Equation Of A Line Parallel To Y=2x+1: Step-by-Step Solution

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Hey guys! Let's dive into a common problem in algebra: finding the equation of a line that's parallel to another line and passes through a specific point. In this case, we want to find the equation of a line that passes through the point (2,6) and is parallel to the line y = 2x + 1. We'll express our answer in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. This is a fundamental concept in coordinate geometry, and understanding it will help you tackle various problems involving linear equations. We'll break down each step, so it's super easy to follow. So, let’s get started and make sure you grasp this concept thoroughly! Remember, practice makes perfect, so feel free to try out similar problems once we're done. Let's make math fun and conquer this together!

Understanding Parallel Lines and Slopes

Before we jump into the calculations, let's quickly recap what it means for lines to be parallel. Parallel lines, guys, are lines that never intersect. They run side by side, maintaining the same distance from each other. The key characteristic of parallel lines is that they have the same slope. The slope, often denoted by m, tells us how steep the line is. It's the ratio of the change in y to the change in x (rise over run). In the slope-intercept form (y = mx + b), m is the coefficient of x. So, in our given line y = 2x + 1, the slope is 2. This means any line parallel to it will also have a slope of 2. Understanding this concept is crucial, as it forms the foundation for solving the problem. It's like knowing the secret ingredient in a recipe – without it, the dish won't turn out right! So, remember, parallel lines share the same slope. This simple fact is what makes solving these types of problems straightforward. We'll use this knowledge to find the equation of our new line, ensuring it runs perfectly parallel to the given one.

Identifying the Slope

The main thing in finding the equation of a line parallel to y = 2x + 1 is recognizing that they share the same slope. As we discussed earlier, the slope-intercept form of a line is y = mx + b, where m represents the slope. Looking at the given equation, y = 2x + 1, we can easily identify the slope. The coefficient of x is 2, so the slope of this line is 2. Since parallel lines have the same slope, the line we're trying to find also has a slope of 2. This is a crucial piece of information! We now know that the m in our new equation will be 2, giving us a starting point of y = 2x + b. All that's left is to find the y-intercept (b). So, we've conquered the first hurdle by understanding parallel lines and identifying the slope. This sets us up perfectly for the next step, where we'll use the given point (2,6) to find the y-intercept. Think of it like building a house – we've laid the foundation, and now we're ready to start constructing the walls!

Using the Point-Slope Form

Now that we know the slope (m = 2) and a point (2,6) that the line passes through, we can use the point-slope form of a linear equation to find the equation of our line. The point-slope form is given by: y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope. In our case, (x₁, y₁) is (2,6) and m is 2. Let's plug these values into the point-slope form: y - 6 = 2(x - 2). This equation represents the line we're looking for, but it's not yet in slope-intercept form. We need to simplify it and rearrange it to look like y = mx + b. The point-slope form is incredibly useful because it allows us to construct the equation of a line using just a single point and the slope. It's like having a map and a compass – you know where you are and which direction to go. By using this form, we've effectively captured all the necessary information to define our line. Now, the next step is to transform this equation into the more familiar slope-intercept form, which will give us the complete picture of our line's equation. So, let's move on to simplifying and rearranging!

Converting to Slope-Intercept Form

We've got our equation in point-slope form: y - 6 = 2(x - 2). Now, let's convert it to slope-intercept form (y = mx + b) to make it easier to read and understand. First, we need to distribute the 2 on the right side of the equation: y - 6 = 2x - 4. Next, we want to isolate y on the left side. To do this, we'll add 6 to both sides of the equation: y - 6 + 6 = 2x - 4 + 6. This simplifies to: y = 2x + 2. Ta-da! We've successfully converted the equation to slope-intercept form. This form clearly shows us the slope (m = 2) and the y-intercept (b = 2). Converting to slope-intercept form is like translating a sentence into a language you understand fluently. It makes the information readily accessible and easy to interpret. We can now confidently say that the equation of the line that passes through the point (2,6) and is parallel to the line y = 2x + 1 is y = 2x + 2. We've solved it, guys! The beauty of mathematics lies in its step-by-step approach, and by following each step carefully, we've arrived at the correct answer.

The Final Equation

So, after all our hard work, we've arrived at the final equation! The equation of the line that passes through the point (2,6) and is parallel to the line y = 2x + 1, written in slope-intercept form, is y = 2x + 2. This equation tells us everything we need to know about our line. The slope is 2, indicating its steepness, and the y-intercept is 2, showing where the line crosses the y-axis. We found this by first recognizing that parallel lines have the same slope, then using the point-slope form to set up the equation, and finally converting it to slope-intercept form. This process is a classic example of how algebra helps us describe and understand geometric relationships. Think of it as finding the perfect recipe – we combined the right ingredients (slope and a point) and followed the steps to create the desired result (the equation of the line). This equation not only satisfies the given conditions but also provides a clear and concise representation of the line. And that’s it, guys! We’ve successfully navigated through the problem and found our solution. Now you’re equipped to tackle similar problems with confidence!