Equation Of A Line: Points (4,1) & (0,3) Explained

Hey guys! Ever stumbled upon a problem asking you to find the equation of a line given two points? It might seem tricky at first, but trust me, it's totally manageable once you break it down. Today, we're going to tackle a classic example and walk through the process step by step. So, let's dive into this math problem together and make sure you nail it every time!

The Problem: Decoding the Line

Let's get started with the problem we're going to solve: Which equation represents the line that passes through the points (4,1) and (0,3)?

We have four options to choose from:

A. $y+1=-\frac{1}{2}(x+4)$ B. $y-1=-\frac{1}{2}(x-4)$ C. $y-4=-\frac{1}{2}(x-1)$ D. $y+4=-\frac{1}{2}(x+1)$

This looks like a multiple-choice question where we need to identify the correct equation of a line. The given options are in point-slope form, which is a handy way to represent a linear equation. To solve this, we'll need to remember a few key concepts, like how to calculate the slope of a line and how to use the point-slope form. Don't worry if you're a bit rusty – we'll review everything together.

Step 1: Calculating the Slope (m)

The slope is the first key piece of information we need to find the equation of a line. Remember, the slope tells us how steep the line is and in what direction it's going. The formula to calculate the slope (often represented as m) between two points is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of our two points. In our case, those points are (4, 1) and (0, 3). Let's plug these values into the formula:

$m = \frac{3 - 1}{0 - 4}$

$m = \frac{2}{-4}$

$m = -\frac{1}{2}$

So, the slope of the line passing through the points (4, 1) and (0, 3) is -1/2. This means that for every 2 units we move to the right along the x-axis, the line goes down 1 unit on the y-axis. Now that we have the slope, we're halfway to finding the equation of the line.

Step 2: Using the Point-Slope Form

The point-slope form is a fantastic tool for writing the equation of a line when you know the slope and one point on the line. It looks like this:

$y - y_1 = m(x - x_1)$

Where:

• m is the slope we just calculated.
• $(x_1, y_1)$ is a point on the line. We have two to choose from: (4, 1) and (0, 3). It doesn't matter which one we pick; the final equation will be the same (or an equivalent form).

Let's use the point (4, 1) first. Plugging in the slope m = -1/2 and the point (4, 1) into the point-slope form, we get:

$y - 1 = -\frac{1}{2}(x - 4)$

Guess what? This matches option B in our list of choices! So, it looks like we might have found our answer. But just to be sure, let's also try using the other point (0, 3) to see if we can get an equivalent equation.

Using the point (0, 3) and the slope m = -1/2, the point-slope form gives us:

$y - 3 = -\frac{1}{2}(x - 0)$

$y - 3 = -\frac{1}{2}x$

This equation doesn't directly match any of the options, but remember, equations can look different and still represent the same line. To see if this is equivalent to option B, we can manipulate option B a bit. Let's distribute the -1/2 on the right side of option B:

$y - 1 = -\frac{1}{2}x + 2$

Now, let's add 1 to both sides:

$y = -\frac{1}{2}x + 3$

If we subtract 3 from both sides, we get:

$y - 3 = -\frac{1}{2}x$

Aha! This is exactly the same equation we got using the point (0, 3). This confirms that option B is indeed the correct answer. Choosing the other point is an excellent way to double-check your work and ensure you're on the right track.

Step 3: Double-Checking and Eliminating Other Options

Even though we're pretty confident with option B, let's quickly look at why the other options are incorrect. This is a good practice to solidify our understanding and avoid making similar mistakes in the future.

• Option A: $y + 1 = -\frac{1}{2}(x + 4)$. This equation uses the point (-4, -1), which is not one of the points our line passes through.
• Option C: $y - 4 = -\frac{1}{2}(x - 1)$. This equation uses the point (1, 4), which is also not on our line.
• Option D: $y + 4 = -\frac{1}{2}(x + 1)$. This equation uses the point (-1, -4), another point not on our line.

By checking these options, we can clearly see that they don't represent the line passing through (4, 1) and (0, 3). This step helps us build confidence in our answer and reinforces the importance of using the correct points in the point-slope form.

The Answer: Option B is the Winner!

After carefully calculating the slope, using the point-slope form, and double-checking our work, we can confidently say that the equation representing the line that passes through the points (4, 1) and (0, 3) is:

B. $y - 1 = -\frac{1}{2}(x - 4)$

Key Takeaways: Mastering Linear Equations

Guys, solving this problem wasn't just about getting the right answer; it was about understanding the process. Let's recap the key steps we took:

1. Calculate the slope (m): Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. The slope is crucial because it tells us the steepness and direction of the line.
2. Use the point-slope form: The formula $y - y_1 = m(x - x_1)$ is your best friend when you have the slope and a point. Plug in the values and you're on your way to the equation.
3. Double-check with the other point: If you have two points, use both in the point-slope form (separately) to see if you get equivalent equations. This is a fantastic way to catch errors.
4. Eliminate incorrect options: Look at the points used in the other equations. If they don't match the given points, you can rule them out.

By following these steps, you'll be able to confidently tackle any problem that asks you to find the equation of a line given two points. Remember, practice makes perfect, so keep working on these types of problems and you'll become a pro in no time!

Practice Makes Perfect: Test Your Skills

Now that we've worked through this example together, it's time to put your knowledge to the test! Try solving similar problems on your own. For instance, you could try finding the equation of a line that passes through the points (2, -1) and (5, 3), or maybe (-3, 0) and (1, -2). The more you practice, the more comfortable you'll become with these concepts.

Consider changing the question slightly to challenge yourself further. What if you were given the slope and a point, and asked to find the equation? Or what if you were given the equation in a different form (like slope-intercept form) and had to convert it to point-slope form? These variations will help you develop a deeper understanding of linear equations and their different representations.

Don't be afraid to make mistakes – they're a crucial part of the learning process. When you encounter a problem you can't solve, take a step back, review the concepts, and try again. And remember, there are tons of resources available online and in textbooks to help you along the way. Keep up the great work, and you'll master linear equations in no time!

Beyond Point-Slope Form: Exploring Other Equation Forms

While we focused on the point-slope form in this problem, it's important to remember that there are other ways to represent linear equations. Two common forms you'll encounter are slope-intercept form and standard form. Understanding these different forms and how to convert between them is a valuable skill in algebra.

• Slope-intercept form: This form is written as $y = mx + b$, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). This form is great for quickly identifying the slope and y-intercept of a line.
• Standard form: This form is written as $Ax + By = C$, where A, B, and C are constants. Standard form is useful for certain types of algebraic manipulations and for representing systems of linear equations.

Knowing how to convert between these forms allows you to choose the most convenient form for a particular problem. For example, you might find the equation in point-slope form first, but then convert it to slope-intercept form to easily graph the line or identify its y-intercept. This flexibility will serve you well as you continue your mathematical journey.

Real-World Applications: Lines are Everywhere!

Linear equations aren't just abstract mathematical concepts; they have tons of real-world applications. From calculating the distance traveled at a constant speed to modeling the relationship between supply and demand in economics, lines are everywhere!

Think about the last time you saw a graph representing a trend or relationship. Chances are, linear equations were involved. Understanding how to work with lines and their equations opens the door to analyzing and interpreting these real-world scenarios.

For example, if you're tracking your savings over time, you might use a linear equation to model your progress. Or if you're planning a road trip, you could use a linear equation to estimate how long it will take you to reach your destination based on your speed and distance. The possibilities are endless!

By mastering linear equations, you're not just learning math; you're developing a powerful tool for understanding and interacting with the world around you. So keep practicing, keep exploring, and keep applying your knowledge to real-world situations. You'll be amazed at how much you can accomplish!

Final Thoughts: Keep Exploring the World of Math

Well, guys, we've reached the end of our journey through this problem. I hope you've found this step-by-step guide helpful and that you're feeling more confident about finding the equation of a line. Remember, math is a journey, not a destination. There's always more to learn and explore, so keep asking questions, keep practicing, and most importantly, keep having fun!

If you ever get stuck on a math problem, don't hesitate to reach out for help. There are tons of resources available, from online tutorials to textbooks to your friendly neighborhood math teacher. And remember, every problem you solve is a step forward on your path to mathematical mastery. So keep up the great work, and I'll see you next time for another math adventure!