Solving SPLTV: Shampoo Volume In Different Packages

Jan 11, 2026 by Tim Redaksi 52 views

Hey guys, let's dive into a fun math problem! We're going to use a system of linear equations in three variables (SPLTV) to figure out the volume of shampoo in different packages. Imagine a shampoo factory that produces shampoo in three different sizes: package A, package B, and package C. We're given some information about the total volume of shampoo when you combine different numbers of these packages. Our goal is to find out the volume of shampoo in each individual package. Sounds interesting, right?

Understanding the Problem: Shampoo Packaging Puzzle

Okay, so the factory has three types of shampoo packages: A, B, and C. We're given three key pieces of information, which we'll use to create our equations. This is where it gets a little bit mathematical, but don't worry, I'll walk you through it. Essentially, we are trying to find three unknowns: the volume of shampoo in package A, the volume in package B, and the volume in package C. We can solve these unknowns because we are provided with the sum of the volumes when combining these packages. Think of it like a puzzle; we have the clues (the total volumes) and we need to put the pieces together (the volumes of each package) to solve it. It's like a real-world application of algebra, pretty cool, huh?

To solve this type of problem, we need to create a system of equations. Each equation will represent one of the pieces of information we've been given about the different package combinations and the total shampoo volumes. It is crucial to set up the equations correctly because this is the foundation for finding the solution. Here is the first equation. The volume of shampoo in 3 packages of A, 1 package of B, and 2 packages of C is 65 mL. We can represent this with the equation: 3A + B + 2C = 65. The volume of shampoo in 2 packages of A, 1 package of B, and 1 package of C is 40 mL. This is written as: 2A + B + C = 40. Lastly, the total volume of shampoo in 1 package of A and 3 packages of C is 45 mL. Here is the last equation: A + 3C = 45. Now, we've got our system of equations! Let's get to work!

Key Takeaway: Understanding the problem and translating the information into mathematical equations is the first and most important step in solving this type of problem. Think of each equation as a clue to help you find the individual volumes.

Setting Up the Equations: Your Math Blueprint

Alright, let's break down how we turn those word problems into mathematical equations. We'll use variables to represent the unknowns – the volume of shampoo in each package. Let:

A = Volume of shampoo in package A
B = Volume of shampoo in package B
C = Volume of shampoo in package C

Now, let's translate the information given into equations:

First Clue: The volume of shampoo in 3 packages A, 1 package B, and 2 packages C is 65 mL. This translates to the equation: 3A + B + 2C = 65
Second Clue: The volume of shampoo in 2 packages A, 1 package B, and 1 package C is 40 mL. This translates to the equation: 2A + B + C = 40
Third Clue: The volume of shampoo in 1 package A and 3 packages C is 45 mL. This translates to the equation: A + 3C = 45

Important Tip: Always double-check that your equations accurately reflect the information provided. Make sure you use the correct coefficients (the numbers in front of the variables) and the correct total volume on the right side of the equation. This will ensure that our solution is accurate. This is the foundation upon which the solution rests, so make sure it's correct!

Solving the System: Finding the Shampoo Volumes

Alright, now that we have our system of equations, let's solve it! There are several methods to solve a system of linear equations, such as the substitution method, the elimination method, or using matrices. For this example, let's use the substitution and elimination methods. These are pretty common and can be a good starting point for solving problems like these.

Step 1: Simplify the Equations. Before we begin, let's rearrange our equations for easier manipulation. From the third equation A + 3C = 45, we can easily express A in terms of C: A = 45 - 3C. This is our equation 4.

Step 2: Substitution. Substitute equation 4 into the second equation 2A + B + C = 40: 2(45 - 3C) + B + C = 40. Simplify this to get 90 - 6C + B + C = 40, which further simplifies to B - 5C = -50. This is our equation 5.

Step 3: More Substitution. Now, substitute equation 4 into the first equation, 3A + B + 2C = 65: 3(45 - 3C) + B + 2C = 65. Simplify this to get 135 - 9C + B + 2C = 65, and further simplify to B - 7C = -70. This is our equation 6.

Step 4: Elimination. Now we have two equations, equation 5 and equation 6, both in terms of B and C. We can eliminate B by subtracting equation 6 from equation 5. Let's do it: (B - 5C) - (B - 7C) = -50 - (-70). This simplifies to 2C = 20, and therefore, C = 10.

Step 5: Back-Substitution. Now that we know C = 10, we can substitute this value back into the equation A = 45 - 3C to solve for A. A = 45 - 3(10), which gives us A = 15.

Step 6: Final Substitution. Finally, substitute the values of A and C into the second original equation: 2A + B + C = 40. Which gives us: 2(15) + B + 10 = 40. Simplify this to solve for B. Which gives us B = 0. This is amazing because it shows that package B does not contain any shampoo! The result is B=0.

So, the volume of shampoo in each package is:

Package A: 15 mL
Package B: 0 mL
Package C: 10 mL

Congratulations, we solved it!

Practical Application and Interpretation

Okay, so we've crunched the numbers and found the volumes. But what does this mean in the real world? This exercise helps us understand how to solve problems that involve multiple variables and equations. In the real world, this could apply to many scenarios, such as in the context of business, finance, and engineering. Think about it: a factory uses complex calculations to decide how much of each type of product to produce to maximize profits.

Interpreting the Results:

Package A: Contains 15 mL of shampoo.
Package B: Contains 0 mL of shampoo. This could mean that the package B does not contain shampoo, or is an empty package.
Package C: Contains 10 mL of shampoo.

This outcome provides valuable information for the factory to control how it packages the shampoo. For example, it helps the factory optimize its production process, inventory management, and even cost control. If a particular package size is empty, the factory can decide to stop producing it to reduce costs. Also, the factory can change its production ratio to fit the demands. The practical applications of systems of equations are vast and varied.

Real-World Relevance and Additional Examples

This type of problem might seem like a pure math exercise, but it has tons of real-world applications. Think about it this way: solving systems of equations helps you figure out how to allocate resources, manage budgets, and even analyze market trends. It is important to know this skill because these skills help you with real-world problems. For example, understanding how to calculate the number of products that must be produced to meet demand and maximize profits. The ability to model real-world scenarios with mathematical equations is a valuable skill in various fields.

Additional Examples:

Mixing Solutions: Imagine you're a chemist mixing solutions. You have three solutions with different concentrations. Using systems of equations, you can determine how much of each solution you need to create a specific mixture with a desired concentration.
Financial Planning: Financial planners use systems of equations to calculate the optimal allocation of investments in different assets to reach a financial goal, such as how to invest money in stocks, bonds, and real estate to maximize returns while minimizing risk.
Supply Chain Management: Companies use systems of equations to determine the most cost-effective way to transport goods from multiple warehouses to various distribution centers. This could involve figuring out how many goods need to be shipped from each warehouse to each distribution center to minimize transportation costs.

Conclusion: Mastering SPLTV and Beyond

Alright, guys, you made it! We successfully solved the shampoo packaging problem using SPLTV. We started with the real-world scenario of the factory, converted the information into mathematical equations, and then used the substitution and elimination methods to find the volumes of each package. Remember, this problem isn't just about finding the answer; it's about understanding the process of problem-solving. It's about translating real-world situations into mathematical models and using those models to find solutions.

This approach to solving problems opens the door to more complex challenges, from calculating the optimal way to invest money to figuring out how to optimize the production of goods and services. So, the next time you encounter a problem with multiple variables, remember this example. You have the tools to tackle it, and with practice, you can become a pro at solving systems of equations!

In summary, we learned:

How to translate word problems into mathematical equations.
How to use the substitution and elimination methods to solve systems of equations.
The real-world applications of systems of equations.

Keep practicing, keep exploring, and you'll find that math can be a powerful tool for solving all kinds of problems!