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

    Let's unravel the mystery of finding the equation of a line! You might be asking, "Why do I need to know this for the College Algebra CLEP Prep Exam?" Well, knowing how to calculate the equation of a line isn't just an algebra exercise; it’s a skill that helps in everything from analyzing data trends to navigating the real world. So, let’s break this down with a real example, shall we?  

    Suppose we have two points: (3, 6) and (1, 5). With these coordinates, our goal is to find the equation of the line that runs through these points. The form we’ll be using is the classic linear equation: **y = mx + b**. Here, *m* stands for the slope of the line, and *b* represents the y-intercept.   

    **Step 1: Calculate the Slope**  
    The first step in our journey is to figure out the slope (*m*). You remember that the slope is defined as the change in y over the change in x, right? So let's perform a little calculation:  

    \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]  

    Plugging in our points (3, 6) and (1, 5), we get:  

    \[ m = \frac{6 - 5}{3 - 1} = \frac{1}{2} \]  

    So our slope is 1/2. Not too tricky, was it?  

    **Step 2: Finding the Y-Intercept (b)**  
    Now that we’ve got our slope, let’s find our y-intercept, *b*. We can do this by substituting one of our points into the equation y = mx + b. Why don’t we use (3, 6) for convenience?  

    So we substitute like this:  

    \[ 6 = \left(\frac{1}{2}\right)(3) + b \]  

    Simplifying that gives us:  

    \[ 6 = \frac{3}{2} + b \]  
    \[ b = 6 - \frac{3}{2} \]  
    \[ b = \frac{12}{2} - \frac{3}{2} \]  
    \[ b = \frac{9}{2} \]  

    Now we have everything we need!  

    **Step 3: The Equation of the Line**  
    With the slope and the y-intercept in hand, we can write our final equation. Substituting back into y = mx + b, we arrive at:  

    \[ y = \frac{1}{2}x + \frac{9}{2} \]  

    Or, simply, that’s:  

    \[ y = 2x + 5 \]  

    And there we have it! That’s the equation of the line passing through our points (3, 6) and (1, 5). It all boils down to that classic form, but now you see how it connects with what you’re studying.  

    **Why It Matters**  
    Understanding how to find the equation of a line isn’t just an exam question. Imagine you're analyzing data for a project or even calculating something simple like how far you need to drive based on your starting and ending points. The applications are endless!  

    Feeling a bit more comfortable with these concepts? Great! Keep practicing, and remember, each step you take adds to your understanding. Algebra doesn’t have to be daunting; it can be an enjoyable puzzle. Just like a good mystery novel, it unfolds piece by piece, and you're the detective!