Math Problem Statement

use matrix

Solution

Let's solve the system of equations using matrix operations.

Given Equations:

We previously had the following system of equations:

20x + 30y + 40z &= 250 \quad \text{(Equation 1)} \\ 30x + 40y + 20z &= 270 \quad \text{(Equation 2)} \\ 40x + 20y + 30z &= 200 \quad \text{(Equation 3)} \end{aligned}$$ This can be represented in matrix form as: $$\begin{pmatrix} 20 & 30 & 40 \\ 30 & 40 & 20 \\ 40 & 20 & 30 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 250 \\ 270 \\ 200 \end{pmatrix}$$ Here: - The coefficient matrix $$A$$ is: $$A = \begin{pmatrix} 20 & 30 & 40 \\ 30 & 40 & 20 \\ 40 & 20 & 30 \end{pmatrix}$$ - The variable matrix $$X$$ is: $$X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$$ - The constants matrix $$B$$ is: $$B = \begin{pmatrix} 250 \\ 270 \\ 200 \end{pmatrix}$$ To find the values of $$x$$, $$y$$, and $$z$$, we need to solve the matrix equation: $$AX = B$$ The solution is given by: $$X = A^{-1}B$$ Let's calculate this using matrix operations.Using matrix operations, the solution to the system is: $$\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 5 \\ 2 \end{pmatrix}$$ ### Therefore: - **Polythene bag (x)**: 1 unit - **Handmade bag (y)**: 5 units - **Newspaper envelope (z)**: 2 units This confirms the earlier result. Would you like more details or have any questions? Here are 5 related questions: 1. How does matrix inversion help in solving systems of linear equations? 2. What happens if the coefficient matrix is singular (non-invertible)? 3. How can you solve larger systems of equations using matrix methods? 4. Can you solve this system using a different matrix method, such as Gaussian elimination? 5. How would errors in the coefficients affect the solution in a real-world scenario? **Tip:** Matrix operations are powerful tools for solving systems of linear equations, especially when dealing with multiple variables.

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Math Problem Analysis

Mathematical Concepts

Matrix Operations
Systems of Linear Equations

Formulas

Matrix equation AX = B
Matrix Inversion

Theorems

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Suitable Grade Level

Advanced High School to College

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