Line Equation: Point (9, -7), Slope 0, Standard Form

Hey guys! Let's dive into a fun math problem today: finding the equation of a line. We've got a specific point that our line passes through, (9, -7), and we know the slope is 0. Plus, we need to express our final answer in standard form. Don’t worry, it sounds trickier than it actually is. We'll break it down step by step so it's super clear. Understanding how to work with slopes and points to define lines is a fundamental concept in algebra and has tons of applications in real-world scenarios, from calculating the steepness of a hill to mapping out financial trends. So, let’s get started!

Understanding Slope and Points

Before we jump into the nitty-gritty of the equation, let's make sure we're all on the same page about what a slope and a point mean in the context of a line. The slope tells us how steep the line is and in what direction it's going. A slope of 0, which we have here, means the line is perfectly horizontal – it doesn't go up or down. Think of it like a flat road; there's no incline. On the other hand, a point, like our (9, -7), gives us a specific location on the coordinate plane that the line passes through. It's like a landmark that the line has to visit. To find the equation, we'll use this information as our guide. Remember, the standard form of a linear equation is usually written as Ax + By = C, where A, B, and C are constants, and x and y are our variables. This form helps us easily identify intercepts and compare different lines, making it a useful tool in various mathematical analyses.

Utilizing the Point-Slope Form

One of the handiest tools we have for this kind of problem is the point-slope form of a linear equation. This form is written as: y - y1 = m(x - x1), where:

• 'y' and 'x' are the variables representing the coordinates of any point on the line.
• 'm' is the slope of the line.
• '(x1, y1)' is a known point on the line. We've got this! It’s our (9, -7).

This form is super useful because it directly incorporates the slope and a point on the line, making it a perfect starting point for our problem. By plugging in the values we know, we can easily manipulate the equation to get it into the standard form we need. It's like having a template that fits our specific situation, making the process much smoother. The point-slope form not only simplifies the initial setup but also gives us a clear understanding of how the slope and a specific point define the line's path across the coordinate plane.

Applying Point-Slope Form

Okay, let’s plug in the values we know into the point-slope form. We have m = 0 (our slope) and (x1, y1) = (9, -7) (our point). So, here’s how it looks:

y - (-7) = 0(x - 9)

Notice how we substituted -7 for y1, 9 for x1, and 0 for m. Now, let’s simplify this a bit. The double negative turns into a positive, and anything multiplied by 0 is just 0. This makes our equation much simpler:

y + 7 = 0

See? We're already getting somewhere! By substituting the given values into the point-slope form, we've created a basic equation that represents our line. The next step is to transform this into the standard form, which will give us a clear and concise representation of the line's properties. Simplifying equations like this is a fundamental skill in algebra, allowing us to work with complex problems more easily and efficiently.

Transforming to Standard Form

Our next step is to get this equation into the standard form, which, as we mentioned earlier, looks like Ax + By = C. We currently have y + 7 = 0. To get this into standard form, we need to isolate the constant term on the right side of the equation. So, let's subtract 7 from both sides:

y + 7 - 7 = 0 - 7

This simplifies to:

y = -7

Now, let's think about how this fits into our standard form Ax + By = C. We don't have an x term, which means A is 0. We have 1y (or just y), so B is 1. And our constant, C, is -7. So, our equation in standard form is:

0x + 1y = -7

But, we can simplify it even further since 0x is just 0. This leaves us with:

y = -7

This is our final equation in standard form! You might notice that this is a horizontal line. That makes sense, right? A slope of 0 means the line doesn't rise or fall, it just runs flat. The transformation into standard form not only provides a clear mathematical representation but also helps in visualizing the line's orientation on the coordinate plane.

The Final Equation and its Meaning

So, guys, the equation of the line that passes through the point (9, -7) and has a slope of 0, written in standard form, is y = -7. This equation represents a horizontal line that crosses the y-axis at -7. No matter what the x-value is, the y-value will always be -7. Understanding this helps us visualize the line on a graph – it’s a straight, horizontal line running across the coordinate plane at y = -7. This type of problem is a great example of how algebraic equations can describe geometric shapes, linking the abstract world of numbers to the visual world of graphs. Mastering these concepts opens doors to more complex mathematical and real-world problem-solving scenarios.

Visualizing the Line

To really solidify our understanding, let’s think about what this line looks like on a graph. Since the equation is y = -7, we know that every point on the line will have a y-coordinate of -7. The x-coordinate can be anything, but the line will always be at the same vertical level. Imagine plotting a few points: (0, -7), (9, -7) (our original point!), (15, -7), (-5, -7). If you connect these points, you'll see a perfectly horizontal line stretching across the graph at y = -7. This visualization helps us connect the equation to its geometric representation, making the concept more intuitive. Understanding how equations translate into graphical forms is crucial for various applications, from designing structures to interpreting data trends.

Key Takeaways

Alright, let’s recap what we’ve learned today. We successfully found the equation of a line given a point and a slope, and we expressed it in standard form. Here’s a quick rundown:

1. We started with the point-slope form: y - y1 = m(x - x1).
2. We substituted our values: m = 0 and (x1, y1) = (9, -7).
3. We simplified the equation to y + 7 = 0.
4. We converted it to standard form: y = -7.

Remember, the key to these problems is understanding the relationship between the slope, points, and the different forms of linear equations. Practice makes perfect, so keep working on these types of problems, and you’ll become a pro in no time! The ability to manipulate and interpret linear equations is a fundamental skill in mathematics and is highly valuable in fields like engineering, economics, and computer science.

Further Practice

If you want to practice more, try working through similar problems. For example, you could try finding the equation of a line with a different slope, or one that passes through a different point. You could also try converting equations from slope-intercept form to standard form, or vice versa. The more you practice, the more comfortable you'll become with these concepts. You can even explore real-world applications of linear equations, such as calculating the cost of a service based on a fixed fee and an hourly rate, or determining the distance traveled at a constant speed. Keep up the great work, and you'll be mastering linear equations in no time!

Great job, everyone! You've tackled a line equation problem and learned how to express it in standard form. Keep practicing, and you'll be a math whiz in no time!