Solving A Linear System Hot Dogs And Chips At The Snack Stand

Hey everyone! Let's dive into a fascinating math problem that involves everyone's favorite game-day snacks: hot dogs and chips! Picture this: a bustling snack stand at a recent game, grilling up delicious hot dogs and serving crispy chips. The hot dogs are priced at a reasonable $2.50, and the chips are a steal at just $0.75. After an exciting game, the snack stand owners tallied up their sales and discovered they had completed 175 transactions, raking in a total of $262.50. Now, the big question is: how many hot dogs and bags of chips did they actually sell? This isn't just about satisfying our curiosity; it's a classic example of how we can use a system of linear equations to solve real-world problems.

To unravel this snack stand mystery, we need to translate the given information into mathematical equations. This is where the magic of algebra comes in! Let's use variables to represent the unknowns we're trying to find. We'll let 'x' represent the number of hot dogs sold and 'y' represent the number of bags of chips sold. Remember, each variable is like a placeholder, waiting for us to discover its true value. Our goal is to find the specific values of 'x' and 'y' that fit the scenario perfectly. Once we have these values, we'll know exactly how many of each item the snack stand sold. So, gear up, math enthusiasts, because we're about to embark on a journey to solve this delectable puzzle!

Setting Up the Equations: Translating Snack Sales into Math

The key to solving this problem lies in carefully translating the information we have into mathematical equations. We have two crucial pieces of information: the total number of transactions and the total revenue earned. Each of these pieces will give us an equation, forming our system of linear equations. The first piece of information tells us that the snack stand had 175 transactions in total. Since each transaction involved either a hot dog or a bag of chips (or both!), we can express this as a simple equation: x + y = 175. This equation is our first step towards solving the mystery. It tells us that the sum of hot dogs sold (x) and chips sold (y) must equal 175. But, that's just one equation with two unknowns. We need another equation to pinpoint the exact values of x and y.

Our second piece of information is the total revenue: $262.50. We know the price of each hot dog ($2.50) and each bag of chips ($0.75). So, the total revenue is the sum of the revenue from hot dogs and the revenue from chips. This gives us our second equation: 2.50x + 0.75y = 262.50. This equation represents the total money earned from selling hot dogs and chips. Notice how the coefficients (2.50 and 0.75) represent the price of each item. Now, we have a system of two linear equations:

1. x + y = 175
2. 2. 50x + 0.75y = 262.50

This system is like a mathematical treasure map, guiding us to the solution. We have two equations with two unknowns, which means we can use various methods to solve for x and y. The next step is to choose a method and start solving!

Cracking the Code: Methods to Solve the System of Equations

Alright, guys, now that we have our system of equations, it's time to choose our weapon of choice! There are several methods we can use to solve a system of linear equations, each with its own strengths and weaknesses. Let's explore two popular methods: substitution and elimination. These are the trusty tools in our mathematical toolkit. Understanding these methods is crucial for tackling a wide range of problems, not just snack stand scenarios! So, let's break them down and see which one best suits our hot dog and chips mystery.

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the problem to a single equation with a single variable, which is much easier to solve. It's like simplifying a complex puzzle piece by piece. On the other hand, the elimination method involves manipulating the equations so that the coefficients of one variable are opposites. When we add the equations together, that variable is eliminated, leaving us with a single equation with one variable. This method is like strategically removing elements to reveal the solution. Both methods are powerful, and the best choice often depends on the specific equations we're dealing with.

In our case, looking at the equations x + y = 175 and 2.50x + 0.75y = 262.50, both methods could work well. However, the substitution method might be slightly easier since we can easily solve the first equation for either x or y. But hey, let's not jump to conclusions just yet! Let's explore both methods and see which one leads us to the solution most efficiently. We're like mathematical detectives, trying to find the best approach to crack the code!

Method 1: Substitution – A Step-by-Step Guide

Let's start with the substitution method. Remember, the key here is to isolate one variable in one equation and then substitute that expression into the other equation. It's like replacing a piece in a puzzle to see how it fits. From our first equation, x + y = 175, it's super easy to solve for either x or y. Let's solve for y because, well, why not? Subtracting x from both sides, we get: y = 175 - x. Ta-da! We've isolated y. Now, this expression (175 - x) represents the value of y in terms of x. This is the magic ingredient we need for substitution.

Now, we take this expression for y and substitute it into our second equation: 2.50x + 0.75y = 262.50. Replacing y with (175 - x), we get: 2.50x + 0.75(175 - x) = 262.50. See what we did there? We've eliminated y and now we have a single equation with only x as the variable. This is a huge step forward! Now, it's just a matter of simplifying and solving for x. We'll distribute the 0.75, combine like terms, and isolate x. It's like unwrapping a present layer by layer to get to the treasure inside. Once we find x, we can plug it back into our expression for y (y = 175 - x) to find the number of bags of chips sold. We're getting closer to solving the mystery, guys! Stay tuned as we crunch the numbers and reveal the solution.

Method 2: Elimination – A Head-on Approach

Now, let's explore the elimination method, another powerful technique for solving systems of equations. This method is all about strategically manipulating the equations to eliminate one variable. It's like a mathematical magic trick where we make one variable disappear! To use elimination effectively, we need to make the coefficients of either x or y opposites in the two equations. This way, when we add the equations together, that variable will cancel out. Let's take a look at our system again:

1. x + y = 175
2. 2. 50x + 0.75y = 262.50

Looking at these equations, it seems easier to eliminate y. To do this, we need to multiply the first equation by -0.75. Why -0.75? Because that will make the y coefficient in the first equation -0.75, which is the opposite of the y coefficient in the second equation. Multiplying the entire first equation by -0.75, we get: -0.75x - 0.75y = -131.25. Remember, we need to multiply every term in the equation to keep it balanced. Now, we have a modified system:

1. -2. 75x - 0.75y = -131.25
2. 50x + 0.75y = 262.50

Now comes the fun part! We add the two equations together. Notice how the -0.75y and +0.75y cancel each other out, eliminating y from the equation. This leaves us with: 1.75x = 131.25. We've successfully eliminated y and now we have a single equation with only x. It's like cutting through the noise to focus on what's important. Now, we just need to divide both sides by 1.75 to solve for x. Once we have x, we can plug it back into either of our original equations to solve for y. We're on the home stretch, guys! The solution is within reach.

The Grand Reveal: How Many Hot Dogs and Chips Were Sold?

Alright, drumroll please! We've battled our way through the equations, and it's time to reveal the answer to our snack stand mystery. Whether you chose the substitution method or the elimination method (or perhaps even a combination of both!), the solution should be the same. Let's recap the steps and unveil the final numbers.

Using either method, we would have found that x = 75. Remember, 'x' represents the number of hot dogs sold. So, the snack stand sold a whopping 75 hot dogs! That's a lot of grilling! But we're not done yet. We still need to find 'y', the number of bags of chips sold. To do this, we can plug our value for x (75) back into either of our original equations. The simplest one to use is x + y = 175. Substituting x = 75, we get: 75 + y = 175. Subtracting 75 from both sides, we find that y = 100. So, the snack stand sold 100 bags of chips!

There you have it! The snack stand sold 75 hot dogs and 100 bags of chips. Mystery solved! This problem demonstrates how powerful systems of linear equations can be in solving real-world scenarios. From snack stands to complex scientific models, these equations help us understand and quantify the relationships between different variables. So, the next time you're at a game enjoying a hot dog and chips, remember the math behind it all!

Real-World Applications: Beyond the Snack Stand

Okay, guys, we've conquered the snack stand problem, but let's not stop there! The beauty of mathematics lies in its ability to be applied to a vast array of real-world situations. Systems of linear equations, in particular, are incredibly versatile tools. They're not just for solving snack stand mysteries; they're used in fields like economics, engineering, computer science, and even medicine. Understanding these applications can make the math we learn in the classroom feel more relevant and engaging. It's like seeing the superpowers that math gives us!

In economics, systems of equations can be used to model supply and demand, predict market trends, and analyze economic policies. For example, economists might use a system of equations to determine the equilibrium price and quantity of a product based on the supply and demand curves. In engineering, these equations are crucial for designing structures, circuits, and systems. Engineers use them to calculate forces, currents, and flows, ensuring the stability and efficiency of their designs. In computer science, systems of equations are used in areas like computer graphics, data analysis, and machine learning. For instance, they can be used to solve for the parameters of a model or to optimize an algorithm. And in medicine, systems of equations can be used to model the spread of diseases, determine drug dosages, and analyze medical data. It's amazing how one mathematical concept can have such far-reaching applications!

Thinking about these real-world applications can make learning math more exciting. It's not just about memorizing formulas and solving abstract problems; it's about developing a powerful toolset that can help us understand and solve real challenges. So, the next time you encounter a system of equations, remember that you're not just doing math; you're building a foundation for a wide range of exciting possibilities. Keep exploring, keep questioning, and keep applying your mathematical skills to the world around you!

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Solving a Linear System Hot Dogs and Chips at the Snack Stand