Solving SPLTV: Shampoo Package Volume Calculation

by Tim Redaksi 50 views
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Hey guys! Let's dive into a fun math problem involving a shampoo factory and different package sizes. We'll be using a method called SPLTV (Sistem Persamaan Linear Tiga Variabel), which is basically a fancy way of saying we're solving a system of three linear equations. Don't worry, it's not as scary as it sounds! Think of it like a puzzle where we need to find the volume of each shampoo package (A, B, and C) based on some clues. This kind of problem is super common in algebra, and it's a great way to understand how math can be used to solve real-world scenarios. We'll break down the problem step-by-step, making it easy to follow along. So, grab a snack, and let's get started!

Understanding the Problem: Shampoo Packaging

Okay, so imagine a shampoo factory that's producing shampoo in three different package sizes: A, B, and C. The factory is mixing and matching these packages, and they've given us some information about the total volume of shampoo in different combinations of these packages. This information is key to solving the problem. The core of this problem revolves around understanding how to translate word problems into mathematical equations. The process of taking a real-world problem and converting it into a mathematical one is an essential skill in mathematics and many other fields. The key here is to recognize the relationships between the unknowns (the volume of each package) and the given information (the total volume of combined packages). We're going to set up a system of equations, and the solution to the system will give us the volume of each package. Remember, the goal is to find the volume of each package (A, B, and C). The problem gives us three different scenarios with varying combinations of packages and their respective total volumes. This will be the foundation for our equations. We'll use these three scenarios to create three equations. Each equation will represent one scenario, with variables representing the volume of each package. Let's make sure we write down all the information from the problem clearly, and then we'll translate each sentence into a mathematical equation. It is also important to pay close attention to detail. This might seem simple, but it is important to carefully read and understand the problem before attempting to solve it. This includes the quantity of each package in each combination and the total volume of those packages. Let's move on to the next section where we'll turn these descriptions into mathematical equations.

The Given Clues

Here's what we know from the problem statement:

  • Clue 1: 3 packages of A + 1 package of B + 2 packages of C = 65 ml
  • Clue 2: 2 packages of A + 1 package of B + 1 package of C = 40 ml
  • Clue 3: 1 package of A + 3 packages of B + 1 package of C = 75 ml

These clues are the key to unlocking the solution. We'll use them to create our equations. Now, let's translate these clues into mathematical equations. We'll use 'a' to represent the volume of package A, 'b' for package B, and 'c' for package C. Remember that these are the variables, and we're trying to find their values.

Setting Up the Equations: From Words to Math

Alright, time to turn those words into equations! This is where we start using algebra to model the problem. We'll represent the unknown volumes of packages A, B, and C with the variables a, b, and c, respectively. It is crucial to set up the equations correctly because any errors at this stage will affect the final result. Understanding the relationship between the quantities and the total volume is key here. By following the clues, we can create a mathematical model of the problem. Let’s translate each of the given clues into an equation: The following equations represent the volume relationship of each package. Let's use the clues to create equations. Then, we will use the process of elimination. The process of elimination means we get rid of variables one by one. Our goal is to isolate the variable, that way, we can figure out the volume of packages A, B, and C. Here are the equations, based on our earlier clues:

  • Equation 1: 3a + b + 2c = 65
  • Equation 2: 2a + b + c = 40
  • Equation 3: a + 3b + c = 75

See? It's not so bad! These equations capture the essence of the problem mathematically. Now, our goal is to solve for a, b, and c. We'll use a method called elimination to do this.

Solving the Equations: Elimination Method

Now, for the fun part! We're going to solve this system of equations using the elimination method. The elimination method involves manipulating the equations so that we can eliminate one variable at a time. The aim is to reduce the system down to a point where we can easily solve for the variables. Let’s start by eliminating 'b'. We'll manipulate Equations 1 and 2 to eliminate 'b'. We'll need to do some subtraction and get an equation with 'a' and 'c' only. The goal is to reduce the complexity by eliminating a variable. We'll then use the resulting equations to eliminate another variable, which will bring us closer to the solution. Here's how we'll do it.

Step 1: Eliminating 'b' from Equations 1 and 2

Subtract Equation 2 from Equation 1. This will eliminate 'b'.

(3a + b + 2c) - (2a + b + c) = 65 - 40 a + c = 25 (Let's call this Equation 4)

Step 2: Eliminating 'b' from Equations 2 and 3

Now, we'll manipulate Equations 2 and 3 to eliminate 'b' again. Multiply Equation 2 by 3 and subtract Equation 3.

3 * (2a + b + c) = 3 * 40 => 6a + 3b + 3c = 120 (6a + 3b + 3c) - (a + 3b + c) = 120 - 75 5a + 2c = 45 (Let's call this Equation 5)

Step 3: Eliminating 'c' from Equations 4 and 5

Now we've got two new equations (4 and 5) with only 'a' and 'c'. Let's eliminate 'c'. Multiply Equation 4 by 2, and then subtract it from Equation 5.

2 * (a + c) = 2 * 25 => 2a + 2c = 50 (5a + 2c) - (2a + 2c) = 45 - 50 3a = -5 a = -5/3

Oops! It seems there might be an error in the original problem statement or the numbers. However, let's proceed with the process to demonstrate how it works. Then we can analyze what may have gone wrong. It's a good reminder that in real-world problems, sometimes the numbers don't perfectly align. Let's work through the solution as though the values are correct. This will show us how to solve the system of equations. In many real-world scenarios, it is common to find scenarios like this. This doesn't mean the method is wrong; it just means we should double-check the values or question the initial assumptions made in the problem. Then, we can calculate and find the volume for b and c. Let’s continue to find the volume of each package and finish the equation.

Step 4: Solving for 'c'

Now that we have 'a', we can use Equation 4 (a + c = 25) to solve for 'c'.

(-5/3) + c = 25 c = 25 + 5/3 c = 80/3

Step 5: Solving for 'b'

Finally, we can plug the values of 'a' and 'c' into any of the original equations to solve for 'b'. Let's use Equation 2: 2a + b + c = 40

2*(-5/3) + b + (80/3) = 40 -10/3 + b + 80/3 = 40 b + 70/3 = 40 b = 40 - 70/3 b = 50/3

The Results: Package Volumes

Based on our calculations, the volumes are:

  • Package A: a = -5/3 ml
  • Package B: b = 50/3 ml
  • Package C: c = 80/3 ml

Important Note: The volume of package A is a negative number, which doesn't make sense in a real-world scenario. This likely indicates an error in the original problem statement, either in the values provided or the setup. In a real-world scenario, you can't have a negative volume. Let's try to verify our answer, and we will find out.

Verifying the Solution: Checking Our Work

It's always a good idea to check your answers! Let's plug our values back into the original equations to see if they hold true. Because package A is negative, we already know something is off. But, let's go ahead and verify our answers, just to be sure. It's important to do this because it helps catch any arithmetic errors or errors in our understanding of the problem. This step ensures that we have the right answer, even though we already know it may not be correct. Let's check our work using the original equations:

  • Equation 1: 3a + b + 2c = 65 3*(-5/3) + (50/3) + 2*(80/3) = -5 + 50/3 + 160/3 = (-15 + 50 + 160)/3 = 195/3 = 65. (This checks out, but it doesn't solve the negative A value)
  • Equation 2: 2a + b + c = 40 2*(-5/3) + (50/3) + (80/3) = -10/3 + 50/3 + 80/3 = (-10 + 50 + 80)/3 = 120/3 = 40. (This checks out, but again, negative A)
  • Equation 3: a + 3b + c = 75 (-5/3) + 3*(50/3) + (80/3) = -5/3 + 150/3 + 80/3 = (-5 + 150 + 80)/3 = 225/3 = 75. (This also checks out, with negative A)

Despite the negative value for package A, the equations do hold true. This still means that the initial conditions or the numbers given in the problem were not perfect. This highlights the importance of the data. However, the process of solving the problem by elimination is correct. Let's analyze what might have gone wrong.

Analyzing the Results: What Went Wrong?

So, what happened? The negative volume for package A tells us that something is amiss. Here are some potential issues:

  • Incorrect Data: There might be an error in the quantities or total volumes given in the original problem statement. This is a very common issue in textbooks. One number might be mistyped or miscalculated. Double-checking the problem statement is always a good idea.
  • Problem Setup: Though less likely, there's a chance that the way the problem was initially described (the relationships between the packages) might have been set up incorrectly. This is unlikely, however, given how the equations check out.
  • Real-World Constraints: It's important to remember that this is a mathematical model of a real-world problem. In reality, volumes can't be negative. This indicates that the mathematical model doesn't perfectly capture the real-world constraints.

Conclusion: Learning from the Process

Well, guys, we successfully navigated an SPLTV problem! Even though we encountered a result that doesn't quite make sense in the real world, we learned a lot. We practiced setting up equations, using the elimination method, and verifying our results. We also learned how to identify when something might be wrong with the problem statement. This is a very valuable skill, and it’s something every mathematician deals with. The most important thing is that we went through the process. That is the point of learning how to solve the problem and apply the SPLTV concept, and how to verify and check answers. Remember to always double-check your work, and don't be afraid to question the data! Keep practicing, and you'll become a pro at solving these types of problems in no time. This kind of mathematical thinking is used in a lot of different fields. Great job, and keep up the awesome work!