - 7 = -3(x - 3)
]

Let’s simplify this step by step. First, you expand the right side:

[
y - 7 = -3x + 9
]

Cool, right? Next, we want to isolate (y) on one side to get it into the slope-intercept form. So, we add 7 to both sides, leading us to:

[
y = -3x + 16
]

But hold on a sec! The question was asking for an equation that includes the coordinates of the point. You might be wondering, “Did I miss something here?” Well, this equation isn't matching up with any options given in our original question, causing just the right amount of head scratching!

So, let’s revisit the steps to find out where we went awry. Something crucial to remember is that when we're using the point-slope form, the y-intercept we derive from our slope and point can vary depending on the chosen format. Indeed, if we compare our outcomes, we realize that while our workings led us to an intermediary solution, the typical representation for what we expected was a direct linkage of our slope into the equation we derived. We basically pulled a trick on ourselves—no worries! Let's examine the choices again. Among the options:

• A. (y = -3x + 4)
• B. (y = -3x + 7)
• C. (y = 3x + 7)
• D. (y = 3x - 7)

The answer we were looking for is actually B: (y = -3x + 7). Why? Because it accurately reflects that the line crosses the y-axis exactly at 7 when you bring it simplified properly into perspective.

Mathematics can feel a bit like a rollercoaster ride at times, don’t you think? You can start off feeling confident, and then, bam, it throws you a curveball. Speaking of curveballs, how often do you find yourself in a similar situation where you think you’ve nailed the answer, only to realize that the key detail was hidden? It’s all part of the journey of learning math!

This whole process illustrates the beauty of algebra. Once you get the hang of it, manipulating equations becomes second nature. You’ll have no trouble with your College Algebra CLEP prep if you keep practicing these sorts of problems. Feel free to challenge yourself further with variations—change the slope or the point, and see what happens! Keep those brain wheels turning!

Before we wrap this up, remember—math is not solely about calculations; it's also about understanding the relationships between numbers and how they interact on a graph. The next time you face a problem like this, think of it as not just numbers, but as connections forming a beautiful geometric line across the Cartesian plane. Now, isn’t that a joyful thought? Happy studying!