Finding The Equation Of A Line Passing Through (1, 4) With Slope -2
Hey guys! Today, we're diving deep into the fascinating world of linear equations, specifically focusing on how to find the equation of a line when you're given a point it passes through and its slope. This is a fundamental concept in algebra and geometry, and mastering it will definitely give you a solid foundation for more advanced math topics. So, let's get started and unravel the mystery behind lines, points, and slopes!
Understanding the Basics
Before we jump into the nitty-gritty, let's quickly recap some essential concepts. A line in a two-dimensional plane is defined by its slope and a point it passes through. The slope, often denoted by m, tells us how steep the line is and in what direction it's inclined. It's essentially the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. A positive slope means the line goes upwards from left to right, a negative slope means it goes downwards, a zero slope indicates a horizontal line, and an undefined slope signifies a vertical line.
The equation of a line is a mathematical expression that describes the relationship between the x- and y-coordinates of all the points lying on that line. There are several forms of linear equations, but the two most commonly used are the slope-intercept form and the point-slope form. The slope-intercept form, y = mx + b, is super handy when you know the slope (m) and the y-intercept (b), which is the point where the line crosses the y-axis. On the other hand, the point-slope form is your go-to choice when you have a point (x₁, y₁) on the line and the slope (m). It's expressed as y - y₁ = m( x - x₁).
Now, let's talk about the specific problem at hand: finding the equation of a line that passes through the point (1, 4) and has a slope of -2. We have a point and we have a slope – sounds like the perfect situation to use the point-slope form, right? Absolutely! This form is tailor-made for this type of problem, and it's going to make our lives so much easier. By plugging in the given values, we can quickly derive the equation and then, if we want, convert it to other forms like the slope-intercept form. Understanding these foundational concepts is key to tackling more complex problems down the road. So, keep these definitions and formulas in your mental toolkit, and you'll be well-equipped to handle any linear equation challenge that comes your way.
Applying the Point-Slope Form
The point-slope form of a linear equation, as we discussed, is given by y - y₁ = m( x - x₁). This nifty formula allows us to construct the equation of a line using just a single point (x₁, y₁) on the line and its slope (m). In our case, we're given that the line passes through the point (1, 4) and has a slope of -2. This means we can directly substitute these values into the point-slope form to get our equation.
Let's break it down step by step. First, we identify our given values: (x₁, y₁) = (1, 4) and m = -2. Now, we plug these values into the formula: y - 4 = -2( x - 1). See how simple that was? We've essentially built the equation of our line using the information provided. But we're not quite done yet. This equation is in point-slope form, which is perfectly valid, but sometimes it's more convenient to have the equation in slope-intercept form ( y = mx + b) or standard form ( Ax + By = C). To get there, we need to do a little algebraic manipulation.
The next step is to simplify the equation we've obtained. We start by distributing the -2 on the right side of the equation: y - 4 = -2x + 2. Now, to isolate y and get the equation into slope-intercept form, we add 4 to both sides: y = -2x + 2 + 4, which simplifies to y = -2x + 6. Voila! We've successfully transformed the equation from point-slope form to slope-intercept form. This form tells us that the line has a slope of -2 (which we already knew) and a y-intercept of 6, meaning it crosses the y-axis at the point (0, 6). This transformation is super useful because the slope-intercept form makes it easy to visualize the line and identify key characteristics like its slope and y-intercept. It's also a great way to double-check our work and ensure that the equation we've derived accurately represents the line passing through (1, 4) with a slope of -2.
Converting to Slope-Intercept Form
As we just saw, the point-slope form is a fantastic starting point for finding the equation of a line, but often, we want to express the equation in slope-intercept form, which is y = mx + b. This form is incredibly useful because it directly reveals the slope (m) and the y-intercept (b) of the line, making it super easy to visualize and analyze. Converting from point-slope to slope-intercept form involves a bit of algebraic manipulation, but it's a straightforward process once you get the hang of it.
We started with the equation in point-slope form: y - 4 = -2( x - 1). The first step in converting to slope-intercept form is to distribute the slope (-2) on the right side of the equation. This means multiplying -2 by both x and -1 inside the parentheses. Doing so, we get: y - 4 = -2x + 2. Notice how we've removed the parentheses and now have a more expanded form of the equation. This is a crucial step because it allows us to isolate y, which is our ultimate goal for achieving the slope-intercept form.
Next, we need to isolate y on the left side of the equation. To do this, we simply add 4 to both sides of the equation. Remember, whatever operation you perform on one side of the equation, you must perform on the other side to maintain the equality. Adding 4 to both sides gives us: y - 4 + 4 = -2x + 2 + 4. Simplifying this, we get: y = -2x + 6. And there you have it! We've successfully converted the equation to slope-intercept form. As we mentioned earlier, this form immediately tells us that the slope of the line is -2 and the y-intercept is 6. This means the line crosses the y-axis at the point (0, 6). The slope-intercept form is not just a different way of writing the equation; it provides valuable information about the line's characteristics at a glance. It's a powerful tool for understanding and working with linear equations, so mastering this conversion is a worthwhile skill.
Understanding the Slope and Y-intercept
Now that we've arrived at the equation y = -2x + 6, which is in slope-intercept form, let's take a moment to truly understand what this equation is telling us. The slope-intercept form, as the name suggests, highlights two key features of a line: its slope and its y-intercept. These two values provide a complete picture of the line's orientation and position in the coordinate plane. So, let's dissect this equation and see what we can learn.
The slope, denoted by m in the equation y = mx + b, represents the steepness and direction of the line. In our equation, y = -2x + 6, the slope m is -2. This negative slope tells us that the line is decreasing as we move from left to right. In other words, for every 1 unit we move horizontally, the line goes down by 2 units vertically. The magnitude of the slope (the absolute value) indicates how steep the line is. A slope of -2 is steeper than a slope of -1, for example. Understanding the slope is crucial for visualizing the line and predicting its behavior. A steeper slope means a more rapid change in the y-value for a given change in the x-value.
The y-intercept, denoted by b in the equation y = mx + b, is the point where the line crosses the y-axis. This is the point where x = 0. In our equation, y = -2x + 6, the y-intercept b is 6. This means the line intersects the y-axis at the point (0, 6). The y-intercept is a fixed point that anchors the line in the coordinate plane. Knowing the y-intercept and the slope allows us to accurately draw the line on a graph. We can start by plotting the y-intercept and then use the slope to find other points on the line. For instance, since the slope is -2, we can move 1 unit to the right from the y-intercept (0, 6) and then 2 units down to find another point on the line, which would be (1, 4). This connection between the equation, the slope, the y-intercept, and the graphical representation is what makes linear equations so powerful and versatile. By understanding these components, you can confidently analyze and manipulate linear equations in various contexts.
Verifying the Solution
After going through the process of finding the equation of a line, it's always a good idea to verify your solution. This helps ensure that you haven't made any mistakes along the way and that the equation you've derived accurately represents the line described in the problem. In our case, we found the equation of a line passing through the point (1, 4) with a slope of -2 to be y = -2x + 6. Now, let's see how we can verify this solution.
There are a couple of ways we can verify our solution. One method is to plug the given point (1, 4) into the equation and see if it satisfies the equation. If the point lies on the line, its coordinates should make the equation true. So, let's substitute x = 1 and y = 4 into our equation: 4 = -2(1) + 6. Simplifying the right side, we get: 4 = -2 + 6, which simplifies further to 4 = 4. This is a true statement, so the point (1, 4) does indeed lie on the line represented by the equation y = -2x + 6. This gives us confidence that our equation is correct.
Another way to verify our solution is to check the slope. We know that the slope of our line should be -2. We can visually verify this by plotting the line on a graph or by finding another point on the line using our equation and calculating the slope between the two points. Let's use our equation to find another point. If we let x = 2, then y = -2(2) + 6 = -4 + 6 = 2. So, the point (2, 2) also lies on our line. Now, we can calculate the slope between the points (1, 4) and (2, 2) using the slope formula: m = ( y₂ - y₁) / (x₂ - x₁). Plugging in our values, we get: m = (2 - 4) / (2 - 1) = -2 / 1 = -2. This confirms that the slope of our line is indeed -2, as required. By verifying both the point and the slope, we can be highly confident that our solution, y = -2x + 6, is the correct equation for the line passing through (1, 4) with a slope of -2.
Conclusion
Alright guys, we've reached the end of our journey through the equation of a line passing through a given point with a specific slope. We've covered a lot of ground, from understanding the basic concepts of slope and linear equations to applying the point-slope form, converting to slope-intercept form, and verifying our solution. Hopefully, you now have a solid grasp of how to tackle these types of problems with confidence and ease. Remember, the key is to break down the problem into manageable steps, understand the underlying principles, and practice, practice, practice!
We started by defining the slope and the equation of a line, highlighting the importance of the point-slope form as a tool for finding the equation when given a point and a slope. We then meticulously applied the point-slope form to our specific problem, substituting the given point (1, 4) and slope -2 into the formula. We didn't stop there, though. We went on to convert the equation from point-slope form to the more versatile slope-intercept form, y = mx + b, which allows us to easily identify the slope and y-intercept of the line.
Understanding the slope and y-intercept is crucial for visualizing and analyzing linear equations. The slope tells us the steepness and direction of the line, while the y-intercept pinpoints where the line crosses the y-axis. We discussed how these two values provide a complete picture of the line's position and orientation in the coordinate plane. Finally, we emphasized the importance of verifying our solution. We demonstrated how to plug the given point into the equation to ensure it holds true and how to calculate the slope using two points on the line to confirm that it matches the given slope. Verification is a critical step in any mathematical problem-solving process, as it helps us catch errors and build confidence in our results.
Linear equations are a fundamental concept in mathematics, and the ability to find the equation of a line given a point and a slope is a valuable skill. It's a building block for more advanced topics in algebra, geometry, and calculus. So, keep practicing, keep exploring, and keep those mathematical gears turning. You've got this!
