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

### Given Equations:
We previously had the following system of equations:

\[
\begin{aligned}
20x + 30y + 40z &= 250 \quad \text{(Equation 1)} \\
30x + 40y + 20z &= 270 \quad \text{(Equation 2)} \\
40x + 20y + 30z &= 200 \quad \text{(Equation 3)}
\end{aligned}
\]

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.

Learn how to solve systems of equations using matrix operations with a step-by-step explanation. This problem involves matrix equations, matrix inversion, and applications to real-world scenarios. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation