Parallel Line Equation: Step-by-Step Solution

by Rajiv Sharma 46 views

Hey guys! Let's dive into a super interesting math problem today that involves finding the equation of a line. This isn't just any line, though! We need to find a line that passes through a specific point and runs parallel to another given line. Sounds like a fun challenge, right? Let's break it down step by step so you can ace these types of problems.

Understanding Parallel Lines and Their Equations

First things first, what does it even mean for lines to be parallel? Well, in simple terms, parallel lines are like train tracks – they run side by side and never intersect. In the world of equations, this translates to a very important characteristic: parallel lines have the same slope. The slope, often denoted as 'm' in the slope-intercept form of a linear equation (y = mx + b), tells us how steep the line is. So, if two lines are parallel, their 'm' values are identical. This is crucial for solving our problem.

Now, let's quickly recap the slope-intercept form: y = mx + b. Here,

  • 'y' and 'x' are the coordinates of any point on the line,
  • 'm' is the slope (the rate at which the line rises or falls),
  • and 'b' is the y-intercept (the point where the line crosses the y-axis).

Knowing this form is super handy because it gives us a clear way to express the equation of a line once we know its slope and y-intercept. In our problem, we're given a line, and we need to find a line parallel to it. This means we can immediately grab the slope from the given line's equation and use it for our new line. The real trick is figuring out the y-intercept for our new line, which we'll do using the given point. Stay tuned, we're getting there!

Let's not forget the point-slope form either: y - y₁ = m(x - x₁). This form is incredibly useful when you have a point (x₁, y₁) and the slope 'm'. You can directly plug these values into the equation and then rearrange it into the slope-intercept form if needed. This is another powerful tool in our arsenal for tackling this problem.

In summary, remember that parallel lines share the same slope, and both the slope-intercept and point-slope forms are your best friends when dealing with linear equations. We'll be using these concepts extensively as we solve our problem, so make sure you've got them down! Next up, we'll dissect the specific problem we're trying to solve.

Deconstructing the Problem: Finding the Right Equation

Okay, let's get our hands dirty with the actual problem! We're tasked with finding the equation of a line that satisfies two key conditions:

  1. It must pass through the point (-3, 2).
  2. It must be parallel to the line y = -2x - 1.

Breaking this down, the first condition gives us a specific location that our line needs to hit. Think of it like a target – our line has to go through this point. The second condition tells us the direction our line should take. Since it's parallel to y = -2x - 1, we know it has to have the same slope as this line. Remember, parallel lines are like twins; they have the same steepness!

Looking at the given line, y = -2x - 1, we can easily identify its slope. Comparing it to the slope-intercept form (y = mx + b), we see that the coefficient of x is -2. This means the slope (m) of the given line is -2. And guess what? This is also the slope of our target line because they're parallel! This is a huge step forward.

So, now we know the slope of our line (m = -2) and a point it passes through (-3, 2). We have all the ingredients we need to cook up the equation of our line. We could use either the slope-intercept form (y = mx + b) or the point-slope form (y - y₁ = m(x - x₁)). Both will lead us to the correct answer, but let's think strategically. Since we have a point and a slope, the point-slope form might be a tad more convenient. It's almost tailor-made for this situation!

Before we jump into the calculations, let's just recap our strategy. We identified the slope from the given parallel line, and we have a point. Now, we'll plug these values into the point-slope form, simplify the equation, and see if it matches any of the answer choices. Easy peasy, right? Let's move on to the solution now!

Step-by-Step Solution: Cracking the Code

Alright, let's put our math skills to the test and find the equation of the line. We've already established that the slope of our line (m) is -2, and it passes through the point (-3, 2). Let's use the point-slope form, which is y - y₁ = m(x - x₁).

Plugging in our values, where (x₁, y₁) is (-3, 2), we get:

y - 2 = -2(x - (-3))

Notice the double negative there! We need to be careful with those. Let's simplify this step by step:

y - 2 = -2(x + 3)

Now, we distribute the -2 on the right side:

y - 2 = -2x - 6

To get the equation into the slope-intercept form (y = mx + b), we need to isolate 'y'. We can do this by adding 2 to both sides:

y = -2x - 6 + 2

Simplifying further:

y = -2x - 4

Boom! We've got our equation. It's y = -2x - 4. Now, let's check if this matches any of the answer choices given in the problem. Looking back at the options, we see that answer choice D, y = -2x - 4, is exactly what we found!

So, we've successfully found the equation of the line that passes through the point (-3, 2) and is parallel to y = -2x - 1. We did it by identifying the slope from the given line, using the point-slope form, and simplifying the equation. Remember, the key here is understanding that parallel lines have the same slope and knowing how to use the point-slope form. With these tools, you can tackle any similar problem with confidence. Up next, we'll recap the entire process and highlight the key takeaways.

Recap and Key Takeaways: Mastering Parallel Lines

Phew! We've successfully navigated through the problem and found the equation of the line. Let's quickly recap the steps we took and highlight the crucial concepts. This will help solidify your understanding and make you a pro at solving these types of problems.

  1. Understand Parallel Lines: The cornerstone of this problem is the understanding that parallel lines have the same slope. This allowed us to directly use the slope from the given line (y = -2x - 1) for our target line.
  2. Identify the Slope: We extracted the slope (m = -2) from the given equation by recognizing the slope-intercept form (y = mx + b). The coefficient of 'x' is the slope, and in this case, it was -2.
  3. Use the Point-Slope Form: Since we had a point (-3, 2) and the slope (-2), the point-slope form (y - y₁ = m(x - x₁)) was our best friend. We plugged in the values and got y - 2 = -2(x + 3).
  4. Simplify the Equation: We simplified the equation step by step. First, we distributed the -2, then we added 2 to both sides to isolate 'y'. This gave us the equation y = -2x - 4.
  5. Match the Answer Choice: Finally, we compared our equation with the given answer choices and found that option D (y = -2x - 4) matched perfectly.

Key Takeaways:

  • Parallel lines have the same slope.
  • The slope-intercept form (y = mx + b) helps you easily identify the slope and y-intercept of a line.
  • The point-slope form (y - y₁ = m(x - x₁)) is super useful when you have a point and a slope.
  • Careful simplification is crucial to avoid errors.

By mastering these concepts and practicing similar problems, you'll become a pro at finding equations of parallel lines. Remember, math is like building blocks – each concept builds upon the previous one. So, make sure you have a solid foundation, and you'll be able to tackle even the trickiest problems. Keep practicing, and you'll get there! Good job, everyone!

Remember guys, practice makes perfect. So, try solving similar problems and you'll become a pro in no time! Keep up the great work!