Finding The Equation Of A Line: A Step-by-Step Guide

Hey there, math enthusiasts! Today, we're diving into a common algebra problem: finding the equation of a line that passes through two given points. Specifically, we'll figure out which of the provided equations correctly represents the line going through the points $(-4, 4)$ and $(8, -2)$. This is super useful, whether you're brushing up on your algebra skills, getting ready for a test, or just curious about how lines are defined in the world of math. So, grab your pencils and let's get started. We'll break down the process step by step, making sure everything is crystal clear. By the end, you'll be a pro at solving these types of problems!

Understanding the Basics: Slope-Intercept Form

Before we jump into the problem, let's quickly review the slope-intercept form of a linear equation. This is the format we'll be using to find our answer. The slope-intercept form is written as: $y = mx + b$, where:

• y is the dependent variable (the output).
• x is the independent variable (the input).
• m is the slope of the line (how steep it is).
• b is the y-intercept (where the line crosses the y-axis).

Our goal is to find the values of m (the slope) and b (the y-intercept) that fit the two points we're given. Think of it like a puzzle: we have some pieces (the points), and we need to fit them into the right equation. Understanding this form is critical because it gives us a clear structure to work with. If you're a bit rusty on this, don't worry! We'll make sure to clarify everything as we go. Remember, the slope tells you the rate of change of y with respect to x, and the y-intercept is where the line begins on the vertical axis. We're essentially trying to decode what this line looks like based on the information we have. This forms the foundation for understanding linear equations. Let's start with calculating the slope using the two points: $(-4, 4)$ and $(8, -2)$. The slope represents the rate of change of y with respect to x. We will begin by calculating the slope of the line.

Step 1: Calculate the Slope

First things first: we need to figure out the slope (m) of the line. The slope tells us how much the line rises or falls for every unit it moves to the right. We can calculate the slope using the following formula, which is a key part of solving this kind of problem:

$m = (y_2 - y_1) / (x_2 - x_1)$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two points. Let's plug in our points, $(-4, 4)$ and $(8, -2)$. We can label $(-4, 4)$ as $(x_1, y_1)$ and $(8, -2)$ as $(x_2, y_2)$.

$m = (-2 - 4) / (8 - (-4))$

Simplify the equation:

$m = -6 / 12$

$m = -1/2$

So, the slope of the line is $-1/2$. This means that for every 2 units we move to the right on the x-axis, the line goes down 1 unit on the y-axis. This is the rise over run concept. The negative sign tells us that the line is sloping downwards from left to right. Now that we have calculated the slope, we can move on to the next step, finding the y-intercept.

Step 2: Find the Y-intercept

Now that we know the slope (m = -1/2), we can find the y-intercept (b). We'll use the slope-intercept form of the equation again, $y = mx + b$. We can choose either of the given points to substitute into the equation, along with the slope we just calculated. Let's use the point $(-4, 4)$. Substituting the values, we get:

$4 = (-1/2) * (-4) + b$

Simplify the equation:

$4 = 2 + b$

Solve for b by subtracting 2 from both sides:

$4 - 2 = b$

$b = 2$

So, the y-intercept is 2. This means that the line crosses the y-axis at the point (0, 2). The y-intercept is where the line intersects the y-axis. Now that we have both the slope and the y-intercept, we can write the equation of the line. It's like putting the final pieces of the puzzle together, creating the complete picture of our line. Now that we've found our slope and y-intercept, it's time to put it all together. Next, we are going to write the final equation of the line.

Step 3: Write the Equation

Now we have all the pieces we need to write the equation of the line in slope-intercept form, $y = mx + b$. We know that $m = -1/2$ and $b = 2$. Substituting these values into the equation, we get:

$y = (-1/2)x + 2$

So, the equation of the line that passes through the points $(-4, 4)$ and $(8, -2)$ is $y = -1/2x + 2$. When you graph this equation, you will find it perfectly matches the two given points. Let's go back to our multiple-choice options and find which one matches our equation. This is the final step, where we compare our hard work with the provided choices to select the correct answer. Congratulations, we're almost there! Let's find the correct answer in the choices.

Step 4: Choose the Correct Answer

Now we look at the provided options:

A. $y = -2x + 14$ B. $y = -2x - 4$ C. $y = -1/2x + 2$ D. $y = -1/2x - 2$

We've calculated that the correct equation is $y = -1/2x + 2$. Therefore, the correct answer is C. We've successfully navigated the process of finding the equation of the line! Remember to always double-check your calculations and make sure your answer makes sense in the context of the problem. This is a very common type of question. So, practice makes perfect. Now that you have learned how to solve this type of problem, you should be able to solve similar problems. Keep up the great work and the practice!

Conclusion

So there you have it! We've successfully found the equation of the line that passes through the points $(-4, 4)$ and $(8, -2)$. We broke down the problem into manageable steps: calculating the slope, finding the y-intercept, and writing the equation in slope-intercept form. It's a great example of how to solve a common math problem. Remember, practice is key. Try working through similar problems on your own to solidify your understanding. You're now equipped with the knowledge and skills to tackle similar linear equation problems. Keep practicing and exploring, and you'll become a math whiz in no time! Keep up the great work and remember to always review your answers and solutions. Have fun learning!