Finding The Equation Of A Line With Slope 4 Passing Through (1, 6)
Hey guys! Today, let's dive into a common problem in mathematics: finding the equation of a line. Specifically, we're going to figure out how to find the equation of a line when we know its slope and a point it passes through. It might sound tricky, but trust me, it's super manageable once you break it down. We'll walk through each step, making sure you understand the logic behind it, not just the formula. So, grab your pencils, and let's get started!
Understanding the Basics: Slope and Point-Slope Form
Before we jump into solving our specific problem, let's make sure we're all on the same page with the fundamentals. The slope of a line, often denoted as m, tells us how steep the line is. A slope of 4, in our case, means that for every 1 unit we move to the right along the x-axis, the line goes up 4 units along the y-axis. Think of it like climbing a steep hill – for every step forward, you go significantly higher. The slope is a crucial concept for understanding linear equations.
Now, the point-slope form is our secret weapon for this type of problem. It's a way to write the equation of a line when you know a point on the line (x₁, y₁) and the slope m. The point-slope form looks like this:
y - y₁ = m(x - x₁)
This formula might seem a bit intimidating at first, but it's really just a way to express the relationship between the slope, a specific point, and any other point (x, y) on the line. Basically, it is derived from the definition of the slope, which is the change in y over the change in x. The point-slope form is incredibly useful because it allows us to build the equation of a line directly from the information we're given. It bridges the gap between a geometrical understanding (slope and a point) and an algebraic representation (the equation of a line).
Why Point-Slope Form is Our Friend
You might be wondering, "Why not just use the slope-intercept form (y = mx + b)?" Well, you totally could, but the point-slope form often saves us a step. With slope-intercept form, you'd have to plug in the point and slope, then solve for the y-intercept (b). Point-slope form lets us bypass that extra calculation. It's all about efficiency, guys! This is one of the primary reasons to learn and internalize the point-slope form – it's a shortcut to finding the equation of a line, especially when you're given a point and a slope. It minimizes the algebraic manipulation required, making the process smoother and less prone to errors.
Applying Point-Slope Form to Our Problem
Okay, now let's get back to our original question. We need to find the equation of a line with a slope of 4 that passes through the point (1, 6). We've got our slope (m = 4) and our point (x₁ = 1, y₁ = 6). Time to plug these values into the point-slope form:
y - y₁ = m(x - x₁)
Substitute m with 4, x₁ with 1, and y₁ with 6:
y - 6 = 4(x - 1)
See? We've already got the equation in a usable form! This is the power of point-slope form in action. We've taken the given information and directly translated it into an equation that describes the line. It's like having a magic formula that instantly converts geometric properties into algebraic expressions.
Converting to Slope-Intercept Form
Now, while y - 6 = 4(x - 1) is a perfectly valid equation for the line, it's not in the typical slope-intercept form (y = mx + b) that we often see. So, let's convert it to that form. This is where we do a little bit of algebra to tidy things up and make the equation look familiar.
First, we'll distribute the 4 on the right side of the equation:
y - 6 = 4x - 4
Next, we want to isolate y on the left side. To do this, we'll add 6 to both sides of the equation:
y - 6 + 6 = 4x - 4 + 6
y = 4x + 2
Boom! We've got our equation in slope-intercept form. This form, y = 4x + 2, tells us the slope (4) and the y-intercept (2) directly. It's a very convenient way to represent a linear equation because you can immediately read off these key characteristics of the line. This conversion from point-slope to slope-intercept form is a common practice, as it makes the equation more readily interpretable and comparable to other linear equations.
Checking Our Answer
It's always a good idea to double-check our work, right? We can do this in a couple of ways. First, we can make sure our slope is indeed 4, which it is in the equation y = 4x + 2. Second, we can plug in the point (1, 6) into our equation and see if it holds true:
6 = 4(1) + 2
6 = 4 + 2
6 = 6
It works! This confirms that our equation is correct. This step of verification is critical in mathematics. It's not enough to just arrive at an answer; you need to be confident that your answer is correct. By plugging the original point back into the equation, we're essentially ensuring that the line we've defined algebraically actually passes through the given point, reinforcing the connection between the geometrical and algebraic representations of the line.
Identifying the Correct Option
Looking back at the options provided, we can see that option C, y = 4x + 2, matches our calculated equation. So, that's our answer!
A. y = 4x - 2 B. y = 4x + 6 C. y = 4x + 2 D. y = 4x - 3
The process of eliminating incorrect options is also a valuable strategy in problem-solving. By systematically comparing the calculated equation with the given options, we can confidently identify the correct answer and avoid common pitfalls. This methodical approach not only helps in arriving at the right solution but also enhances our understanding of the concepts involved.
Key Takeaways and Practice
So, what have we learned today? We've successfully found the equation of a line using the point-slope form and then converted it to slope-intercept form. Remember these key steps:
- Identify the slope (m) and the point (x₁, y₁).
- Plug the values into the point-slope form: y - y₁ = m(x - x₁).
- Convert to slope-intercept form (y = mx + b) if needed.
- Check your answer!
The key to mastering this skill is practice. Try working through similar problems with different slopes and points. The more you practice, the more comfortable you'll become with the process. You'll start to recognize the patterns and see how the point-slope form makes these problems much easier to solve. Also, try visualizing these lines on a graph. Understanding the visual representation can further solidify your grasp of the concepts. The interplay between algebraic manipulation and geometric visualization is a powerful tool in learning mathematics.
Practice Problems
To help you get started, here are a couple of practice problems:
- Find the equation of a line with a slope of -2 that passes through the point (3, 1).
- Find the equation of a line with a slope of 1/2 that passes through the point (-2, 4).
Work through these problems using the steps we've discussed. Don't be afraid to make mistakes – that's how we learn! Check your answers by plugging the point back into your equation. The process of solving problems independently is crucial for developing mathematical fluency. It's not just about knowing the formulas; it's about being able to apply them in different contexts and situations. The more you practice, the more confident and proficient you'll become in handling these types of problems.
Conclusion
Finding the equation of a line doesn't have to be a daunting task. By understanding the concepts of slope and point-slope form, and by following a systematic approach, you can confidently solve these problems. So, go forth and conquer those lines! Remember, practice makes perfect, and the more you work with these concepts, the more intuitive they'll become. And most importantly, have fun with it! Mathematics is a fascinating world, and the more you explore, the more you'll discover its beauty and power.
I hope this explanation was helpful, guys! Keep practicing, and you'll be line-equation masters in no time. Good luck, and happy problem-solving!
