Subway Capacity: $6x + 2y = 120$

Hey guys! Let's dive into the world of subway capacity and figure out what this equation, $6x + 2y = 120$, is all about. This equation is a sweet way to model how many people can fit onto a subway train, considering both folks who get a seat and those who are standing. Here, '$x$' represents the seating capacity, meaning the number of seats available, and '$y$' stands for the standing capacity, which is how many people can squeeze in while standing. The total capacity is capped at 120, so this equation basically tells us the limit of how many people can be on board. It's a linear equation, which is super common in math for modeling real-world relationships like this. You'll see these kinds of equations pop up in all sorts of places, from figuring out how much stuff you can pack in a car to how many hours you can work to meet a certain income goal. The cool thing about linear equations is that they show a constant rate of change. In this case, for every seat you add, the standing capacity might have to decrease, or vice-versa, to stay within the total limit of 120. We're going to break down this equation and see which of the given options truly represents the same relationship. Get ready to do some algebra, because we're going to rearrange this bad boy into different forms to see which one matches!

Understanding the Equation: $6x + 2y = 120$

Alright, let's get our hands dirty with the equation $6x + 2y = 120$. This equation is telling us a story about the subway's capacity. Think of it like this: every seat '$x$' takes up 'space' in the overall capacity, and every standing spot '$y$' also takes up 'space'. The numbers 6 and 2 are like coefficients, telling us how much 'weight' each type of capacity has towards the total limit of 120. So, for example, maybe a seated passenger takes up more 'capacity value' than a standing passenger, hence the 6 for seats and 2 for standing. Our main goal here is to see which of the answer choices, when rearranged, looks exactly like this relationship. Often, these kinds of problems want you to express one variable in terms of the other. The most common way to do this is to isolate '$y$', which is called writing the equation in slope-intercept form ($y = mx + b$). This form is super useful because it directly shows you the y-intercept and the slope of the line, which can tell you a lot about the relationship between '$x$' and '$y$'. We'll be doing just that – isolating '$y$' – to compare it with the given options. It's like solving a puzzle where each piece needs to fit perfectly. The total capacity of 120 is the maximum limit, and the equation defines all the possible combinations of seating and standing spots that can reach this limit. It's a fundamental concept in understanding constraints and trade-offs in resource allocation, whether it's subway seats or anything else!

Solving for '$y$' to Find the Equivalent Relationship

So, how do we find the equation that represents the same relationship? The trick is to manipulate the original equation, $6x + 2y = 120$, until it looks like one of the answer choices. Most of the answer choices have '$y$' isolated on one side. This means we need to do the same for our original equation. Let's get '$y$' by itself. First, we want to move the term with '$x$' to the other side. To do this, we subtract $6x$ from both sides of the equation:

$6x + 2y - 6x = 120 - 6x$

This simplifies to:

$2y = 120 - 6x$

Now, '$y$' is almost alone, but it's being multiplied by 2. To get '$y$' completely by itself, we need to divide every single term on both sides by 2:

rac{2y}{2} = rac{120}{2} - rac{6x}{2}

And voilà! We get:

$y = 60 - 3x$

This is our equation with '$y$' isolated. Now, we just need to compare this result to the given options to see which one matches perfectly. Remember, the goal is to find the exact same relationship, just potentially written in a different format. It’s like finding a different way to say the same thing – the meaning stays the same!

Analyzing the Options:

Okay, guys, we've done the heavy lifting by rearranging our original subway capacity equation to $y = 60 - 3x$. Now comes the fun part: checking which of the provided options matches this result. Let's go through them one by one:

• A. $y = 20 - 3x$: Does this look like our equation? Nope. While the '-3x' part is the same, the constant term is 20, not 60. So, this one is out.
• B. y = 60 - rac{1}{3}x: This one has the correct constant term (60), but the coefficient of '$x$' is -rac{1}{3}, not $-3$. So, this isn't it either.
• C. $y = 60 - 3x$: Bingo! Look at this one. The constant term is 60, and the coefficient of '$x$' is -3. This exactly matches the equation we derived from the original subway capacity model. This is our winner, folks!
• D. $y = 20 - Discussion category : mathematics: This option seems incomplete. It looks like the text got cut off, and even if it were complete, it doesn't match our derived equation. We can definitively rule this one out.

So, after comparing our rearranged equation, $y = 60 - 3x$, with the given choices, option C is the only one that accurately represents the same relationship as the initial subway capacity model. It's all about understanding how to manipulate equations to see different perspectives of the same underlying relationship. Pretty neat, right?

Why This Matters: Real-World Math Applications

Understanding how to rearrange equations like $6x + 2y = 120$ and finding equivalent forms, such as $y = 60 - 3x$, isn't just about passing a math test. This skill is super valuable in the real world, especially when you're dealing with constraints and trade-offs. Think about planning a budget, optimizing cargo space, or even scheduling tasks. For our subway example, the equation $y = 60 - 3x$ tells us something really interesting: for every additional seat ($x$) you decide to have, you have to give up three standing spots ($y$) to stay within the total capacity limit of 120. This is because the slope is -3. This kind of information is crucial for transit authorities when they're designing new train cars or planning routes. They need to balance comfort (seating) with maximum passenger numbers (standing). Knowing these relationships helps them make informed decisions that affect thousands of commuters every day. The '-3' slope indicates a direct trade-off: more seats mean significantly less standing room. If the slope had been, say, -0.5, it would mean each extra seat only cost half a standing spot, a much smaller sacrifice. So, the numbers in these linear equations aren't just abstract figures; they represent concrete consequences and choices. It’s a powerful demonstration of how mathematics provides a framework for solving practical problems and understanding the dynamics of systems around us. Keep an eye out for these kinds of relationships in your everyday life – math is everywhere!