Slope Of Parallel Line: Y = (2/7)x + 9 Explained

Hey guys! Ever wondered how to find the slope of a line that's parallel to another line? It's a pretty common question in math, especially when you're dealing with linear equations. Today, we're going to break down a specific example: finding the slope of a line parallel to y = (2/7)x + 9. We'll go through the concepts step by step, so you'll not only get the answer but also understand the why behind it. Let's dive in!

Understanding Parallel Lines and Slopes

First off, let's make sure we're all on the same page about what parallel lines are. Parallel lines are lines in the same plane that never intersect. Think of train tracks running side by side – they go on forever without ever meeting. The key characteristic of parallel lines that we need to focus on is their slopes. The slopes of parallel lines are always equal. This is a fundamental concept in coordinate geometry, and it's crucial for solving problems like the one we're tackling today.

So, what is slope anyway? Slope is a measure of the steepness and direction of a line. It tells us how much the line rises (or falls) for every unit of horizontal change. Mathematically, slope is often represented by the letter 'm', and it's calculated as the “rise over run.” If you have two points on a line, (x1, y1) and (x2, y2), the slope (m) can be found using the formula: m = (y2 - y1) / (x2 - x1). However, when a linear equation is given in slope-intercept form, which is y = mx + b, identifying the slope becomes super easy. The 'm' in the equation is the slope, and 'b' is the y-intercept (the point where the line crosses the y-axis). Understanding these basics will make solving our problem much smoother, and it’s a great foundation for more advanced math concepts later on.

Identifying the Slope of the Given Line

Okay, now let's get back to our specific problem. We're given the equation y = (2/7)x + 9, and we need to figure out the slope of a line that's parallel to it. Remember what we just discussed? The slopes of parallel lines are equal. So, our first task is to identify the slope of the line y = (2/7)x + 9. This is where the slope-intercept form of a linear equation comes in handy. As we mentioned earlier, the slope-intercept form is y = mx + b, where 'm' represents the slope and 'b' represents the y-intercept.

Looking at our equation, y = (2/7)x + 9, we can directly compare it to the slope-intercept form. It's pretty clear that the number in front of the 'x' term is the slope. In this case, the slope (m) is 2/7. The '+ 9' part tells us that the y-intercept is 9, but that's not what we're concerned with right now. We're focused on the slope because that's what determines whether lines are parallel. So, the slope of the given line is 2/7. This is a crucial piece of information because it's going to directly tell us the slope of any line parallel to it. We've identified the slope of the original line, which is the first key step in solving our problem. Now, we can use this information to find the slope of a parallel line.

Determining the Slope of a Parallel Line

Here's the magic part! Since parallel lines have the same slope, the slope of any line parallel to y = (2/7)x + 9 will also be 2/7. That's it! There's no complicated calculation needed. Once you've identified the slope of the original line, you know the slope of all lines parallel to it. This is a fundamental property of parallel lines, and it makes these types of problems surprisingly straightforward once you understand the concept. Think about it this way: if two lines have the same steepness and direction, they will run alongside each other without ever meeting – that's the essence of being parallel. So, in our case, any line with a slope of 2/7 will be parallel to the line y = (2/7)x + 9, regardless of its y-intercept. This principle holds true for all parallel lines, making it a handy shortcut in geometry and algebra problems. Now, let's take a look at why the other options might be incorrect to reinforce our understanding.

Why the Other Options Are Incorrect

Let's quickly go through the other options to understand why they're not the correct answer. This will help solidify our understanding of parallel lines and slopes. We have the following options:

• B. -7/2
• C. 9
• D. -1/9
• E. -2/7
• F. 1/9

Option B, -7/2, is the negative reciprocal of 2/7. Lines with slopes that are negative reciprocals of each other are perpendicular, not parallel. Perpendicular lines intersect at a 90-degree angle, so this is the slope of a line that would be perpendicular to our original line. Option C, 9, is the y-intercept of the given line, not the slope. It tells us where the line crosses the y-axis, but it doesn't tell us anything about the line's steepness or direction. Options D, -1/9, E, -2/7, and F, 1/9, are just different numbers that don't have a direct relationship to the slope of the original line. They aren't the same as 2/7, and they aren't the negative reciprocal of 2/7, so they don't represent the slope of a parallel or perpendicular line.

The key takeaway here is that only lines with the same slope are parallel. Understanding this distinction is crucial for correctly answering questions about parallel lines and their slopes. By eliminating the incorrect options, we reinforce our understanding of the core concept and build confidence in our ability to solve similar problems in the future.

Conclusion: The Slope of a Parallel Line

So, to wrap it all up, the slope of a line parallel to y = (2/7)x + 9 is indeed 2/7. We arrived at this answer by understanding the fundamental principle that parallel lines have the same slope. We identified the slope of the given line using the slope-intercept form (y = mx + b), and then we applied the parallel line rule. Remember, guys, this is a key concept in coordinate geometry, and mastering it will help you tackle a wide range of math problems. Keep practicing, and you'll become a slope-finding pro in no time!

By understanding the relationship between parallel lines and their slopes, you can easily solve these types of problems. The next time you see a question like this, you'll be ready to tackle it with confidence. And that’s a win for everyone! Keep up the great work, and remember, math can be fun when you understand the underlying concepts. Now go out there and conquer those slopes!