Finding Equations Of Parallel Lines A Step-by-Step Guide

Hey guys! Let's dive into a cool math problem today: finding the equation of a line that's parallel to another line and passes through a specific point. This is a classic problem in coordinate geometry, and we're going to break it down step by step.

Understanding Parallel Lines and Their Slopes

Before we jump into the problem, let's quickly recap what it means for lines to be parallel. Parallel lines, in simple terms, are lines that run in the same direction and never intersect. The most important thing to remember about parallel lines is that they have the same slope. Slope, often denoted as m, tells us how steep a line is. It's the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. The slope-intercept form of a line, which you've probably seen before, is $y = mx + b$, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). Understanding the relationship between parallel lines and their slopes is crucial for solving this type of problem. If you've got two lines and they have the exact same slope value, you immediately know they're cruising along parallel paths. It's like they're on the same trajectory, heading in the same direction at the same angle, so they'll never bump into each other. Now, why is this so important when we're trying to figure out the equation of a line? Well, imagine we're given a line and we need to find another line that's parallel to it. The first thing we latch onto is the slope of the given line. Since parallel lines share the same slope, this gives us a crucial piece of information for our new line. We already know a key part of its equation! This knowledge becomes the foundation upon which we build the rest of our solution. Think of it as having one of the main ingredients for a recipe – we're not quite there yet, but we've got a solid start. So, keeping the slope connection in mind is going to make the whole process of finding the equation of a parallel line way smoother and more intuitive. It's the secret sauce that makes this type of problem much less daunting. The slope is the key characteristic that dictates the direction and steepness of a line. When two lines share the same slope, they are destined to be parallel, forever traveling side-by-side without ever meeting.

The Problem: A Step-by-Step Solution

Okay, so here's the problem we're tackling: Write an equation of a line that passes through the point $(2,3)$ and is parallel to the line $y=\frac{3}{2} x+5$. We have four options to choose from:

A. $y=-\frac{2}{3} x$ B. $y=-\frac{2}{3} x-7$ C. $y=\frac{3}{2} x$ D. $y=\frac{3}{2} x+7$

Let's break this down step by step.

Step 1: Identify the Slope of the Given Line

The given line is in slope-intercept form: $y=\frac{3}{2} x+5$. Remember that the slope-intercept form is $y = mx + b$, where m is the slope and b is the y-intercept. By comparing our given equation to the slope-intercept form, we can easily see that the slope of the given line is $\frac{3}{2}$. The slope of a line is the measure of its steepness and direction on a coordinate plane. It's represented by the letter 'm' in the slope-intercept form of a linear equation, which is $y = mx + b$. In this form, 'm' tells us how much the line rises (or falls) for every unit it runs horizontally. A positive slope indicates that the line goes upwards from left to right, while a negative slope means it goes downwards. The larger the absolute value of the slope, the steeper the line. The constant 'b' in the equation represents the y-intercept, which is the point where the line crosses the vertical y-axis. Understanding the slope is crucial because it dictates the line's inclination and direction. It's a fundamental concept in linear algebra and is used extensively in various fields, including physics, engineering, and economics, to model relationships between variables. Now, when we look at our specific problem, the given equation $y=\frac{3}{2} x+5$ is already in slope-intercept form. This makes our job much easier. We don't have to rearrange anything or perform any calculations. We simply need to compare the equation to the standard form $y = mx + b$ and identify the coefficient of 'x', which represents the slope. In this case, the coefficient of 'x' is $\frac{3}{2}$. This tells us that the given line has a positive slope, meaning it rises from left to right, and for every 2 units we move horizontally, the line rises 3 units vertically. This is a crucial piece of information, because as we discussed earlier, parallel lines have the same slope. So, any line parallel to this one must also have a slope of $\frac{3}{2}$. Identifying the slope is always the first step when dealing with problems involving parallel or perpendicular lines, as it provides the foundation for determining the equation of the new line.

Step 2: Determine the Slope of the Parallel Line

Since parallel lines have the same slope, the line we're looking for also has a slope of $\frac{3}{2}$. This is the key property of parallel lines that makes this problem solvable. Remember, parallel lines are like train tracks – they run side by side and never meet. This is because they have the same steepness, or slope. The slope is a number that describes how much a line tilts upwards or downwards. It's often thought of as "rise over run," which means how much the line goes up (or down) for every unit you move to the right. Now, why is this slope business so important when we're trying to find the equation of a parallel line? Well, imagine you have a line already drawn on a graph, and you need to draw another line that's parallel to it. The first thing you'd want to make sure is that the new line has the same tilt as the original. That's exactly what having the same slope means. If two lines have the same slope, they're tilting upwards or downwards at the same rate, so they'll run alongside each other without ever crossing. In the context of our problem, we're given a line with a specific slope ($\frac{3}{2}$), and we need to find another line that's parallel to it. This is where the "same slope" rule comes in handy. We know right away that the line we're looking for must also have a slope of $\frac{3}{2}$. This is a crucial piece of information, because it narrows down our options considerably. We're not just looking for any line; we're looking for a line with a very specific characteristic. Think of it like a detective solving a mystery. The fact that parallel lines share the same slope is a major clue that helps us identify the culprit – in this case, the correct equation of the line. Without this key piece of knowledge, we'd be wandering around in the dark, trying to guess the answer. So, remember, when you're dealing with parallel lines, the slope is your best friend. It's the secret ingredient that makes the puzzle come together. Knowing that the slope remains constant between parallel lines allows us to transfer this information directly to the new line we're trying to define. This simplifies the problem significantly and sets the stage for the next steps in finding the full equation.

Step 3: Use Point-Slope Form

We know the slope of the parallel line (rac{3}{2}) and a point it passes through $(2,3)$. This is perfect for using the point-slope form of a line, which is: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the given point and m is the slope. Plugging in our values, we get: $y - 3 = \frac{3}{2}(x - 2)$. The point-slope form is a fantastic tool when you have, well, a point and a slope! It allows us to write the equation of a line using minimal information. It's derived directly from the definition of slope, which is the change in y divided by the change in x. If we rearrange that definition slightly, we arrive at the point-slope form. Now, let's think about why this form is so useful in our specific situation. We've already figured out the slope of our new line – it's the same as the slope of the given line because they're parallel. We also have a specific point that our new line needs to pass through: $(2, 3)$. This is like having a target we need to hit on the coordinate plane. The point-slope form lets us take this target and the slope, and directly construct the equation of the line that satisfies both conditions. It's like a mathematical GPS, guiding us to the correct line. In our equation, $(x_1, y_1)$ represents the coordinates of the point we know our line passes through, and m is the slope we've already determined. We simply substitute these values into the formula, and voila, we have an equation that describes the line. This equation isn't quite in the familiar slope-intercept form ($y = mx + b$) yet, but it's a very short step away. The point-slope form is like the scaffolding that supports the construction of the line's equation. It provides a solid framework that we can then manipulate to get the equation into the form we prefer. Think of it as a powerful intermediate step that simplifies the process of finding the line's equation. Without the point-slope form, we might be stuck trying to guess the y-intercept or use more complicated methods. But with this handy tool, we can quickly and efficiently write the equation of the line based on the information we have.

Step 4: Convert to Slope-Intercept Form

Now, let's convert the equation from point-slope form to slope-intercept form ($y = mx + b$). To do this, we need to distribute the $\frac{3}{2}$ and then isolate y. Starting with $y - 3 = \frac{3}{2}(x - 2)$, we distribute the $\frac{3}{2}$: $y - 3 = \frac{3}{2}x - 3$. Next, we add 3 to both sides to isolate y: $y = \frac{3}{2}x$. Converting to slope-intercept form is like taking a rough draft and polishing it into a final, presentable document. The point-slope form gave us the basic equation of the line, but the slope-intercept form puts it in a standard format that's easy to read and understand. The slope-intercept form, $y = mx + b$, is a powerhouse of information. As we've discussed, m tells us the slope of the line, and b tells us the y-intercept – the point where the line crosses the y-axis. This form makes it incredibly easy to visualize the line on a graph. We can immediately see its steepness and where it intersects the y-axis. So, why do we bother converting from point-slope to slope-intercept form? Well, it's partly about convention – the slope-intercept form is a widely recognized and used standard. But it's also about clarity and ease of use. In the slope-intercept form, the key features of the line are immediately apparent. To convert, we simply need to do a little algebraic tidying up. We distribute any terms outside the parentheses, and then we isolate y on one side of the equation. This usually involves adding or subtracting constants from both sides. The goal is to get the equation into the clean and simple $y = mx + b$ format. In our case, we started with $y - 3 = \frac{3}{2}(x - 2)$. After distributing and adding 3 to both sides, we arrived at $y = \frac{3}{2}x$. This final equation tells us everything we need to know about our line. It has a slope of $\frac{3}{2}$ (the same as the original line, as expected) and a y-intercept of 0 (meaning it passes through the origin). Converting to slope-intercept form is the finishing touch that makes our equation ready for prime time. It transforms a perfectly valid but slightly clunky equation into a sleek and informative representation of the line.

Step 5: Choose the Correct Option

Our final equation is $y = \frac{3}{2}x$, which matches option C. Therefore, the correct answer is C. $y=\frac{3}{2} x$. Choosing the correct option is the final step in our problem-solving journey. We've done all the hard work – we've identified the key concepts, applied the appropriate formulas, and performed the necessary calculations. Now, we just need to compare our result to the given options and select the one that matches. This step might seem simple, but it's crucial to double-check your work and make sure you're selecting the correct answer. It's easy to make a small mistake along the way, and this final comparison can help you catch any errors. In our case, we've arrived at the equation $y = \frac{3}{2}x$. Now, we carefully examine the options provided: A. $y=-\frac{2}{3} x$, B. $y=-\frac{2}{3} x-7$, C. $y=\frac{3}{2} x$, and D. $y=\frac{3}{2} x+7$. By comparing our equation to the options, we can clearly see that option C, $y=\frac{3}{2} x$, matches our result perfectly. The other options have either the wrong slope (options A and B) or the wrong y-intercept (option D). Once we've identified the matching option, we can confidently select it as our answer. It's always a good idea to take a moment to celebrate your success! You've tackled a challenging problem and arrived at the correct solution. This final step is the culmination of your efforts, and it's satisfying to see all the pieces come together. So, remember to take a deep breath, double-check your work, and confidently choose the option that matches your solution. You've earned it!

Why This Matters: Real-World Applications

You might be thinking, "Okay, that's cool, but when will I ever use this in real life?" Well, understanding parallel lines and their equations has many practical applications. For example, architects and engineers use these concepts when designing buildings and structures. Parallel lines ensure stability and aesthetic appeal in designs. Also, in computer graphics and video game development, parallel lines are used to create realistic perspectives and 3D environments. These concepts pop up in unexpected places, making it super useful to have a grasp on them.

Practice Makes Perfect

To really nail this down, try practicing with similar problems. Change the point and the original line equation, and see if you can still find the equation of the parallel line. The more you practice, the more comfortable you'll become with these concepts.

Keep up the great work, and I'll catch you in the next math adventure!