Find The Equation Of A Line Passing Through (1,-1) And (3,5)
Hey guys! Today, we're diving into a classic math problem: finding the equation of a line. Specifically, we want to find the equation of a line that passes through the points (1, -1) and (3, 5). This is a super important concept in algebra, and once you get the hang of it, you'll be solving these problems like a pro. We'll break it down step-by-step so it's crystal clear. So, grab your pencils and let's get started!
Understanding Slope-Intercept Form
Before we jump into solving the problem, let's make sure we're all on the same page about slope-intercept form. This is a super common way to write the equation of a line, and it looks like this:
y = mx + b
Where:
yis the dependent variable (usually plotted on the vertical axis)xis the independent variable (usually plotted on the horizontal axis)mis the slope of the line, which tells us how steep the line is and whether it's going uphill or downhill as you move from left to right.bis the y-intercept, which is the point where the line crosses the y-axis. In other words, it's the value ofywhenxis 0.
Think of the slope (m) as the rate of change of the line. A positive slope means the line is going upwards, a negative slope means it's going downwards, a slope of zero means it's a horizontal line, and an undefined slope means it's a vertical line. The larger the absolute value of the slope, the steeper the line. The y-intercept (b) is our starting point on the y-axis. It's where the line begins its journey across the coordinate plane. Understanding these two components is key to unlocking the secrets of linear equations. Mastering slope-intercept form isn't just about memorizing y = mx + b; it's about understanding what each part represents. The slope m is the line's inclination, its steepness and direction, while the y-intercept b is its anchor, the point where it intersects the vertical axis. When you look at a line on a graph, can you immediately picture its slope and y-intercept? That's the level of understanding we're aiming for. The beauty of slope-intercept form lies in its simplicity. It neatly packages all the essential information about a line into a compact equation. By simply glancing at the equation, you can tell where the line crosses the y-axis and how sharply it rises or falls. This makes it incredibly useful for graphing lines, comparing their steepness, and solving real-world problems involving linear relationships. The power of the slope lies in its ability to describe the constant rate of change between two variables. In everyday scenarios, this could represent the speed of a car, the growth of a plant, or the cost per unit of a product. The steeper the line, the faster the change. By understanding the slope, we can make predictions and draw conclusions about the relationship between the variables. The y-intercept is equally important, as it represents the starting value or initial condition. In our car analogy, it could be the distance the car had already traveled before we started measuring. In the plant example, it could be the initial height of the seedling. The y-intercept provides a baseline for understanding the overall behavior of the linear relationship. So, remember guys, slope-intercept form is your friend! It's a powerful tool for understanding and working with linear equations. Get comfortable with it, and you'll be well on your way to mastering algebra. Once you've grasped this fundamental concept, you'll be able to tackle a wide range of problems involving lines, from finding equations to graphing and interpreting their behavior. It's a stepping stone to more advanced mathematical concepts, so make sure you have a solid foundation here. Let's move on and apply this knowledge to our specific problem.
Step 1: Finding the Slope (m)
The first thing we need to do is find the slope (m) of the line. Remember, the slope tells us how much the line goes up or down for every unit it moves to the right. We can calculate the slope using the following formula:
m = (y2 - y1) / (x2 - x1)
Where:
(x1, y1)and(x2, y2)are the coordinates of two points on the line.
In our case, we have the points (1, -1) and (3, 5). So, let's plug those values into the formula:
m = (5 - (-1)) / (3 - 1)
m = (5 + 1) / 2
m = 6 / 2
m = 3
So, the slope of our line is 3. This means that for every 1 unit we move to the right along the line, we move 3 units up. A positive slope of 3 indicates a fairly steep line that rises from left to right. We've already made significant progress in finding the equation of our line. We now know the m in our y = mx + b equation. This is a crucial piece of the puzzle. Finding the slope using the formula m = (y2 - y1) / (x2 - x1) is a fundamental skill in algebra. It allows us to quantify the steepness and direction of a line given any two points on it. The formula itself is derived from the concept of "rise over run," where the rise is the vertical change (difference in y-coordinates) and the run is the horizontal change (difference in x-coordinates). The beauty of this formula is its generality. It works for any two points on any non-vertical line. By plugging in the coordinates, we can quickly and accurately determine the slope, which is the foundation for understanding the line's behavior. It's important to be careful with the signs when using the formula. Subtracting a negative number is the same as adding a positive number, so make sure you're paying close attention to the order of operations. A common mistake is to mix up the order of subtraction in the numerator and denominator, which will result in the wrong sign for the slope. So, always double-check your calculations! Now that we've found the slope, we're halfway to our goal. We know that our equation will look something like y = 3x + b. The only thing left to do is find the y-intercept (b). We'll tackle that in the next step. Just remember guys, finding the slope is often the first and most important step in determining the equation of a line. Once you have the slope, you have a good sense of the line's direction and steepness, which makes the rest of the problem much easier. Practice using the slope formula with different pairs of points, and you'll become a slope-calculating master in no time! Let's move on and conquer the y-intercept next!
Step 2: Finding the y-intercept (b)
Now that we know the slope (m = 3), we can plug it into our slope-intercept form equation:
y = 3x + b
To find the y-intercept (b), we can use either of the points we were given, (1, -1) or (3, 5). Let's use the point (1, -1). This means we'll substitute x = 1 and y = -1 into the equation:
-1 = 3(1) + b
-1 = 3 + b
Now, we just need to solve for b:
-1 - 3 = b
-4 = b
So, the y-intercept (b) is -4. This means the line crosses the y-axis at the point (0, -4). Finding the y-intercept is like finding the line's starting point. It tells us where the line begins its journey on the coordinate plane. We've now discovered that our line crosses the y-axis at -4, which is a crucial piece of information. Substituting the coordinates of a known point into the equation and solving for b is a clever way to find the y-intercept. It allows us to leverage the information we already have (the slope and a point on the line) to uncover the missing piece of the puzzle. It's a technique that's widely used in algebra and other areas of mathematics. We could have used the other point, (3, 5), and we would have gotten the same answer for b. Try it out yourself to verify! This shows that the method is consistent and doesn't depend on which point we choose. The y-intercept has a significant meaning in many real-world applications. For example, if we're modeling the cost of a service based on the number of hours worked, the y-intercept might represent the fixed cost or the initial fee. Understanding the y-intercept can provide valuable insights into the context of the problem. We're almost there guys! We've found both the slope and the y-intercept. Now, all that's left to do is put them together to form the final equation of the line. It's like the final brushstroke on a masterpiece. Are you ready to see the complete picture? Let's move on to the final step!
Step 3: Writing the Equation in Slope-Intercept Form
We've found the slope (m = 3) and the y-intercept (b = -4). Now we can plug these values into the slope-intercept form equation:
y = mx + b
y = 3x + (-4)
y = 3x - 4
So, the equation of the line in slope-intercept form is y = 3x - 4. And that's our answer! We've successfully found the equation of the line that passes through the points (1, -1) and (3, 5). Writing the equation in slope-intercept form is the culmination of our efforts. It's the final answer that neatly summarizes the line's characteristics. The equation y = 3x - 4 tells us everything we need to know about this line. It has a slope of 3, meaning it rises steeply from left to right, and it crosses the y-axis at -4. With this equation, we can easily graph the line, find other points on the line, and solve related problems. We can be proud of ourselves for solving this problem step-by-step. We broke it down into manageable chunks, found the slope, found the y-intercept, and then put it all together. This is a great strategy for tackling any math problem. Remember guys, the slope-intercept form is a powerful tool. It allows us to represent lines in a concise and informative way. By understanding the slope and y-intercept, we can unlock the secrets of linear equations and their applications in the real world. We've reached the finish line! We've successfully found the equation of the line. Give yourselves a pat on the back! But don't stop here. The more you practice, the better you'll become at solving these types of problems. Try working through some similar examples, and you'll soon be a master of linear equations. Let's celebrate our success and keep learning!
Conclusion
Therefore, the equation of the line that passes through the points (1, -1) and (3, 5) in slope-intercept form is y = 3x - 4. The correct answer is C. We found the slope to be 3 and the y-intercept to be -4, and plugging these values into the slope-intercept form equation gave us our final answer. We've demonstrated a clear and methodical approach to solving this problem. We started by understanding the slope-intercept form, then we found the slope using the formula, then we found the y-intercept by substituting a point into the equation, and finally, we wrote the complete equation. This step-by-step process is a valuable tool for tackling any mathematical challenge. We've reinforced the key concepts of slope and y-intercept and their role in defining a line. Understanding these concepts is fundamental to success in algebra and beyond. We've also highlighted the importance of careful calculations and attention to detail. A small mistake can lead to a wrong answer, so it's crucial to double-check your work. Remember guys, math is a journey, not a destination. Every problem we solve is a step forward on that journey. By practicing and persevering, we can unlock our full mathematical potential. So, keep learning, keep exploring, and keep having fun with math! We've successfully conquered this problem, and we're ready for the next challenge. Let's continue our mathematical adventure!
