Parallel Line Equation: Are Trish & Demetri Correct?
Let's dive into a classic geometry problem where we need to find the equation of a line parallel to a given line and passing through a specific point. Our students, Trish and Demetri, have come up with their own answers, and it's our job to figure out who's on the right track (or if they both are!). So, if you're ready to put on your geometry hats, let's get started!
Understanding the Problem: Parallel Lines and Equations
Before we jump into Trish's and Demetri's solutions, let's quickly recap some fundamental concepts about parallel lines and their equations. This will give us a solid foundation for evaluating their work. When we talk about parallel lines, we're referring to lines that run in the same direction and never intersect. A crucial property of parallel lines is that they have the same slope. This is the key idea we'll use to solve our problem.
Now, let's think about how we represent lines mathematically. One common way is using the slope-intercept form, which looks like this: y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). Another useful form is the point-slope form, which is expressed as y - y₁ = m(x - x₁). In this form, m is still the slope, and (x₁, y₁) is a specific point on the line. The point-slope form is particularly handy when we know a point on the line and its slope, which, as you'll see, is exactly the kind of information we have in our problem.
So, to recap, parallel lines have the same slope, and we can use either slope-intercept or point-slope form to represent the equation of a line. Got it? Great! Now, let's get back to Trish and Demetri.
Analyzing the Given Information
Okay, guys, let's break down the information we've got. We're given a line: y - 3 = -(x + 1). The first thing we need to do is figure out its slope. To do that, let's rewrite the equation in slope-intercept form (y = mx + b). Adding 3 to both sides, we get: y = -(x + 1) + 3. Now, distribute the negative sign: y = -x - 1 + 3. Finally, simplify: y = -x + 2. Aha! Now it's clear. The slope of the given line is -1. Remember, parallel lines have the same slope, so any line parallel to this one will also have a slope of -1.
We also know that the line we're looking for passes through the point (4, 2). This is our (x₁, y₁) point. So, we have a slope (m = -1) and a point (4, 2). This is perfect for using the point-slope form! Now, let's see what Trish and Demetri did with this information.
Trish's Solution: y - 2 = -1(x - 4)
Trish states that the equation of the parallel line is y - 2 = -1(x - 4). Let's analyze her solution. Looking at her equation, we can see that she's used the point-slope form: y - y₁ = m(x - x₁). She correctly identified the slope as -1 (the same as the given line) and plugged in the point (4, 2) as (x₁, y₁). So, her equation matches the point-slope form perfectly!
To be absolutely sure, we can rewrite Trish's equation in slope-intercept form to make it easier to compare with Demetri's answer. Let's distribute the -1: y - 2 = -x + 4. Now, add 2 to both sides: y = -x + 6. This tells us that Trish's line has a slope of -1 (which is what we want for a parallel line) and a y-intercept of 6. So far, so good for Trish!
Demetri's Solution: y = -x + 6
Demetri claims that the parallel line is represented by the equation y = -x + 6. This equation is already in slope-intercept form, which makes it easy to analyze. We can immediately see that the slope is -1, which is exactly what we need for a line parallel to the original line. Also, if we substitute the point (4, 2) into Demetri's equation, we get: 2 = -(4) + 6, which simplifies to 2 = 2. This is a true statement, meaning the point (4, 2) does indeed lie on the line represented by Demetri's equation.
So, Demetri's equation also represents a line with a slope of -1 that passes through the point (4, 2). It looks like Demetri might be on the right track too!
Are Trish and Demetri Correct? A Side-by-Side Comparison
Now comes the moment of truth! Are Trish and Demetri both correct? To answer this, let's put their solutions side-by-side:
- Trish's solution: y - 2 = -1(x - 4)
- Demetri's solution: y = -x + 6
Wait a minute… Didn't we already rewrite Trish's equation in slope-intercept form? Yes, we did! And guess what we got? y = -x + 6! So, Trish's equation y - 2 = -1(x - 4) is actually just another way of writing Demetri's equation y = -x + 6. They are equivalent equations representing the same line.
Think of it like this: Trish used the point-slope form to express the equation, while Demetri used the slope-intercept form. Both forms are perfectly valid ways to represent a line, and in this case, they both lead to the same line.
The Verdict: Both Students Are Correct!
So, the answer is… drumroll please… both Trish and Demetri are correct! They both found the equation of a line that is parallel to y - 3 = -(x + 1) and passes through the point (4, 2). Trish expressed the equation in point-slope form, while Demetri used slope-intercept form. This problem beautifully illustrates how different forms of an equation can represent the same line. Cool, right?
Key Takeaways and Why This Matters
This problem highlights several important concepts in geometry and algebra. First, it reinforces the understanding of parallel lines and their equal slopes. Second, it showcases the usefulness of both the point-slope and slope-intercept forms of a linear equation. Third, it demonstrates that equivalent equations can look different but represent the same relationship.
Why is this important? Well, understanding these concepts is crucial for a variety of reasons. In mathematics, it lays the foundation for more advanced topics like systems of equations and linear transformations. In the real world, these skills can be applied to problems involving navigation, engineering, and even computer graphics. So, mastering the basics of linear equations and parallel lines is a valuable investment in your problem-solving toolkit.
Practice Makes Perfect: Try These Problems!
Now that we've successfully navigated this problem, how about trying a few more to solidify your understanding? Here are a couple of challenges for you:
- Find the equation of a line parallel to y = 2x - 1 that passes through the point (-1, 3).
- Determine the equation of a line parallel to 3x + y = 5 that passes through the point (0, -2).
Remember, the key is to identify the slope of the given line, use that same slope for the parallel line, and then use either the point-slope or slope-intercept form to find the equation. Give it a shot, and let me know how it goes!
By working through problems like these, you'll build your confidence and skills in geometry and algebra. And who knows? Maybe you'll even discover your own cool ways to solve these kinds of problems. Keep exploring, keep questioning, and most importantly, keep having fun with math!