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

Hey guys! Today, we're diving into a super common problem in algebra: finding the equation of a line that's parallel to a given line and passes through a specific point. It might sound tricky at first, but trust me, once you get the hang of it, it's a piece of cake. So, let's break it down step by step.

Understanding Parallel Lines and Their Slopes

Before we jump into the equation, let's make sure we're all on the same page about parallel lines. The key concept here is that parallel lines have the same slope. Think of it like this: they're going in the exact same direction, just maybe starting from different points on the graph. So, if we know the slope of one line, we automatically know the slope of any line parallel to it. This is crucial for solving this type of problem.

The slope-intercept form of a linear equation, y = mx + b, is our best friend here. The m represents the slope, and the b represents the y-intercept (where the line crosses the y-axis). When we're given an equation in this form, it's super easy to identify the slope. For instance, in the equation y = -6/5x + 10, the slope is simply -6/5. This means any line parallel to this one will also have a slope of -6/5. Remember this, it's the golden rule for parallel lines! Identifying the slope from the given equation is the first, and arguably the most important, step in finding the equation of our parallel line. Once we have the slope, we're halfway there. We can then use this slope, along with the given point, to figure out the rest of the equation. So, keep that slope in mind as we move on to the next step!

Using the Point-Slope Form

Now that we understand slopes, let's talk about how to use a point and a slope to actually create the equation of a line. This is where the point-slope form comes in super handy. The point-slope form is written as: y - y1 = m(x - x1). Don't let the letters scare you! It's actually quite simple. Here, m is still the slope, and (x1, y1) is the point that the line passes through. This form is perfect for our situation because we have both the slope (which we got from the parallel line) and a point (the one given in the problem).

Let's say we have a line with a slope of m and it passes through the point (x1, y1). To use the point-slope form, we just plug in these values. So, instead of y - y1 = m(x - x1), we'd have something like y - 2 = 3(x - 1) if our point was (1, 2) and our slope was 3. See? Not too scary! This gives us an equation that represents a line with the correct slope that also goes through the specified point. The beauty of this form is that it directly incorporates the information we have – the slope and a point – making it a straightforward way to build our equation. After plugging in the values, we usually simplify the equation to get it into slope-intercept form (y = mx + b), which is often the final form we're looking for. But the point-slope form is the crucial stepping stone that gets us there. It’s the bridge between the information we have and the equation we need.

Converting to Slope-Intercept Form

Okay, we've got our equation in point-slope form, which is a great start! But often, we want our final answer in the slope-intercept form, which, as we discussed earlier, is y = mx + b. This form is super useful because it immediately tells us the slope (m) and the y-intercept (b) of the line. So, how do we get from point-slope form to slope-intercept form? It's all about a little bit of algebraic manipulation.

The main steps involve distributing and isolating y. First, we distribute the slope (m) across the terms inside the parentheses. Then, we add the y1 value to both sides of the equation to get y by itself. This process might seem a bit abstract, so let's look at an example. Suppose we have the equation y - 2 = 3(x - 1) (which we got from the point-slope form in the previous section). First, we distribute the 3: y - 2 = 3x - 3. Then, we add 2 to both sides: y = 3x - 1. Boom! We're in slope-intercept form. We can now clearly see that the slope is 3 and the y-intercept is -1. Mastering this conversion is essential for fully understanding and utilizing linear equations. It allows us to easily read off key information about the line, like its slope and where it crosses the y-axis. So, practice this process until it feels second nature. Once you can confidently switch between point-slope and slope-intercept form, you'll be a linear equation whiz!

Solving the Problem: A Step-by-Step Approach

Alright, guys, let's put all this knowledge together and tackle the actual problem. We're given a line and a point, and we need to find the equation of a line that's parallel to the given line and passes through that point. Remember, it's all about breaking it down into manageable steps.

1. Identify the slope of the given line: This is our starting point. Look at the equation of the given line and find the coefficient of x. That's your slope! Since parallel lines have the same slope, this is also the slope of the line we're trying to find.
2. Use the point-slope form: Now that we have the slope and the given point, we can plug these values into the point-slope form equation: y - y1 = m(x - x1). This gives us an equation that represents a line with the correct slope that also passes through the given point.
3. Convert to slope-intercept form: Our final step is to get the equation into the more familiar slope-intercept form: y = mx + b. To do this, we distribute the slope and isolate y, as we discussed earlier.

By following these steps carefully, you can confidently solve any problem of this type. Each step builds upon the previous one, leading you to the final answer. And remember, practice makes perfect! The more you work through these problems, the more comfortable and confident you'll become. So, let's apply this process to the specific example you provided and see how it works in action.

Applying the Steps to the Example

Okay, let's get down to the nitty-gritty and apply our step-by-step approach to the problem at hand. We need to find the equation of a line that's parallel to a given line and passes through the point (12, -2). To do this effectively, we'll work through each step meticulously, ensuring we understand the logic behind each action. Let's consider the given line equation is: y = -6/5x + 10.

Step 1: Identify the slope of the given line

The first thing we need to do is figure out the slope of the given line. Remember, the slope is the coefficient of x in the slope-intercept form (y = mx + b). Looking at the equation y = -6/5x + 10, we can clearly see that the slope (m) is -6/5. This is a crucial piece of information because it tells us that any line parallel to this one will also have a slope of -6/5. We've essentially unlocked the direction of our new line!

Step 2: Use the point-slope form

Now that we have the slope (-6/5) and the point (12, -2), we can use the point-slope form to create an equation for our parallel line. The point-slope form is y - y1 = m(x - x1). We plug in our values: m = -6/5, x1 = 12, and y1 = -2. This gives us: y - (-2) = -6/5(x - 12). Notice how we're carefully substituting each value into its correct place. It's really important to pay attention to signs here, especially when dealing with negative numbers. Simplifying the equation a bit, we get: y + 2 = -6/5(x - 12). This is the equation of our line in point-slope form. It represents a line with a slope of -6/5 that passes through the point (12, -2). We're well on our way to finding the final equation!

Step 3: Convert to slope-intercept form

We've got our equation in point-slope form, but to make it truly shine, we need to convert it to slope-intercept form (y = mx + b). This will allow us to easily see the slope and y-intercept of our line. To do this, we'll distribute the -6/5 and then isolate y. Starting with our equation from Step 2: y + 2 = -6/5(x - 12), we first distribute the -6/5: y + 2 = -6/5x + 72/5. Remember, we're multiplying -6/5 by both x and -12. A common mistake is to forget to distribute to both terms, so be careful! Next, we want to get y by itself, so we subtract 2 from both sides: y = -6/5x + 72/5 - 2. To combine the constant terms, we need a common denominator. We can rewrite 2 as 10/5: y = -6/5x + 72/5 - 10/5. Finally, we combine the fractions: y = -6/5x + 62/5. And there you have it! This is the equation of the line in slope-intercept form. We can clearly see that the slope is -6/5 (which we already knew it had to be) and the y-intercept is 62/5. We've successfully found the equation of the line parallel to the given line and passing through the specified point!

Conclusion

So, there you have it, guys! We've walked through the process of finding the equation of a line parallel to a given line and passing through a specific point. Remember the key concepts: parallel lines have the same slope, the point-slope form is your friend, and converting to slope-intercept form makes your answer clear and easy to understand. Practice these steps, and you'll be solving these problems like a pro in no time! Keep up the awesome work, and don't hesitate to ask questions if you get stuck. You've got this!