Hey guys! Today, we're diving into a deliciously tricky math problem involving everyone's favorite treat: yogurt! Imagine a yogurt shop selling three sizes – small, medium, and large – each with its own price tag. We've got a mystery to solve, a puzzle wrapped in creamy goodness. So, grab your thinking caps, and let's embark on this mathematical adventure together!
The Yogurt Shop Scenario
Our scene is a bustling yogurt shop, the kind that makes your mouth water the moment you step inside. They've got all the flavors, all the toppings, and of course, all the sizes. Now, here’s the setup: Small yogurts are priced at a cool $2, medium yogurts go for $3, and those large yogurts will set you back $4. During a single hour, this shop was a whirlwind of activity, selling a total of 27 yogurts and raking in a sweet $98 in sales. But wait, there's more! We also know that the shop sold five more large yogurts than small yogurts. This is where things get interesting, guys. We need to figure out how many of each size yogurt were sold. It's like being a detective, but with a mathematical twist!
Setting Up the Equations: Our Detective Tools
To crack this case, we need to translate the information into mathematical equations. Think of these as our detective tools, each one helping us uncover a piece of the puzzle. Let's use some variables to represent the unknowns:
- Let 'x' be the number of small yogurts sold.
- Let 'y' be the number of medium yogurts sold.
- Let 'z' be the number of large yogurts sold.
Now, let's build our equations based on the information we have:
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Total Yogurts Sold: We know the shop sold 27 yogurts in total. This gives us our first equation: x + y + z = 27
This equation tells us that the sum of small, medium, and large yogurts must equal 27. It’s a straightforward start, but crucial for our solution. We're essentially mapping out the total quantity of our creamy culprits, setting the stage for further investigation. Think of it as gathering the initial evidence – we know a crime (or, in this case, a sale) occurred, and we have the total count of items involved. This forms the bedrock of our mathematical model, a foundation upon which we'll build the rest of our solution strategy. Without this equation, we'd be wandering in the dark, with no clear grasp of the overall scale of yogurt transactions. So, let's file this one away as our primary lead – the sum of all yogurts equals 27. We're on our way to cracking this case, one equation at a time, guys!
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Total Sales Revenue: The total sales amounted to $98. We can express this as: 2x + 3y + 4z = 98
This equation represents the monetary aspect of our yogurt mystery. Each small yogurt contributes $2 to the total, each medium yogurt adds $3, and the large ones chip in $4. When we add up all these contributions, we should arrive at the grand total of $98. It's like tracing the money trail in a detective movie, except our money is measured in yogurt sales. This equation gives us a weighted sum, where the weights are the prices of each yogurt size. It reflects the economic dimension of the yogurt shop's business during that bustling hour. It’s a vital piece of the puzzle because it introduces the financial constraints into our mathematical model. Without it, we'd only be counting yogurts, but not accounting for their monetary value. This equation allows us to consider the impact of each yogurt size on the total revenue. So, let's add this crucial clue to our arsenal – the weighted sum of yogurt sales equals $98. We're getting closer to unraveling this creamy conundrum!
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Relationship Between Large and Small Yogurts: There were five more large yogurts than small yogurts. This can be written as: z = x + 5
This equation introduces a direct relationship between the number of large and small yogurts sold. It tells us that for every small yogurt sold, five more large yogurts were purchased. It's like discovering a key witness statement that links two seemingly disparate pieces of the puzzle. This equation is particularly valuable because it reduces the number of unknowns we need to solve for. Instead of treating 'z' as an independent variable, we can express it in terms of 'x'. This simplification will make our algebraic manipulations much easier. It's a strategic move, like connecting the dots in a detective's investigation, bringing us closer to a clearer picture. Without this equation, we'd have three independent variables, making the system much harder to solve. So, let's seize this crucial link between large and small yogurts – the number of large yogurts equals the number of small yogurts plus 5. We're honing in on the solution, guys, and the yogurt mystery is starting to crack!
The Matrix Representation: Organizing Our Clues
Now, here's where things get a little more formal. We can represent these equations in matrix form. A matrix is simply a rectangular array of numbers, and it's a powerful tool for organizing and solving systems of equations. Our system can be represented by the following matrix equation:
| 1 1 1 | | x |
| 2 3 4 | | y | = | 27 |
| -1 0 1 | | z | | 5 |
Let's break this down, guys. On the left side, we have three matrices. The first matrix contains the coefficients of our variables (x, y, and z) from our equations. The second matrix is a column matrix containing our variables. When we multiply these two matrices, we get a new column matrix. This resulting matrix is then equated to the column matrix on the right side, which contains the constants from our equations (27, 98, and 5). It might look intimidating at first, but it's just a neat way of packaging our equations. Think of it as a super-organized filing system for our clues!
Understanding the Matrix Components
- The first row
| 1 1 1 |corresponds to our first equation (x + y + z = 27). The numbers 1, 1, and 1 are the coefficients of x, y, and z, respectively. - The second row
| 2 3 4 |corresponds to our second equation (2x + 3y + 4z = 98). The numbers 2, 3, and 4 are the coefficients of x, y, and z in this equation. - The third row
| -1 0 1 |comes from rearranging our third equation (z = x + 5) to -x + z = 5. Notice the -1 for the x coefficient, 0 for the y coefficient (since y is not present in the equation), and 1 for the z coefficient. - The column matrix
| x |simply lists our variables, x, y, and z.| y || z | - The column matrix
| 27 |contains the constants on the right-hand side of our equations.| 98 || 5 |
Why Use a Matrix? Efficiency and Clarity
So, why bother with matrices? Well, guys, matrices provide a compact and efficient way to represent and solve systems of linear equations. They allow us to use powerful techniques like Gaussian elimination or matrix inversion to find the values of our variables. It's like having a super-charged calculator specifically designed for solving these types of problems. Furthermore, the matrix representation provides a clear and organized view of the system, making it easier to spot patterns and relationships. It's like having a well-structured blueprint of our problem, making it easier to navigate the solution process. In essence, matrices are our secret weapon for tackling complex mathematical puzzles. They streamline the process, enhance clarity, and pave the way for efficient solutions. So, let's embrace the power of matrices and continue our quest to decode the yogurt mystery!
Solving the System: Cracking the Code
Now comes the exciting part – actually solving for x, y, and z! There are several methods we can use, such as substitution, elimination, or even matrix inversion. For simplicity, let's use substitution. We already have z expressed in terms of x (z = x + 5), so let's substitute that into our other equations:
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Substitute z = x + 5 into the first equation (x + y + z = 27): x + y + (x + 5) = 27 2x + y = 22
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Substitute z = x + 5 into the second equation (2x + 3y + 4z = 98): 2x + 3y + 4(x + 5) = 98 6x + 3y = 78
Now we have a system of two equations with two variables:
- 2x + y = 22
- 6x + 3y = 78
We can simplify the second equation by dividing by 3:
- 2x + y = 22
- 2x + y = 26
Wait a minute, guys! We've hit a snag. Notice that the left sides of these two equations are identical, but the right sides are different. This means the system is inconsistent and has no solution. Uh oh! It seems like there might be an error in the problem statement or the information provided. This happens sometimes in the real world too – data can be inconsistent or contradictory. It's like a detective finding conflicting evidence at a crime scene. What do we do now?
Dealing with Inconsistency: A Real-World Lesson
This unexpected twist actually teaches us a valuable lesson. In real-world problems, things aren't always neat and tidy. Sometimes, the data we have is flawed, inconsistent, or incomplete. When this happens, we can't simply force a solution. We need to step back, re-examine our assumptions, and look for the source of the inconsistency. It's like a detective realizing a witness might be unreliable or that some evidence might be tampered with. In our yogurt shop scenario, the inconsistency suggests that the numbers provided (27 yogurts, $98 total, 5 more large than small) might not be compatible. Perhaps there was a mistake in recording the sales, or maybe the prices are slightly different than stated. It's a reminder that math, while precise, is often applied to messy, real-world situations. And sometimes, the most important skill isn't just solving equations, but also recognizing when something doesn't add up, guys! So, while we couldn't find a numerical solution in this case, we learned a valuable lesson about data consistency and the importance of critical thinking.
Conclusion: Math as a Tool for Understanding
So, while our yogurt shop mystery didn't have a straightforward solution this time, it highlighted the power of mathematics as a tool for understanding the world around us. We used equations and matrices to model a real-world scenario, and even when we encountered an inconsistency, we were able to use our mathematical reasoning to identify the problem. Math isn't just about getting the right answer; it's about developing a way of thinking, a way of approaching problems logically and systematically. And that's a skill that will serve you well in any situation, guys, whether you're solving a math problem or trying to figure out what flavor of yogurt to order!