Equation Of A Parallel Line: Step-by-Step Solution
Have you ever wondered how to find the equation of a line that runs parallel to another, especially when given a specific point it needs to pass through? It might sound like a daunting task, but don't worry, guys! It's actually quite straightforward once you break it down. In this article, we'll walk through a classic problem: finding the equation of a line parallel to x - 3y = 8 and passing through the point (6, -1). So, grab your math hats, and let's dive in!
Understanding Parallel Lines
Before we jump into the solution, let's quickly recap what parallel lines are. Parallel lines are lines that run in the same direction and never intersect. This means they have the same slope. The slope, often denoted as m, tells us how steep a line is. If two lines have the same slope, they're parallel. This is a crucial concept to remember as we tackle our problem.
Now, let's talk about linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. A common form for a linear equation is the slope-intercept form: 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 super helpful because it directly shows us the slope and y-intercept of the line. Another useful form is the point-slope form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. This form is particularly handy when we know a point on the line and its slope, which is exactly the situation we're facing in our problem!
Understanding these concepts—parallel lines, slopes, and linear equation forms—is fundamental to solving our problem effectively. We know that we need to find a line with the same slope as the given line and that it must pass through a specific point. This knowledge sets the stage for a clear and methodical approach to finding the equation of the parallel line. With a solid grasp of these basics, we can confidently move forward and tackle the problem step by step. Remember, mathematics often involves breaking down complex problems into smaller, manageable parts, and that’s exactly what we’re doing here.
Step 1: Finding the Slope of the Given Line
Our first task is to find the slope of the line x - 3y = 8. To do this, we need to rewrite the equation in slope-intercept form (y = mx + b). This will allow us to easily identify the slope m. Let's rearrange the equation:
- Start with the equation: x - 3y = 8
- Subtract x from both sides: -3y = -x + 8
- Divide both sides by -3: y = (1/3)x - 8/3
Now our equation is in the form y = mx + b. By comparing our rearranged equation y = (1/3)x - 8/3 with the slope-intercept form, we can see that the slope m of the given line is 1/3. This is a critical piece of information because any line parallel to this one will have the same slope. Remember, parallel lines have the same slope, so the line we're trying to find will also have a slope of 1/3. This makes our task much easier, as we now have the slope for our new line.
Finding the slope is often the first and most important step in solving problems involving parallel and perpendicular lines. Once we have the slope, we can use it along with other given information, such as a point on the line, to determine the full equation of the line. In our case, we’ve successfully found the slope of the given line, and we know that the parallel line we’re looking for shares this slope. This sets us up perfectly for the next step, where we’ll use this slope and the given point to find the equation of the parallel line. Understanding how to manipulate equations into slope-intercept form is a fundamental skill in algebra, and it’s one that will serve you well in many mathematical problems. So, with our slope in hand, let's move on to the next step and bring this parallel line to life!
Step 2: Using the Point-Slope Form
We know the slope of our parallel line is 1/3, and we also know it passes through the point (6, -1). This is where the point-slope form of a linear equation comes to the rescue. The point-slope form is given by: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line. We have all the pieces we need, so let's plug them in:
- m = 1/3
- (x₁, y₁) = (6, -1)
Substituting these values into the point-slope form, we get:
y - (-1) = (1/3)(x - 6)
Now, let's simplify this equation. First, we can rewrite y - (-1) as y + 1:
y + 1 = (1/3)(x - 6)
This equation is a perfectly valid representation of our line, but often we prefer to have the equation in slope-intercept form (y = mx + b) or standard form (Ax + By = C). So, let's continue simplifying to get it into slope-intercept form. To do this, we'll distribute the 1/3 on the right side of the equation:
y + 1 = (1/3)x - 2
Now, subtract 1 from both sides to isolate y:
y = (1/3)x - 2 - 1
y = (1/3)x - 3
Voila! We now have the equation of our parallel line in slope-intercept form. This form clearly shows us the slope (1/3) and the y-intercept (-3). Using the point-slope form is a powerful technique because it allows us to construct the equation of a line directly from its slope and a point it passes through. It’s especially useful in situations like this, where we have a slope derived from a parallel or perpendicular line and a specific point. This step-by-step process highlights how we can take a general form of a linear equation and tailor it to fit specific conditions, ultimately leading us to the exact equation we need. So, with the equation in hand, let's move on to the final step and make sure everything is perfect!
Step 3: Verifying the Solution
We've found the equation of the line parallel to x - 3y = 8 and passing through (6, -1), which is y = (1/3)x - 3. But, like good mathematicians, we should always verify our solution to make sure it's correct. There are a couple of ways we can do this.
First, let's check if the slope is correct. We know that parallel lines have the same slope. The slope of the original line x - 3y = 8 is 1/3, and the slope of our new line y = (1/3)x - 3 is also 1/3. So, the slopes match, which is a good sign! This confirms that our line is indeed parallel to the original line.
Next, we need to make sure our line passes through the point (6, -1). To do this, we'll substitute x = 6 and y = -1 into our equation and see if it holds true:
-1 = (1/3)(6) - 3
-1 = 2 - 3
-1 = -1
The equation holds true! This confirms that the point (6, -1) lies on our line. By verifying both the slope and the point, we can be confident that our solution is correct. Verification is a crucial step in any mathematical problem-solving process. It’s like the final seal of approval that ensures our answer is accurate and reliable. In this case, we’ve double-checked our work and confirmed that the equation y = (1/3)x - 3 perfectly satisfies the given conditions: it's parallel to the original line, and it passes through the specified point. This thorough approach not only gives us the right answer but also enhances our understanding of the underlying concepts and techniques. So, with our solution verified and our understanding solidified, we can confidently say we've mastered this problem!
Conclusion
There you have it, guys! We've successfully found the equation of the line parallel to x - 3y = 8 and passing through (6, -1). The equation is y = (1/3)x - 3. We did this by first finding the slope of the given line, then using the point-slope form to find the equation of the parallel line, and finally, verifying our solution to ensure accuracy. This problem illustrates how understanding key concepts like parallel lines, slopes, and linear equation forms can help us solve mathematical problems in a systematic and effective way. Remember, math isn't just about memorizing formulas; it's about understanding the underlying principles and applying them logically. By breaking down complex problems into smaller, manageable steps, we can tackle even the most challenging questions with confidence. So, keep practicing, keep exploring, and most importantly, keep having fun with math!
This step-by-step approach is applicable to many similar problems. Whether you're dealing with parallel lines, perpendicular lines, or other geometric relationships, the key is to break down the problem into manageable steps and use the appropriate formulas and techniques. And remember, verification is always your friend! It's the safety net that catches errors and ensures your solution is rock solid. So, the next time you encounter a problem like this, remember our journey today, and you’ll be well-equipped to find the solution. Keep those math muscles flexed and those problem-solving skills sharp, and you’ll be amazed at what you can achieve! Until next time, happy calculating!