Matrix Equation For Laundry Service Prices

Oct 16, 2025 by ADMIN 43 views

Hey guys! Ever stumbled upon a seemingly simple problem that needs a bit of matrix magic to solve? Let's dive into a laundry price puzzle that's perfect for flexing our linear algebra muscles. This isn't just about clean clothes; it's about clean math too! We're going to break down how to translate a real-world scenario—like figuring out laundry service costs—into a matrix equation. So, grab your thinking caps, and let's get started!

Understanding the Laundry Price Problem

Okay, so here's the deal. Imagine you're looking at a laundry price list, and it says:

Wash, Iron, and Fold: Rp. 6,000 per kg
Wash and Fold: Rp. 4,500 per kg

Now, we want to figure out the individual prices for washing/folding and ironing. Let's say:

The price for wash and fold per kg = x
The price for ironing per kg = y

The big question is: how do we represent this information as a matrix equation so we can solve for x and y? This is where the fun begins! We need to translate these word problems into mathematical equations and then represent those equations in matrix form. Think of it like decoding a secret message, but instead of spies and gadgets, we have laundry prices and matrices.

Setting Up the Equations

First, we need to convert the given information into algebraic equations. This is a crucial step because it lays the foundation for our matrix representation. Let’s break it down:

Wash, Iron, and Fold (Rp. 6,000/kg): This means the cost includes both the wash-fold service (x) and the ironing service (y). So, we can write this as:

x + y = 6,000

This equation tells us that the combined cost of washing, folding, and ironing is Rp. 6,000 per kilogram.
Wash and Fold (Rp. 4,500/kg): This cost only includes the wash-fold service, which we've defined as x. So, this is simply:

x = 4,500

This equation is straightforward: the cost for washing and folding is Rp. 4,500 per kilogram.

Now we have two equations:

x + y = 6,000
x = 4,500

These equations form a system of linear equations. Our next step is to represent this system in matrix form, which will allow us to use matrix operations to solve for x and y. Think of it as organizing our information in a way that makes it easier to manipulate and solve. This is where the real power of matrices comes into play!

Constructing the Matrix Equation

Alright, time to turn these equations into a matrix equation! This might sound intimidating, but trust me, it's like fitting puzzle pieces together once you get the hang of it. We’re going to create three matrices:

Coefficient Matrix (A): This matrix contains the coefficients of our variables (x and y) from the equations.
Variable Matrix (X): This matrix contains our variables, x and y.
Constant Matrix (B): This matrix contains the constants from the right side of our equations.

Let’s build them step by step.

1. Coefficient Matrix (A):

Our equations are:

x + y = 6,000
x = 4,500

In the first equation, the coefficient of x is 1, and the coefficient of y is 1. In the second equation, the coefficient of x is 1, and the coefficient of y is 0 (since y is not present). So, our coefficient matrix A looks like this:

| 1  1 |
| 1  0 |

2. Variable Matrix (X):

This matrix simply contains our variables, x and y. It’s a 2x1 matrix:

| x |
| y |

3. Constant Matrix (B):

This matrix contains the constants on the right side of our equations:

6,000
4,500

So, the constant matrix B is:

| 6000 |
| 4500 |

Now, we can combine these matrices to form our matrix equation:

A X = B

Which translates to:

| 1  1 |   | x |   | 6000 |
| 1  0 | * | y | = | 4500 |

This matrix equation represents our system of linear equations. The next step is to solve this equation to find the values of x and y. We’re one step closer to cracking the laundry price code!

Solving for x and y Using Matrices

Okay, so we've got our matrix equation AX = B. Now, the exciting part: solving for X, which will give us the values of x and y. There are a couple of ways we can do this, but one common method involves finding the inverse of matrix A. Think of it like the matrix version of division!

Finding the Inverse of Matrix A

To solve for X, we need to find the inverse of matrix A, denoted as A⁻¹. The inverse of a matrix, when multiplied by the original matrix, gives us the identity matrix (a matrix with 1s on the diagonal and 0s elsewhere). This is super handy for isolating our variable matrix X.

Our matrix A is:

| 1  1 |
| 1  0 |

For a 2x2 matrix, the inverse can be found using a simple formula. If we have a matrix:

| a  b |
| c  d |

The inverse is:

(1 / (ad - bc))  *  |  d  -b |
                     | -c   a |

So, for our matrix A:

a = 1
b = 1
c = 1
d = 0

Let's plug these values into the formula:

Calculate the determinant (ad - bc):

(1 * 0) - (1 * 1) = 0 - 1 = -1

Apply the inverse formula:

A⁻¹ = (1 / -1) *

|  0  -1 |
| -1   1 |

A⁻¹ = -1 *

|  0  -1 |
| -1   1 |

A⁻¹ =

|  0   1 |
|  1  -1 |

So, the inverse of matrix A is:

|  0   1 |
|  1  -1 |

Now that we have A⁻¹, we can use it to solve for X.

Solving for X

To find X, we multiply both sides of our matrix equation AX = B by A⁻¹ on the left:

A⁻¹ AX = A⁻¹ B

Since A⁻¹ A equals the identity matrix (I), we have:

IX = A⁻¹ B

And IX is simply X, so:

X = A⁻¹ B

Now, let's plug in our matrices:

X =

|  0   1 |   | 6000 |
|  1  -1 | * | 4500 |

To multiply these matrices, we perform the following calculations:

(0 * 6000) + (1 * 4500) = 0 + 4500 = 4500
(1 * 6000) + (-1 * 4500) = 6000 - 4500 = 1500

So, our variable matrix X is:

| 4500 |
| 1500 |

This means:

x = 4500
y = 1500

Therefore, the price for wash and fold per kg is Rp. 4,500, and the price for ironing per kg is Rp. 1,500. We did it! We used matrices to solve a real-world problem. How cool is that?

The Correct Matrix Equation: Putting It All Together

Alright, let’s circle back to the original question. We needed to figure out which matrix equation correctly represents our laundry price problem. We’ve done all the heavy lifting, so this should be a breeze!

We started with the following information:

Wash, Iron, and Fold: Rp. 6,000/kg (x + y = 6,000)
Wash and Fold: Rp. 4,500/kg (x = 4,500)

We then translated this into the matrix equation:

| 1  1 |   | x |   | 6000 |
| 1  0 | * | y | = | 4500 |

This equation perfectly captures the relationships between the prices. The coefficient matrix represents the coefficients of x and y in our equations, the variable matrix represents x and y, and the constant matrix represents the total costs.

So, the correct matrix equation to determine the values of x and y is:

| 1  1 |   | x |   | 6000 |
| 1  0 | * | y | = | 4500 |

This is the equation that allows us to use matrix operations to solve for the individual prices of washing/folding and ironing. We’ve successfully navigated the world of matrices and applied it to a practical problem. Give yourselves a pat on the back, guys!

Why This Matters: Real-World Applications of Matrix Equations

Now, you might be thinking, “Okay, this is a cool math trick, but when am I ever going to use this in real life?” Well, you’d be surprised! Matrix equations aren't just abstract mathematical concepts; they're powerful tools with tons of applications in various fields.

Think about it: any situation where you have multiple variables and multiple equations relating them can potentially be solved using matrices. Here are a few examples:

Economics: Economists use matrices to analyze economic models, such as supply and demand. They can represent systems of equations that describe the relationships between different economic variables and use matrix operations to find equilibrium points and predict economic trends.
Engineering: Engineers use matrices extensively in structural analysis, circuit analysis, and control systems. For example, when designing a bridge, engineers use matrices to calculate the forces and stresses acting on different parts of the structure.
Computer Graphics: Matrices are the backbone of computer graphics. They are used to perform transformations like rotations, scaling, and translations of objects in 3D space. Without matrices, your favorite video games and animated movies wouldn't be possible!
Cryptography: Matrices play a crucial role in encryption and decryption. Certain encryption algorithms use matrix operations to scramble and unscramble data, ensuring secure communication.
Data Analysis: In data analysis and machine learning, matrices are used to represent datasets and perform various operations like dimensionality reduction, clustering, and classification. Techniques like Principal Component Analysis (PCA) rely heavily on matrix computations.

Our laundry price problem was a simplified example, but it illustrates the basic principles of using matrices to solve systems of equations. The same concepts can be applied to much more complex problems in these diverse fields. So, understanding matrix equations isn't just about acing your math test; it's about gaining a valuable problem-solving tool that can be applied in countless real-world scenarios. Keep those matrix skills sharp, guys!

Conclusion: Laundry Prices and Matrix Magic

So, there you have it! We’ve taken a seemingly simple problem—figuring out laundry service prices—and transformed it into a matrix equation masterpiece. We've walked through setting up the equations, constructing the matrices, finding the inverse, and solving for our unknowns. Not only did we crack the laundry price code, but we also got a glimpse into the power and versatility of matrix equations.

Remember, the key takeaways are:

Translate the problem: Convert real-world scenarios into algebraic equations.
Build the matrices: Organize the coefficients, variables, and constants into matrices.
Solve the equation: Use matrix operations, like finding the inverse, to solve for the unknowns.

This journey through laundry prices and matrix magic highlights the importance of math in everyday life. From economics to engineering, matrices are essential tools for solving complex problems. So, the next time you see a system of equations, don't shy away! Think of it as a puzzle waiting to be solved with the power of matrices. And who knows, maybe you’ll even use your newfound skills to calculate the best laundry deals in town! Keep exploring, keep learning, and keep those matrices multiplying!