Lines L1 & L2: Finding And Verifying Equations
Hey guys! Let's dive into some math and figure out the equations of lines from a graph. This is a super important skill in algebra, and we're going to break it down step by step. We'll be looking at two lines, L1 and L2, and using the points they pass through to find their equations. Then, we'll verify a specific statement about one of the lines. So, grab your thinking caps, and let's get started!
Understanding the Graph and Given Points
First things first, let's visualize the graph. We've got two lines, L1 and L2, plotted on a coordinate plane. Line L1 zips through the points (-1, 0) and (0, 1), while line L2 cruises through (0, 2) and (2, 0). These points are our clues, the breadcrumbs that will lead us to the equations of the lines. Remember, every point on a line satisfies its equation, so we can use these points to our advantage. Think of the coordinate plane as a map, and these points as landmarks that help us chart the course of each line. Knowing the coordinates allows us to calculate the slope and y-intercept, which are the key ingredients in the equation of a line. So, let’s use the given points effectively to figure out the equation for each line.
The coordinate plane acts as a visual representation of the relationship between x and y values. Each point is defined by its x and y coordinates, allowing us to pinpoint its exact location. In this case, the points (-1, 0), (0, 1), (0, 2), and (2, 0) give us the crucial information needed to determine the slope and y-intercept for each line. For instance, the points (-1, 0) and (0, 1) on line L1 tell us how the y-value changes as the x-value changes. This change is what we call slope, and it is a fundamental aspect of linear equations. The y-intercept, on the other hand, is the point where the line crosses the y-axis, which is an important anchor point for writing the equation. By carefully analyzing these points, we can unveil the unique characteristics of each line and accurately express them in equation form. We can use the slope-intercept form, y = mx + b, where 'm' is the slope and 'b' is the y-intercept, to represent these lines algebraically. Now, let's calculate the slopes and intercepts for lines L1 and L2 to determine their equations.
Understanding these points is essential because they're the foundation for finding the line's equation. We’ll use these points to calculate the slope and then plug them into the slope-intercept form (y = mx + b) to find the equation. Remember, the slope (m) tells us how steep the line is and in what direction it's going, while the y-intercept (b) tells us where the line crosses the y-axis. These two values completely define a line, so once we have them, we've cracked the code! Thinking about it this way makes the process less intimidating and more like a puzzle. So, let’s use these points to piece together the equations of lines L1 and L2. This step-by-step approach will ensure we don't miss any important details and arrive at the correct equations. By thoroughly understanding the points and their significance, we set ourselves up for success in determining the equations of the lines.
Finding the Equation of Line L1
Let's focus on line L1 first. We know it passes through the points (-1, 0) and (0, 1). To find the equation, we'll use the trusty slope-intercept form: y = mx + b. The first thing we need to do is calculate the slope (m). Remember, the slope is the change in y divided by the change in x (rise over run). So, using our two points, the slope (m) is (1 - 0) / (0 - (-1)) = 1 / 1 = 1. Now we know that m = 1, so our equation is starting to look like y = 1x + b, or simply y = x + b. But we're not done yet! We still need to find the y-intercept (b).
To find the y-intercept (b), we can plug one of our points into the equation y = x + b. Let's use the point (0, 1) because it's super easy to work with. Plugging in x = 0 and y = 1, we get 1 = 0 + b. This simplifies to b = 1. Now we have all the pieces of the puzzle! We know the slope (m) is 1 and the y-intercept (b) is 1. So, we can confidently say that the equation of line L1 is y = x + 1. See? Not too scary, right? By breaking it down into smaller steps, we can easily find the equation of any line given two points. The key is to remember the slope formula and the slope-intercept form. Once you've got those down, you're golden! So, with the equation of line L1 in our grasp, let's move on to tackling line L2. We'll use the same approach, and soon we'll have both equations figured out!
The equation y = x + 1 is a linear equation, which means it represents a straight line on the coordinate plane. The slope (m = 1) tells us that for every one unit we move to the right on the x-axis, the line moves one unit up on the y-axis. The y-intercept (b = 1) tells us that the line crosses the y-axis at the point (0, 1). This equation is a concise way to describe the relationship between the x and y coordinates of every point on line L1. To further verify this equation, we can plug in the other given point, (-1, 0), to see if it satisfies the equation. Plugging in x = -1, we get y = -1 + 1 = 0, which matches the y-coordinate of the point. This confirms that the equation y = x + 1 accurately represents line L1. Understanding the equation of a line allows us to make predictions about its behavior and relationships with other lines. Now that we have successfully found the equation of line L1, we can proceed to find the equation of line L2, using the same principles and methods.
Finding the Equation of Line L2
Now, let's tackle line L2. We know it passes through the points (0, 2) and (2, 0). Just like before, we'll use the slope-intercept form (y = mx + b) to find the equation. First up, let's calculate the slope (m). Using our points, the slope (m) is (0 - 2) / (2 - 0) = -2 / 2 = -1. Aha! We have a negative slope this time, which means the line is sloping downwards. So, our equation is starting to look like y = -1x + b, or simply y = -x + b. We're on a roll! Now we just need to find the y-intercept (b).
Finding the y-intercept (b) for line L2 is actually super easy because we already have a point where x = 0. Remember, the y-intercept is the y-value when x = 0. So, looking at our points, we see that line L2 passes through the point (0, 2). This means that the y-intercept (b) is 2! Awesome! We've got the slope (m = -1) and the y-intercept (b = 2). Putting it all together, the equation of line L2 is y = -x + 2. High five! We've successfully found the equations of both lines. See how using the slope-intercept form and the given points makes it a breeze? Now we can confidently describe these lines with algebraic equations. Understanding these equations allows us to analyze the relationship between the lines, such as where they intersect or if they are parallel or perpendicular. So, with both equations in hand, we've accomplished our goal of representing lines L1 and L2 mathematically.
Knowing the equation of line L2 (y = -x + 2) gives us a complete picture of its behavior. The negative slope (m = -1) indicates that the line descends as we move from left to right. For every one unit we move to the right on the x-axis, the line moves one unit down on the y-axis. The y-intercept (b = 2) tells us that the line crosses the y-axis at the point (0, 2). To double-check our work, we can plug in the other given point, (2, 0), into the equation. Plugging in x = 2, we get y = -2 + 2 = 0, which matches the y-coordinate of the point. This confirms the accuracy of our equation. Now that we have the equations for both lines, we can analyze their characteristics, such as their slopes and intercepts, and understand their relationship on the coordinate plane. Finding the equations of lines is a fundamental skill in algebra, and it opens the door to solving many real-world problems involving linear relationships.
Verifying the Statement about Line L1
Okay, so the statement we need to verify is: "The equation of line L1 is y = x + 1." Guess what? We already found that the equation of line L1 is indeed y = x + 1! So, the statement is absolutely correct. Woohoo! We did it! By carefully calculating the slope and y-intercept using the given points, we were able to confirm the equation of line L1. This is a great example of how mathematical statements can be verified using logical steps and calculations. There's a certain satisfaction in knowing you've correctly solved a problem and verified a statement. It's like being a math detective and cracking the case! So, next time you encounter a statement about a line's equation, remember the steps we took today: calculate the slope, find the y-intercept, and put it all together in the slope-intercept form. You'll be verifying statements like a pro in no time!
Verifying the statement is an important step in the problem-solving process. It ensures that our solution is accurate and reliable. By comparing our calculated equation with the given statement, we can confirm whether they match or not. In this case, our calculations perfectly align with the statement, which gives us confidence in our understanding of the problem and the methods we used. Verification also helps us catch any potential errors in our calculations or assumptions. If our calculated equation didn't match the statement, we would need to go back and review our steps to identify the mistake. Therefore, verifying statements is not just a formality but a crucial part of mathematical reasoning and problem-solving. It reinforces the logical flow of our work and helps us build a strong foundation in mathematical concepts. So, always remember to verify your solutions whenever possible to ensure accuracy and deepen your understanding.
In conclusion, we've successfully navigated the world of lines and equations! We started by understanding the graph and the given points. Then, we used the slope-intercept form to find the equations of lines L1 and L2. Finally, we verified the statement about line L1 and found it to be correct. You guys rock! Keep practicing these skills, and you'll become line equation masters in no time! Remember, math can be fun and rewarding when you break it down into manageable steps. So, keep exploring, keep learning, and keep those math muscles strong!