Finding The Equation Of A Line Parallel To 2x+5y=10 And Passing Through (-5,1)

Hey everyone! Today, we're diving into a classic algebra problem: finding the equation of a line that's parallel to a given line and passes through a specific point. This might sound intimidating, but don't worry, we'll break it down step by step so it's super clear. Our mission is to figure out the equation of a line that's just like the line $2x + 5y = 10$, but hanging out somewhere else on the graph because it needs to pass through the point $(-5, 1)$. We have some options to choose from, and we're going to make sure we pick the right ones. Let's get started!

Understanding Parallel Lines

So, what exactly does it mean for lines to be parallel? In the simplest terms, parallel lines are like train tracks – they run side by side and never intersect. The key characteristic of parallel lines is that they have the same slope. Remember the slope-intercept form of a line, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept? That slope, $m$, is what we need to focus on first. To kick things off, we need to figure out the slope of our given line, $2x + 5y = 10$. To do this, we need to rearrange the equation into slope-intercept form. This means we want to get $y$ all by itself on one side of the equation. Let's subtract $2x$ from both sides: $5y = -2x + 10$. Now, to get $y$ alone, we divide both sides by 5: y = -rac{2}{5}x + 2. Aha! Now we can clearly see that the slope of the given line is -rac{2}{5}. This is crucial because any line parallel to this one will also have a slope of $-\frac{2}{5}$. We've nailed the first part – understanding and finding the slope of the parallel line. Remember, this slope will be the magic number we use to find the equations of the parallel lines from the given options. Keep this slope in your mind as we move on to the next step, where we'll use this slope and the given point to pinpoint the correct equations. Stay with me, guys, we're making great progress!

Using the Point-Slope Form

Now that we know the slope of our parallel line is $-\frac{2}{5}$, and we know it passes through the point $(-5, 1)$, we can use the point-slope form of a linear equation to find its equation. The point-slope form is a super handy tool, and it looks like this: $y - y_1 = m(x - x_1)$, where $m$ is the slope, and $(x_1, y_1)$ is the point the line passes through. In our case, $m = -\frac{2}{5}$ and $(x_1, y_1) = (-5, 1)$. Let's plug those values into the point-slope form: $y - 1 = -\frac{2}{5}(x - (-5))$. Simplify that a bit, and we get: $y - 1 = -\frac{2}{5}(x + 5)$. Guess what? We've already found one of the correct answers! Option E, $y - 1 = -\frac{2}{5}(x + 5)$, matches perfectly. But we're not done yet. Often, you'll need to convert from point-slope form to slope-intercept form ($y = mx + b$) or standard form ($Ax + By = C$) to match the answer choices. So, let's keep going and see if we can find any other equations that represent the same line. This is where our algebra skills really come into play, as we'll need to manipulate the equation to see if it matches any of the other given options. Don't be intimidated by the algebra – we'll take it one step at a time and make sure we understand each move. Remember, the goal here is not just to find one answer, but to understand how different forms of the equation can represent the same line. This is a key concept in algebra, and mastering it will help you tackle all sorts of problems. Let's continue on our quest to uncover all the equations of this parallel line!

Converting to Slope-Intercept Form

Alright, let's take the equation we found in point-slope form, $y - 1 = -\frac{2}{5}(x + 5)$, and convert it to slope-intercept form ($y = mx + b$). This will help us see if any of the other answer choices match our line. To do this, we need to distribute the $-\frac{2}{5}$ on the right side of the equation: $y - 1 = -\frac{2}{5}x - \frac{2}{5} * 5$. Simplifying that, we get: $y - 1 = -\frac{2}{5}x - 2$. Now, to get $y$ by itself, we add 1 to both sides: $y = -\frac{2}{5}x - 2 + 1$. And finally, we have: $y = -\frac{2}{5}x - 1$. Boom! We've got another match! Option A, $y = -\frac{2}{5}x - 1$, is also a correct answer. See how converting between different forms of the equation helped us find another solution? This is why it's so important to be comfortable manipulating equations. We're not just solving for $x$ or $y$ here; we're uncovering different ways to represent the same relationship between them. It's like having different outfits that still make you, you. Each form of the equation gives us a slightly different perspective on the line. The point-slope form highlights a specific point and the slope, while the slope-intercept form makes the slope and y-intercept crystal clear. Now, let's move on and see if we can find even more equations hiding in the answer choices. We've tackled point-slope and slope-intercept forms, so the next logical step is to explore standard form.

Converting to Standard Form

Let's take our equation in slope-intercept form, $y = -\frac{2}{5}x - 1$, and transform it into standard form, which looks like $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ is usually positive. This might seem like a lot of rules, but don't worry, we'll follow them step by step. First, we want to get rid of the fraction. To do this, we can multiply both sides of the equation by 5: $5y = 5(-\frac{2}{5}x - 1)$. Distributing the 5 on the right side, we get: $5y = -2x - 5$. Next, we want to get the $x$ and $y$ terms on the same side of the equation. To do this, we can add $2x$ to both sides: $2x + 5y = -5$. And there we have it! We've successfully converted the equation to standard form. Now, let's check the answer choices. Option B, $2x + 5y = -5$, matches perfectly! We've found yet another equation of the same line. But hold on, let's not stop here. We still have option D to consider. Sometimes, there might be a slight variation in the standard form, like a multiple of the equation we just found. This is where our careful checking and algebraic skills come in handy. We need to compare option D, $2x + 5y = -15$, to our standard form equation, $2x + 5y = -5$, and see if they represent the same line or not. Remember, parallel lines have the same slope, but they have different y-intercepts (otherwise, they'd be the same line). So, let's analyze this last option and make sure we've truly uncovered all the possible equations.

Checking the Final Option

We've already found that options A, B, and E are correct. Now, let's carefully examine option D, $2x + 5y = -15$, to see if it represents the same line as the others. We know that the lines we're looking for are parallel to $2x + 5y = 10$ and pass through the point $(-5, 1)$. We've already established that the equation $2x + 5y = -5$ (option B) fits the bill. The easiest way to check if option D is correct is to see if the point $(-5, 1)$ satisfies the equation. Let's plug in $x = -5$ and $y = 1$ into the equation $2x + 5y = -15$: $2(-5) + 5(1) = -10 + 5 = -5$. Since $-5$ is not equal to $-15$, the point $(-5, 1)$ does not lie on the line represented by option D. Therefore, option D is incorrect. We've done it! We've systematically gone through each option, used our knowledge of parallel lines, and applied our algebraic skills to find all the correct equations. This wasn't just about finding the answers; it was about understanding the connections between different forms of linear equations and how they represent the same line. We used the slope-intercept form, the point-slope form, and the standard form, and we saw how to convert between them. This is a powerful set of tools that will help you tackle a wide range of algebra problems. So, give yourselves a pat on the back – you've conquered this challenge!

Final Answer

The correct answers are A. $y=-\frac{2}{5}x-1$, B. $2x+5y=-5$, and E. $y-1=-\frac{2}{5}(x+5)$. We found these by understanding the properties of parallel lines, using the point-slope form, and converting between different forms of linear equations. Great job, everyone!