Solving Winchester's Price Puzzle: Silver Bullets & Holy Water

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Hey guys, ready to crack a case? Forget demons and monsters for a second; we're diving into a math problem worthy of the Winchesters themselves. This isn't about hunting down the supernatural; it's about figuring out the cost of their gear. Specifically, we're going to solve a classic word problem about Dean and Sam's shopping habits, focusing on silver bullets and holy water. Sounds fun, right? Let's get started!

Setting the Scene: The Winchester's Shopping Spree

So, here’s the deal, straight from the pages of our mystery: Dean Winchester went shopping and snagged 4 silver bullets and 2 vials of holy water, shelling out a cool $52.00. Then, his brother, Sam, decided to get in on the action, grabbing 2 silver bullets and 4 vials of holy water for $48.00. The burning question: What’s the price of each item? This kind of puzzle is a perfect example of a system of linear equations, a common topic in algebra. We're going to use this mathematical approach to unravel this mystery. It's like a little puzzle where you have to find out the individual values of the items. Think of each item as a variable, and the total cost as the result of a calculation. By using the information we know, like the number of items each person bought and how much they spent, we can solve for those variables. We can do this using substitution, elimination, or even graphing if you want to get fancy with it. The main goal is to use the given information to create equations. These equations will use two variables. For example, we'll use 'x' for the price of silver bullets and 'y' for the price of holy water. By combining the equations, we can find the individual prices. It's all about turning a word problem into mathematical expressions, and the Winchesters would definitely approve.

We'll treat silver bullets and holy water as our unknowns. This means we will use algebra to solve this problem. So, let’s get those brain cells fired up and work through it together. We'll break down the problem and show you exactly how to solve it step-by-step.

Translate the problem into algebraic equations:

Let's denote the price of a silver bullet as 'x' and the price of a vial of holy water as 'y'.

  • Dean's purchase: 4x + 2y = 52
  • Sam's purchase: 2x + 4y = 48

Now, we have two equations with two variables. We can use either substitution or elimination to solve this system of equations. Let's go with elimination, as it often simplifies the process.

Decoding the Equations: A Step-by-Step Solution

Alright, buckle up, because we're about to get into the nitty-gritty of solving this. First, we'll take a look at what we know and figure out the right way to go about finding the solutions. The goal is to make the equations simpler to solve. This involves manipulating the equations to isolate one variable. Here’s how we'll break it down, step-by-step. We’re going to use the method of elimination, which involves getting rid of one variable so we can solve for the other. It's like magic, but with numbers! Our goal here is to isolate one of the variables, and then use that to solve for the other.

Elimination Method

  1. Multiply Equations: To eliminate one variable, we need to make the coefficients of either 'x' or 'y' the same (but with opposite signs) in both equations. Let’s multiply the first equation (Dean's purchase) by -1. It does not matter which equation you manipulate, the goal is to reduce the complexity to isolate a variable.

    • -1 * (4x + 2y = 52) => -4x - 2y = -52
    • The second equation (Sam's purchase) remains as is: 2x + 4y = 48

  2. Combine Equations: Now, add the modified first equation to the second equation:

    • (-4x - 2y = -52) + (2x + 4y = 48)
    • This simplifies to: -2x + 2y = -4

  3. Eliminate One Variable (Again!): Let's multiply the first equation by 2 to eliminate x.

    • 2 * (2x + 4y = 48) => 4x + 8y = 96

    • 2 * (4x + 2y = 52) => 8x + 4y = 104

  4. Solve for a Variable: Now that we've modified the equations, let's use elimination one more time!

    • (8x + 4y = 104) - (4x + 8y = 96)
    • 4x - 4y = 8. Therefore, x - y = 2

  5. Solve for the remaining variable: Let's add this new equation to equation -2x + 2y = -4

    • (x - y = 2) + (-2x + 2y = -4)

    • -x + y = -2

    • Then, we can see that the result will be the price of y = 0, so let's substitute 0 for y in any of the equations. Let's pick x - y = 2.

      • x - 0 = 2, so x = 2.

Now, we've got our answers! Let's interpret what all these numbers mean in terms of our story.

The Verdict: Unraveling the Prices

So, after all that mathematical sleuthing, what do we have? The price of each silver bullet is $2.00, and each vial of holy water costs $0.00. Yep, you read that right! It appears the holy water was a freebie or, perhaps, wasn't factored into the initial cost by the clerk. Considering the usual costs of supplies, we can use this to guess the cost of Dean's supply run. Dean, with his purchase of 4 silver bullets, would have spent 4 * $2.00 = $8.00 on the bullets. He also purchased 2 vials of holy water, but, as we saw, this was a freebie. Sam, on the other hand, bought 2 silver bullets, which at $2.00 each, came out to $4.00. He purchased 4 vials of holy water, which was also a freebie.

Final Answer

  • Silver Bullet: $2.00
  • Holy Water: $0.00

This case, solved! It’s kind of cool, isn't it? That using a bit of math, we can figure out these prices, and now you have the skills to solve many other problems. Think about how you could use this in your own life. Maybe you want to compare prices at the store, or understand how interest works. The more you practice and use it, the better you'll become. Now go forth and solve some problems! Just like our favorite monster-hunting brothers, you're ready to face any challenge.

Checking our work:

To make sure our answer is correct, we can plug the prices back into the original equations. Remember the original equations:

  • Dean's purchase: 4x + 2y = 52
  • Sam's purchase: 2x + 4y = 48

With x = $2.00 and y = $0.00

  • Dean's purchase: (4 * $2.00) + (2 * $0.00) = $8.00. This doesn't match the original equation, meaning the holy water was not free. This also means, the equation is incorrect.

Correcting the equation:

There seems to be an error in our equations, as the purchase totals would not make sense with holy water being free. This highlights an important lesson in problem-solving: always double-check your work and the assumptions you make. Let's go back and rework the equation and properly resolve the issue.

Corrected Equations:

Let's re-evaluate our original equations, as the holy water was not free.

  • Dean's purchase: 4x + 2y = 52
  • Sam's purchase: 2x + 4y = 48

Elimination Method

  1. Multiply Equations: To eliminate one variable, we need to make the coefficients of either 'x' or 'y' the same (but with opposite signs) in both equations. Let’s multiply the first equation (Dean's purchase) by -1. It does not matter which equation you manipulate, the goal is to reduce the complexity to isolate a variable.

    • -1 * (4x + 2y = 52) => -4x - 2y = -52
    • The second equation (Sam's purchase) remains as is: 2x + 4y = 48

  2. Combine Equations: Now, add the modified first equation to twice the second equation:

    • (-4x - 2y = -52) + 2(2x + 4y = 48)
    • (-4x - 2y = -52) + (4x + 8y = 96)
    • This simplifies to: 6y = 44
    • y = 44/6 or $7.33

  3. Eliminate One Variable (Again!): Let's multiply the first equation by 2 to eliminate x.

    • 2 * (2x + 4y = 48) => 4x + 8y = 96
    • 2 * (4x + 2y = 52) => 8x + 4y = 104

  4. Solve for a Variable: Now that we've modified the equations, let's use elimination one more time!

    • (8x + 4y = 104) - (4x + 8y = 96)

    • 4x - 4y = 8

    • Solve for x: x = (8 + 4y) / 4

    • x = (8 + 4(7.33))/4

    • x = 10

Final Answer

  • Silver Bullet: $10.00
  • Holy Water: $7.33

Conclusion: The Cost of Protection

Well, guys, we've successfully solved the mystery of the Winchester's shopping spree. It turns out silver bullets are the pricier item, costing $10 each, while holy water is a bit more affordable at $7.33 per vial. This exercise demonstrates how a system of equations can be used to determine the individual prices of items, even when only the total costs are known. It's a practical application of algebra that's as useful in real life as it is in solving a hypothetical case for the Winchesters. The cool thing about solving these problems is that it helps train your mind to think logically and systematically. These skills come in handy whether you're dealing with a math problem or life’s challenges. So, the next time you encounter a similar puzzle, remember the Winchesters, grab your pen and paper, and get ready to solve! You got this!