Parallel Line Equation Through A Point: A Step-by-Step Guide

by Admin 61 views
Parallel Line Equation Through a Point: A Step-by-Step Guide

Finding the equation of a line that's parallel to another line and passes through a specific point is a common problem in algebra. Let's break down how to solve it, step by step. We'll tackle this problem: find the equation of the line that is parallel to the given line 3x - 4y = -17 and passes through the point (-3, 2).

Understanding Parallel Lines

Before we dive into the solution, let's quickly recap what it means for lines to be parallel. Parallel lines have the same slope but different y-intercepts. This is a crucial concept because it tells us that the line we're trying to find will have the same slope as the given line, 3x - 4y = -17. Think of it like train tracks; they run in the same direction and never meet because they have the same slope, maintaining a constant distance from each other. Understanding this concept is vital for tackling these kinds of problems. Why is it important that they have the same slope? Imagine if the slopes were different; the lines would eventually intersect! This characteristic of parallel lines is the key that unlocks the door to solving this type of problem. By determining the slope of the given line, we automatically know the slope of the line we want to find. The only thing left to figure out is the y-intercept, which we can find by using the given point that the line passes through. This simple but powerful idea allows us to navigate through these problems with ease and confidence. Mastering this concept will not only help you solve similar problems but also deepen your understanding of linear equations and their properties. So, always remember: parallel lines, same slope, different y-intercepts. Keep this in mind, and you'll be well-equipped to handle any challenge involving parallel lines that comes your way!

Step 1: Find the Slope of the Given Line

To find the slope of the line 3x - 4y = -17, we need to rewrite it in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. Let's isolate y in the given equation:

  • 3x - 4y = -17
  • -4y = -3x - 17
  • y = (3/4)x + (17/4)

From this, we can see that the slope of the given line is 3/4. Remember, the line we are looking for will also have this slope, because it's parallel. Now, let's talk about why converting to slope-intercept form is so important. It's all about making the slope easily identifiable. When the equation is in the form y = mx + b, the coefficient of x is directly the slope. No extra calculations needed! This is incredibly useful, especially when you're under pressure during an exam. But what if you're not given the equation in this form? That's where your algebraic skills come into play. You need to manipulate the equation, isolating y on one side. This might involve adding or subtracting terms, multiplying or dividing, but the goal is always the same: to get the equation into that beautiful y = mx + b format. And once you do, you've instantly unlocked the slope! Another thing to consider is that understanding slope-intercept form isn't just about finding the slope. It also gives you the y-intercept, which can be useful for graphing the line. So, by mastering this form, you're not just solving for the slope; you're gaining a deeper understanding of the line itself. Remember to practice converting equations to slope-intercept form. The more you do it, the faster and more comfortable you'll become. This will not only help you with problems involving parallel lines but also with a wide range of other algebraic challenges. So, keep practicing, and soon you'll be a slope-intercept form master!

Step 2: Use the Point-Slope Form

Now that we know the slope of the parallel line (m = 3/4) and a point it passes through (-3, 2), we can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the given point. Plug in the values:

  • y - 2 = (3/4)(x - (-3))
  • y - 2 = (3/4)(x + 3)

This equation represents the line we're looking for, but we can simplify it further. Let's talk a little more about the point-slope form and why it's such a handy tool. The point-slope form, y - y1 = m(x - x1), is a powerful way to represent a linear equation when you know the slope (m) and a point (x1, y1) that the line passes through. It's particularly useful in situations like this, where you're given a point and need to find the equation of a line parallel (or perpendicular) to another line. The beauty of the point-slope form is that it directly incorporates the given information into the equation. You simply plug in the values for m, x1, and y1, and you have an equation that represents the line. No need to solve for the y-intercept first! It's a direct and efficient way to get to the equation you need. But where does this formula come from? It's derived from the definition of slope itself. The slope is the change in y divided by the change in x. So, if you have two points on a line, (x1, y1) and (x, y), the slope can be expressed as m = (y - y1) / (x - x1). If you multiply both sides of this equation by (x - x1), you get y - y1 = m(x - x1), which is the point-slope form. Understanding the derivation of the formula can help you remember it and appreciate its elegance. The point-slope form is not only useful for finding the equation of a line, but it can also be used to graph a line. If you have the equation in point-slope form, you know the slope and a point on the line. You can then use this information to plot the point and draw the line using the slope to find other points. Learning to use the point-slope form effectively will significantly improve your ability to solve problems involving linear equations. It's a valuable tool in your mathematical arsenal!

Step 3: Simplify to Slope-Intercept Form (Optional)

While the equation y - 2 = (3/4)(x + 3) is perfectly valid, we often want to express the equation in slope-intercept form y = mx + b or standard form Ax + By = C. Let's convert it to slope-intercept form:

  • y - 2 = (3/4)x + (9/4)
  • y = (3/4)x + (9/4) + 2
  • y = (3/4)x + (9/4) + (8/4)
  • y = (3/4)x + (17/4)

So, the equation of the line parallel to 3x - 4y = -17 and passing through (-3, 2) is y = (3/4)x + (17/4). We can also convert this to standard form by multiplying everything by 4 and rearranging:

  • 4y = 3x + 17
  • -3x + 4y = 17
  • 3x - 4y = -17

Wait a minute! Why did we end up with the same equation we started with? Well, there must have been a problem in the original question. Let's plug in the point (-3, 2) in to the original equation 3x - 4y = -17 to check if it is on the line.

  • 3(-3) - 4(2) = -17
  • -9 - 8 = -17
  • -17 = -17

Since the point satisfies the equation, that means that the parallel line to 3x - 4y = -17 that passes through the point (-3, 2) is the line itself 3x - 4y = -17.

Alternative Solution

Because of the mistake, the best solution to choose from the choices given is 3x - 4y = -17, since it would be the same line, but this is not an option. Let's see if the point is on any of the other lines.

  • 3x - 4y = -20
    • 3(-3) - 4(2) = -20
    • -9 - 8 = -20
    • -17 = -20. No
  • 4x + 3y = -2
    • 4(-3) + 3(2) = -2
    • -12 + 6 = -2
    • -6 = -2. No
  • 4x + 3y = -6
    • 4(-3) + 3(2) = -6
    • -12 + 6 = -6
    • -6 = -6. Yes

This means that our parallel line will be 4x + 3y = -6, we just need to find a line that is parallel to 3x - 4y = -17.

  • 3x - 4y = -17 converted to slope intercept form y = (3/4)x + (17/4) which means the slope is 3/4.

Now let's convert 4x + 3y = -6 to slope intercept form

  • 4x + 3y = -6
  • 3y = -4x - 6
  • y = (-4/3)x - 2 which means the slope is -4/3

Since the slopes are not the same, this is not the solution.

Key Takeaways

  • Parallel lines have the same slope. This is the foundation of solving these problems.
  • Point-slope form is your friend. It makes it easy to plug in the slope and a point to get the equation of the line.
  • Slope-intercept form is useful for visualizing the line. It clearly shows the slope and y-intercept.
  • Double-check your work. Mistakes can happen, so always verify your solution.

By following these steps, you can confidently find the equation of a line parallel to a given line and passing through a specific point. Keep practicing, and you'll master this skill in no time!

Additional Tips and Tricks

  • Practice, practice, practice! The more you work through these types of problems, the more comfortable you'll become.
  • Visualize the problem. Sketching a quick graph can help you understand the relationships between the lines and points.
  • Don't be afraid to ask for help. If you're stuck, reach out to your teacher, classmates, or online resources for assistance.

Finding the equation of a parallel line doesn't have to be daunting. With a clear understanding of the concepts and a systematic approach, you can solve these problems with confidence. Happy calculating!

Mastering the process of finding the equation of a line parallel to another and passing through a given point is a fundamental skill in algebra. By understanding the core concepts—the relationship between slopes of parallel lines, the utility of point-slope form, and the clarity of slope-intercept form—you gain the tools to confidently tackle these problems. The more you practice and apply these techniques, the more fluent you become in the language of linear equations. Remember to always double-check your work and seek help when needed. With dedication and perseverance, you can unlock the door to a deeper understanding of mathematics.