Line Equation: (-3,-4) To (6,-10) In Slope-Intercept Form
Hey there, math enthusiasts! Today, we're diving into the fascinating world of linear equations. Specifically, we'll tackle a common problem: finding the equation of a line when you're given two points on that line. Our mission, should we choose to accept it, is to determine the equation of a line that passes through the points (–3, –4) and (6, –10), and express it in the ever-popular slope-intercept form. This form, as you probably know, is represented as y = mx + b, where 'm' is the slope of the line and 'b' is the y-intercept (the point where the line crosses the y-axis). It's a fundamental concept in algebra and geometry, and mastering it will unlock a whole new level of understanding for plotting lines, analyzing data, and solving a myriad of mathematical problems. So, grab your notebooks, sharpen your pencils, and let's embark on this algebraic adventure together! We'll break down the process step-by-step, ensuring that by the end of this article, you'll feel confident and capable of tackling similar problems on your own. Understanding how to derive this equation is more than just a mathematical exercise; it's about grasping the fundamental relationships between points, lines, and their graphical representations. Think of it as learning the language of the coordinate plane. We'll start by calculating the slope, which tells us how steep our line is and in which direction it's heading. Then, we'll use that slope along with one of the given points to find the y-intercept. It's a logical progression, and each step builds upon the previous one, creating a clear path to our final answer. Don't worry if math sometimes feels a bit daunting; we're here to make it accessible and even enjoyable! We'll use clear explanations and avoid jargon where possible, focusing on the core concepts that matter. So, let's get ready to unravel the mystery of this line's equation!
Calculating the Slope (m)
The first crucial step in finding the equation of our line in slope-intercept form (y = mx + b) is to calculate the slope, often denoted by the letter 'm'. The slope tells us the steepness and direction of a line. A positive slope indicates a line that rises from left to right, while a negative slope means it falls. The magnitude of the slope tells us how much it rises or falls for every unit of horizontal change. To find the slope when given two points, (x1, y1) and (x2, y2), we use the formula: m = (y2 - y1) / (x1 - x1). This formula essentially measures the 'rise' (the change in the y-coordinates) over the 'run' (the change in the x-coordinates). It's a ratio that captures the essence of the line's inclination. Let's plug in our given points: (–3, –4) and (6, –10). We can assign (x1, y1) to the first point and (x2, y2) to the second point. So, x1 = –3, y1 = –4, x2 = 6, and y2 = –10. Now, let's substitute these values into our slope formula:
m = (–10 – (–4)) / (6 – (–3))
Notice the double negative when subtracting the y1 and x1 values. This is a common place for errors, so pay close attention! Simplifying the numerator: –10 – (–4) = –10 + 4 = –6. And simplifying the denominator: 6 – (–3) = 6 + 3 = 9. So, our slope 'm' becomes:
m = –6 / 9
This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. Therefore:
m = –2 / 3
This means that for every 3 units we move to the right along the x-axis, our line goes down by 2 units. The slope of our line is –2/3. It's important to remember that it doesn't matter which point you assign as (x1, y1) and which as (x2, y2); you'll arrive at the same slope. For instance, if we chose (6, –10) as (x1, y1) and (–3, –4) as (x2, y2):
m = (–4 – (–10)) / (–3 – 6)
m = (–4 + 10) / (–9)
m = 6 / –9
m = –2 / 3
As you can see, the result is identical. This consistency is a hallmark of mathematical formulas and gives us confidence in our calculation. Understanding the slope is the bedrock of determining the line's equation, and we've successfully laid that foundation.
Finding the Y-Intercept (b)
Now that we've successfully calculated the slope (m = –2/3), the next step is to find the y-intercept, represented by 'b' in our slope-intercept form equation (y = mx + b). The y-intercept is the y-coordinate of the point where the line crosses the y-axis. At this point, the x-coordinate is always zero. To find 'b', we can use the slope-intercept form equation itself and plug in the slope we just found, along with the coordinates of either of the two points given. This is because both points lie on the line, so they must satisfy its equation. Let's choose the first point, (–3, –4). Here, x = –3 and y = –4. We already know m = –2/3. So, we substitute these values into y = mx + b:
–4 = (–2/3) * (–3) + b
Let's simplify the multiplication: (–2/3) * (–3). The negative signs cancel each other out, and the 3 in the denominator cancels with the 3 in the multiplier, leaving us with just 2. So the equation becomes:
–4 = 2 + b
To isolate 'b', we need to subtract 2 from both sides of the equation:
–4 – 2 = b
–6 = b
So, our y-intercept is –6. This means our line crosses the y-axis at the point (0, –6). To be absolutely sure and to reinforce our understanding, let's try using the other point, (6, –10). Here, x = 6 and y = –10, and m = –2/3. Plugging these into y = mx + b:
–10 = (–2/3) * (6) + b
Now, let's simplify the multiplication: (–2/3) * 6. We can think of this as (–2 * 6) / 3 = –12 / 3, which equals –4. So the equation becomes:
–10 = –4 + b
To find 'b', we add 4 to both sides of the equation:
–10 + 4 = b
–6 = b
Again, we arrive at b = –6. This consistency confirms our calculation. We now have both the slope (m = –2/3) and the y-intercept (b = –6), which are the two essential components for writing the equation of the line in slope-intercept form.
Writing the Equation in Slope-Intercept Form
We've reached the final and most satisfying step: writing the equation of the line in slope-intercept form. We have all the necessary pieces: the slope 'm' and the y-intercept 'b'. Our target form is y = mx + b. We found that m = –2/3 and b = –6. Simply substitute these values into the general form:
y = (–2/3)x + (–6)
This can be written more cleanly as:
y = –2/3x – 6
And there you have it! This is the equation of the line that passes through the points (–3, –4) and (6, –10), expressed in slope-intercept form. This equation is incredibly useful. If you wanted to know the y-value for any given x-value on this line, you could just plug that x into the equation and solve for y. For example, if you wanted to know the y-value when x = 0, you'd get y = –2/3(0) – 6 = –6, which is our y-intercept, as expected. If you wanted to know the y-value when x = 3, you'd get y = –2/3(3) – 6 = –2 – 6 = –8. So, the point (3, –8) also lies on this line. This equation encapsulates the entire line and all the points that lie upon it. It's a concise and powerful way to represent linear relationships. Remember, the process involved finding the slope using the two-point formula and then using that slope and one of the points to solve for the y-intercept. Each step is logical and builds towards the final answer. This skill is fundamental in many areas of mathematics, from graphing to solving systems of equations and understanding real-world linear models.
Visualizing the Line
To truly appreciate the equation y = –2/3x – 6, let's quickly visualize it. We know our line passes through (–3, –4) and (6, –10). We also know it crosses the y-axis at (0, –6). If you were to plot these three points on a graph, you would see they all fall perfectly on a straight line. The slope of –2/3 means that starting from any point on the line, if you move 3 units to the right (increase x by 3), you must move 2 units down (decrease y by 2) to reach another point on the line. This consistent ratio is what defines the line's path. The negative slope confirms that the line descends as you move from left to right, which aligns with our y-values decreasing as our x-values increase from –3 to 6. The y-intercept of –6 tells us exactly where the line intersects the vertical y-axis. This visual confirmation helps solidify the abstract algebraic result. Imagine starting at the y-intercept (0, -6). If you move 3 units right (to x=3), you'd move 2 units down to y=-8, confirming the point (3, -8). If you move another 3 units right (to x=6), you'd move another 2 units down to y=-10, confirming our second given point (6, -10). This step-by-step movement, governed by the slope, is the essence of a linear relationship. Understanding this visual representation can make the abstract concept of an equation much more concrete and intuitive. It transforms numbers and formulas into a tangible line on a graph.
Conclusion
We've successfully navigated the process of finding the equation of a line given two points. By first calculating the slope (m) using the formula m = (y2 - y1) / (x2 - x1), we found m = –2/3. Then, using this slope and one of the given points, we substituted the values into the slope-intercept form y = mx + b to solve for the y-intercept (b), discovering that b = –6. Finally, we combined these values to write the equation of the line in its definitive slope-intercept form: y = –2/3x – 6. This method is a cornerstone of coordinate geometry and has wide-ranging applications. Whether you're analyzing trends in data, modeling physical phenomena, or solving complex mathematical problems, understanding how to represent lines algebraically is an invaluable skill. Keep practicing these steps with different sets of points, and you'll become proficient in no time! Remember, the journey of learning mathematics is often about mastering these fundamental building blocks. If you're interested in exploring more about linear equations and their properties, I highly recommend checking out resources from Khan Academy for further study and practice exercises. They offer a wealth of information and tutorials that can deepen your understanding.
