Find The Equation Of A Line Parallel To Y=2x+5 Passing Through (-4,3)

Hey guys! In this article, we're going to dive into a common problem in algebra: finding the equation of a line that passes through a specific point and is parallel to another given line. This might sound intimidating, but don't worry, we'll break it down step by step so it's super easy to understand.

Understanding Parallel Lines

Before we jump into the problem, let's quickly recap what it means for lines to be parallel. Parallel lines are lines that run in the same direction and never intersect. Think of train tracks – they run alongside each other without ever meeting. The crucial thing about parallel lines is that they have the same slope. This is the key concept we'll use to solve our problem. When trying to find the equation of a line, it's important to consider that parallel lines share the same slope, making the process straightforward once you identify the slope of the given line. This shared slope is what ensures they never intersect, maintaining their parallel relationship across the coordinate plane. Remember, the slope represents the steepness and direction of a line, so identical slopes mean the lines have the exact same inclination. Understanding this basic principle simplifies finding equations for parallel lines significantly. Keep this concept in mind as we proceed through the steps to solve our specific problem, and you'll see how easily we can apply this rule to find the equation of our desired line. Parallel lines and their slopes are fundamental concepts in coordinate geometry, and mastering them will greatly enhance your problem-solving skills in various mathematical contexts. By grasping this core idea, you'll be well-equipped to tackle similar problems with confidence and accuracy.

Problem Statement: Finding the Equation

Okay, let's get to the problem at hand. We need to find the equation of a line that passes through the point $(-4, 3)$ and is parallel to the line $y = 2x + 5$. This is a classic example of a problem that combines the concepts of slope, points, and parallel lines. To tackle this, we'll use the fact that parallel lines have the same slope, and we'll leverage the point-slope form of a linear equation. When you're asked to find the equation of a line parallel to another, always start by identifying the slope of the given line. In this case, the equation $y = 2x + 5$ is in slope-intercept form, which makes it easy to see the slope. Once we have the slope, we can use the coordinates of the point $(-4, 3)$ to determine the unique equation of the parallel line that passes through it. This approach simplifies what might initially seem like a complex task into a series of manageable steps. Remember, the key to solving these types of problems lies in understanding the relationships between different forms of linear equations and how they can be manipulated to suit our needs. By breaking down the problem into smaller, more digestible parts, we can systematically arrive at the solution. Now, let's move on to extracting the necessary information from the given line's equation and applying it to our problem. This structured approach will not only help us solve this specific problem but also equip you with a solid strategy for similar challenges in the future.

Step 1: Identify the Slope of the Given Line

The line $y = 2x + 5$ is in slope-intercept form, which is $y = mx + b$, where $m$ represents the slope and $b$ is the y-intercept. So, what's the slope of our given line? That's right, it's 2! Remember, the slope is the coefficient of $x$ when the equation is in slope-intercept form. Identifying the slope is the first crucial step when you find the equation of a line parallel to another because parallel lines share the same slope. This initial step allows us to quickly determine one of the key parameters needed for our new equation. Recognizing the slope-intercept form and being able to extract the slope directly from it is a fundamental skill in algebra. It lays the groundwork for more complex problem-solving scenarios involving linear equations. By mastering this basic concept, you can efficiently tackle a wide range of problems related to parallel and perpendicular lines, as well as various other applications of linear functions. Now that we've identified the slope of our given line, we're well on our way to finding the equation of the parallel line that passes through the specified point. This systematic approach makes the entire process much more manageable and ensures we don't miss any critical information.

Step 2: Use the Same Slope for the Parallel Line

Since we're looking for a line parallel to $y = 2x + 5$, our new line will also have a slope of 2. Remember, parallel lines have the same slope. This is a fundamental principle in coordinate geometry, and it's what makes solving this type of problem possible. When you find the equation of a line that's parallel to another, the shared slope is your starting point. This common characteristic simplifies the process significantly, as it narrows down the possibilities for the new equation. Understanding this relationship between parallel lines and their slopes is essential for solving a wide range of mathematical problems. It's a concept that crops up frequently in algebra and geometry, so mastering it is a valuable investment in your mathematical toolkit. Now that we've confirmed that our parallel line will also have a slope of 2, we're ready to move on to the next step: incorporating the point that the line must pass through. This is where the point-slope form of a linear equation comes into play, allowing us to precisely define the line we're looking for.

Step 3: Use the Point-Slope Form

The point-slope form of a linear equation is given by $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line. We know the slope ($m = 2$) and a point on the line $(-4, 3)$. Let's plug these values into the point-slope form: $y - 3 = 2(x - (-4))$. This equation is a powerful tool when you find the equation of a line because it directly incorporates both the slope and a specific point the line passes through. The point-slope form is particularly useful when you have limited information but need to define a line uniquely. It provides a straightforward way to relate the slope, a point on the line, and the variables $x$ and $y$ that represent all the points on the line. Understanding and being comfortable using the point-slope form is a key skill in algebra. It's a versatile equation that can be applied in various scenarios, making it an indispensable tool in your mathematical arsenal. Now that we've plugged in our values, we have the equation in point-slope form. The next step is to simplify this equation to get it into a more standard form, like slope-intercept form.

Step 4: Simplify to Slope-Intercept Form (Optional)

Let's simplify the equation we got in the last step: $y - 3 = 2(x + 4)$. First, distribute the 2: $y - 3 = 2x + 8$. Next, add 3 to both sides to isolate $y$: $y = 2x + 11$. This is the equation of our line in slope-intercept form! When you find the equation of a line, presenting it in slope-intercept form is often the most intuitive and easily understood way to do so. The slope-intercept form, $y = mx + b$, clearly shows the slope ($m$) and the y-intercept ($b$), making it easy to visualize the line and understand its characteristics. Simplifying to slope-intercept form is not always strictly necessary, but it's a common practice that can make your answer more accessible and easier to work with. It's also a good exercise in algebraic manipulation, reinforcing your skills in solving equations. Now that we have our equation in slope-intercept form, we've successfully found the equation of the line that passes through the point $(-4, 3)$ and is parallel to the line $y = 2x + 5$.

Final Answer

The equation of the line that passes through $(-4, 3)$ and is parallel to the line $y = 2x + 5$ is $y = 2x + 11$. Woohoo! We did it! Remember, the key steps are identifying the slope of the given line, using that same slope for the parallel line, plugging the slope and point into the point-slope form, and then simplifying (if needed). This systematic approach will help you confidently find the equation of a line in similar problems. By breaking down the problem into manageable steps and understanding the underlying principles, you can tackle even seemingly complex mathematical challenges with ease. Remember, practice makes perfect, so try applying this method to other problems to solidify your understanding. Keep up the great work, and you'll be a master of linear equations in no time! Understanding how to find the equation of a parallel line is a valuable skill in algebra and geometry, and it opens the door to more advanced concepts in mathematics. So, keep exploring, keep learning, and keep challenging yourself to grow your mathematical abilities.