Hey there, math enthusiasts! Today, we're diving into a classic geometry problem: finding the equation of a line parallel to another, given a point it passes through. This might sound tricky, but don't worry, we'll break it down step by step. So, grab your pencils and let's get started!
Understanding the Problem
Our problem states that a given line passes through the points (0,-3) and (2,3). The main goal here is to figure out the equation, in point-slope form, of a new line. This new line has two important characteristics: it's parallel to our original line and it goes through the point (-1,-1). Before we jump into the calculations, let's make sure we're on the same page with some key concepts. First, what does it mean for two lines to be parallel? Parallel lines, as you might remember, are lines that never intersect. They run alongside each other, maintaining the same distance apart. The crucial aspect here is that parallel lines have the same slope. This is our golden rule for solving this problem. Next, let's talk about the point-slope form of a linear equation. This form is super handy when we know a point on the line and the slope of the line. The point-slope form is given by: y - y1 = m(x - x1) where: m is the slope of the line. (x1, y1) is a point on the line. We'll be using this form to express our final answer. Now that we have a solid grasp of the basics, let's roll up our sleeves and get to the solution!
Step 1: Finding the Slope of the Given Line
The first thing we need to do is determine the slope of the line that passes through the points (0, -3) and (2, 3). Remember, the slope (often denoted by m) tells us how steep the line is and in which direction it's inclined. The formula to calculate the slope between two points (x1, y1) and (x2, y2) is: m = (y2 - y1) / (x2 - x1). This formula essentially calculates the “rise over run” – the change in the y-coordinate divided by the change in the x-coordinate. Now, let's plug in our points (0, -3) and (2, 3) into this formula. We can label (0, -3) as (x1, y1) and (2, 3) as (x2, y2). So, we have: x1 = 0, y1 = -3, x2 = 2, and y2 = 3. Substituting these values into the slope formula, we get: m = (3 - (-3)) / (2 - 0). Let's simplify this expression: m = (3 + 3) / 2. This further simplifies to: m = 6 / 2. Finally, we find that the slope of the given line is: m = 3. So, the line passing through the points (0, -3) and (2, 3) has a slope of 3. This is a crucial piece of information, as it will help us determine the slope of the parallel line.
Step 2: Determining the Slope of the Parallel Line
Alright, we've successfully found the slope of our original line, which is 3. Now comes the easy part! Remember our golden rule: parallel lines have the same slope. This means that the line we're trying to find, the one that's parallel to the given line, also has a slope of 3. That's it! We've got our slope for the parallel line. This principle is fundamental in coordinate geometry and makes our task much simpler. Since the lines are parallel, their steepness and direction of inclination are identical. Visually, you can imagine two lines running side-by-side, never converging or diverging. Mathematically, this translates directly into equal slopes. So, to reiterate, the slope (m) of our parallel line is also 3. We’re halfway there! Now that we know the slope, we just need to incorporate the point that the parallel line passes through to complete our equation in point-slope form.
Step 3: Using the Point-Slope Form
We're on the home stretch now! We know the slope of our parallel line is 3, and we also know that it passes through the point (-1, -1). This is exactly the information we need to use the point-slope form of a linear equation. As a quick refresher, the point-slope form is: y - y1 = m(x - x1) where: m is the slope of the line (x1, y1) is a point on the line. We have all these pieces! We know m = 3 and (x1, y1) = (-1, -1). Let's plug these values into the point-slope form: y - (-1) = 3(x - (-1)). Now, let's simplify this equation. Remember that subtracting a negative number is the same as adding a positive number. So, we can rewrite the equation as: y + 1 = 3(x + 1). And there you have it! This is the equation of the line that is parallel to the given line and passes through the point (-1, -1), expressed in point-slope form. This form is particularly useful because it directly shows the slope of the line (3) and a point it passes through (-1, -1).
Final Answer
The equation of the line that is parallel to the line passing through (0, -3) and (2, 3), and also passes through the point (-1, -1), in point-slope form, is: y + 1 = 3(x + 1). We successfully found the equation of the parallel line in point-slope form. We started by calculating the slope of the original line, then used the fact that parallel lines have the same slope to determine the slope of our new line. Finally, we plugged the slope and the given point into the point-slope form to get our answer. Remember, practice makes perfect, so keep working on these types of problems to build your skills and confidence!
Introduction
Hello, math lovers! Today, we’re going to tackle a classic problem in geometry: finding the equation of a line that’s parallel to another line, and passes through a specific point. This might sound a bit daunting, but trust me, we’ll break it down into simple, manageable steps. So, grab your favorite notebook, and let’s dive in!
Understanding the Basics
In this guide, we’ll solve a problem where we’re given that a certain line passes through the points (0, -3) and (2, 3). Our mission is to find the equation—in point-slope form—of a new line. This new line has two key characteristics: it’s parallel to the original line and it goes through the point (-1, -1). Before we jump into the calculations, it’s crucial that we’re all on the same page with some fundamental concepts. Let’s start with parallelism. What does it mean for two lines to be parallel? Simply put, parallel lines are lines that never intersect; they run alongside each other, maintaining a constant distance. The most important thing to remember here is that parallel lines share the same slope. This is the golden rule that will guide us through this problem. Next, let's discuss the point-slope form of a linear equation. This form is incredibly useful when we know a point on the line and the line’s slope. The point-slope form is expressed as: y - y1 = m(x - x1) where: m represents the slope of the line, and (x1, y1) is a known point on the line. We’ll use this form to express our final answer, so it’s important to keep it in mind. Now that we have a clear understanding of the basics, let’s put on our thinking caps and get started with the solution!
Step 1 Calculating the Slope of the Given Line
The very first step in our journey is to find the slope of the line that passes through the points (0, -3) and (2, 3). Remember, the slope, often denoted as m, tells us how steep the line is and the direction of its inclination. To calculate the slope between two points (x1, y1) and (x2, y2), we use the formula: m = (y2 - y1) / (x2 - x1). This formula is essentially the “rise over run” – the change in the y-coordinate divided by the change in the x-coordinate. Let's plug in our points (0, -3) and (2, 3) into this formula. We can assign (0, -3) as (x1, y1) and (2, 3) as (x2, y2). This gives us: x1 = 0, y1 = -3, x2 = 2, and y2 = 3. Now, substituting these values into the slope formula, we get: m = (3 - (-3)) / (2 - 0). Let’s simplify this expression step by step. First, we have: m = (3 + 3) / 2. This simplifies further to: m = 6 / 2. And finally, we find that the slope of the given line is: m = 3. So, the line that passes through the points (0, -3) and (2, 3) has a slope of 3. This is a critical piece of information, as it will help us determine the slope of the parallel line we’re trying to find. Understanding how to calculate slope is fundamental in linear algebra, and this step is a perfect example of its practical application.
Step 2 Finding the Slope of the Parallel Line
Great job! We’ve successfully calculated the slope of our original line, which turned out to be 3. Now for the easy part! Let's recall our golden rule: parallel lines have the same slope. What does this mean for us? It means that the line we're trying to find—the one that's parallel to the given line—also has a slope of 3. Yes, it’s that straightforward! We’ve just found the slope of our parallel line. This principle is a cornerstone of coordinate geometry and makes our task much simpler. Because the lines are parallel, their steepness and direction of inclination are identical. Think of it visually: two lines running side-by-side, never meeting or diverging. Mathematically, this translates directly into equal slopes. To reiterate, the slope (m) of our parallel line is also 3. We’re halfway there! Now that we have the slope, we only need to incorporate the point that the parallel line passes through to complete our equation in point-slope form. So, let’s move on to the next step where we’ll put all the pieces together.
Step 3: Putting it All Together Using Point-Slope Form
We’re on the home stretch now! We know the slope of our parallel line is 3, and we also know that it passes through the point (-1, -1). This is exactly the information we need to use the point-slope form of a linear equation. As a quick refresher, the point-slope form is: y - y1 = m(x - x1) where: m is the slope of the line and (x1, y1) is a point on the line. We have all these pieces ready! We know m = 3 and (x1, y1) = (-1, -1). Let’s substitute these values into the point-slope form: y - (-1) = 3(x - (-1)). Now, let's simplify this equation step by step. Remember that subtracting a negative number is the same as adding a positive number. So, we can rewrite the equation as: y + 1 = 3(x + 1). And there you have it! This is the equation of the line that is parallel to the given line and passes through the point (-1, -1), expressed in point-slope form. This form is particularly useful because it directly shows the slope of the line (3) and a point it passes through (-1, -1). It's a clear and concise way to represent the line's characteristics.
Conclusion
So, to wrap things up, the equation of the line that is parallel to the line passing through (0, -3) and (2, 3), and also passes through the point (-1, -1), in point-slope form, is: y + 1 = 3(x + 1). We successfully navigated through the problem, step by step, to find the equation of the parallel line. We started by calculating the slope of the original line, then we used the crucial fact that parallel lines have the same slope to determine the slope of our new line. Finally, we plugged the slope and the given point into the point-slope form, simplifying to get our answer. Remember, practice is key, so keep tackling these types of problems to enhance your skills and boost your confidence in mathematics! Great job, everyone, and keep up the fantastic work! Remember, math is like a puzzle – each piece fits perfectly to reveal the solution!