Equation Of A Line Passing Through Two Points (-6, 7) And (-3, 6)
Let's dive into the world of linear equations and figure out which one represents the line passing through the points (-6, 7) and (-3, 6). It might sound tricky at first, but don't worry, we'll break it down step by step so it's super easy to understand. We’ll cover everything from the basic formulas to the practical application of finding the right equation. So, grab your thinking caps, and let's get started!
Understanding Linear Equations
Before we jump into the problem, let's quickly recap what linear equations are all about. Think of a linear equation as a straight line drawn on a graph. The most common form you'll see is y = mx + b, where:
- y is the vertical coordinate.
- x is the horizontal coordinate.
- m is the slope of the line (how steep it is).
- b is the y-intercept (where the line crosses the y-axis).
Understanding this form is crucial because it gives us a framework to work with. When we know two points on a line, we can use this information to find the equation of the line. The slope m tells us the direction and steepness of the line, while the y-intercept b anchors the line to a specific point on the graph. So, with these pieces of information, we can accurately describe any straight line using an equation.
Now, why is this important? Well, in many real-world scenarios, linear relationships pop up everywhere. From calculating the cost of a taxi ride based on distance to predicting the growth of a plant over time, linear equations help us model and understand these situations. So, by mastering the basics, we’re not just solving math problems; we’re gaining a powerful tool for analyzing the world around us.
So, let's keep this y = mx + b in mind as we tackle our problem. We’ll use it to find the slope and y-intercept, and ultimately, the correct equation for the line that passes through the given points. Are you ready? Let’s move on to the next step!
Step 1: Calculate the Slope (m)
The slope (m) is the heart and soul of a linear equation, telling us how much the line rises or falls for every step we take to the right. To find the slope when we have two points, we use a nifty little formula:
m = (y₂ - y₁) / (x₂ - x₁)
Where:
- (x₁, y₁) are the coordinates of the first point.
- (x₂, y₂) are the coordinates of the second point.
In our case, we have the points (-6, 7) and (-3, 6). Let’s plug these values into the formula:
- x₁ = -6
- y₁ = 7
- x₂ = -3
- y₂ = 6
So, our equation becomes:
m = (6 - 7) / (-3 - (-6))
Let's simplify this step-by-step. First, subtract the y-coordinates: 6 - 7 = -1. Then, subtract the x-coordinates: -3 - (-6) = -3 + 6 = 3. Now, we have:
m = -1 / 3
Ta-da! We've found the slope. The slope, m, is -1/3. This tells us that for every 3 units we move to the right on the graph, the line goes down 1 unit. Think of it like walking downhill – for every few steps forward, you lose a little bit of altitude.
But why is calculating the slope so crucial? Well, it's the foundation upon which we build the rest of the equation. Without the slope, we can't determine the direction and steepness of the line, making it impossible to accurately represent the line with an equation. The slope is the key that unlocks the relationship between the x and y coordinates on our line.
So, with our slope in hand, we're one giant step closer to finding the equation of the line. Next up, we'll use this slope and one of our points to find the y-intercept. Let's keep going!
Step 2: Find the Y-Intercept (b)
Now that we've conquered the slope, it's time to find the y-intercept (b). Remember, the y-intercept is the point where the line crosses the y-axis. To find it, we're going to use the slope-intercept form of a linear equation, which we talked about earlier:
y = mx + b
We already know the slope (m = -1/3), and we have two points to choose from: (-6, 7) and (-3, 6). Let's pick the point (-6, 7) for this step. We'll plug in the values of x, y, and m into the equation and solve for b:
7 = (-1/3) * (-6) + b
First, let's simplify the multiplication: (-1/3) * (-6) = 2. So, our equation becomes:
7 = 2 + b
Now, to isolate b, we subtract 2 from both sides of the equation:
7 - 2 = b
5 = b
Eureka! We've found the y-intercept. The y-intercept, b, is 5. This means the line crosses the y-axis at the point (0, 5).
But why go through all this trouble to find the y-intercept? Well, the y-intercept is like the line's anchor on the y-axis. It tells us exactly where the line starts its journey across the graph. Without it, we'd know the slope, but we wouldn't know where to position the line. The y-intercept completes the picture, giving us all the information we need to define the line.
Now that we have both the slope and the y-intercept, we're ready to write the equation of the line. This is where all our hard work comes together. Are you excited? Let's do it!
Step 3: Write the Equation
We've done the heavy lifting – calculating the slope and finding the y-intercept. Now comes the satisfying part: putting it all together to write the equation of the line. Remember our trusty slope-intercept form:
y = mx + b
We know:
- m = -1/3 (the slope)
- b = 5 (the y-intercept)
Let's plug these values into the equation:
y = (-1/3)x + 5
And there you have it! The equation of the line that passes through the points (-6, 7) and (-3, 6) is y = (-1/3)x + 5.
But what does this equation really tell us? It tells us the relationship between any x-coordinate and its corresponding y-coordinate on the line. If you give me any x value, I can plug it into the equation and instantly find the y value that makes that point lie on the line. It's like a magic formula that describes every single point on the line!
Why is this so powerful? Well, with this equation, we can now predict where the line will go. We can extend it infinitely in both directions and know exactly which points it will pass through. This is incredibly useful in many real-world applications, from predicting trends to designing structures.
So, let's take a moment to appreciate what we've accomplished. We started with two points and, through a series of logical steps, we've arrived at the equation of the line that connects them. This is a fundamental skill in algebra, and you've just mastered it!
Now, let’s make sure we choose the correct answer from the options provided.
Step 4: Choose the Correct Option
We've found that the equation of the line is:
y = (-1/3)x + 5
Now, let's look at the options given and see which one matches our equation:
A. y = -1/3 x + 9 B. y = -1/3 x + 5 C. y = -3x - 11y D. y = -3x + 25
Comparing our equation with the options, we can see that option B, y = -1/3 x + 5, is an exact match!
So, the correct answer is B. Woohoo! We did it!
But wait, why is it so important to check our answer against the options? Well, in a multiple-choice setting, there might be subtle variations in the equations that can trip us up. By carefully comparing our result with the options, we make sure we haven't made any mistakes and that we're choosing the most accurate answer.
Also, this step gives us a chance to double-check our work. If our equation doesn't match any of the options, it's a red flag that we might have made a mistake somewhere along the way. This is a great opportunity to go back and review our calculations to make sure everything is correct.
So, choosing the correct option isn't just about picking the right letter; it's about confirming our understanding and ensuring our accuracy. It's the final piece of the puzzle that completes our journey from two points to the equation of a line.
Conclusion
Guys, we've successfully navigated the world of linear equations and found the equation of the line that passes through the points (-6, 7) and (-3, 6). We broke down the problem into manageable steps:
- Understanding Linear Equations
- Calculating the slope (m)
- Finding the y-intercept (b)
- Writing the equation
- Choosing the correct option
We learned how to use the slope-intercept form (y = mx + b) and how to apply it to solve a real problem. Remember, the slope tells us the direction and steepness of the line, while the y-intercept tells us where the line crosses the y-axis. Together, they give us all the information we need to define any straight line.
But the journey doesn't end here! The skills you've learned today are just the beginning. Linear equations are a fundamental concept in mathematics, and they pop up in all sorts of contexts, from physics and engineering to economics and computer science. By mastering these basics, you're building a strong foundation for future learning.
So, what's next? Maybe you want to explore different forms of linear equations, like the point-slope form or the standard form. Or perhaps you're curious about systems of linear equations and how they can be used to solve real-world problems. The possibilities are endless!
Keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is vast and fascinating, and you've got the tools to explore it. And remember, every problem you solve is a step forward on your journey to mathematical mastery. You've got this!
