or the following systems,
a. Write the matrix that represents the system
b. Use the RREF function on a calculator to write the matrix in row-reduced echelon form
c. State the solution for the system
7𝑥 + 22𝑦 = 371
4𝑥 − 2𝑦 = 8

Let's solve the system of linear equations step by step.

### Given System of Equations:

1. \( 7x + 22y = 371 \)
2. \( 4x - 2y = 8 \)

### Part a: Write the Matrix that Represents the System

To write the system in matrix form, we express it as \( A\mathbf{x} = \mathbf{b} \), where \( A \) is the coefficient matrix, \( \mathbf{x} \) is the column vector of variables, and \( \mathbf{b} \) is the constant column vector.

\[
\begin{pmatrix}
7 & 22 \\
4 & -2
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
371 \\
8
\end{pmatrix}
\]

This can also be written as an augmented matrix:

\[
\text{Augmented Matrix: } \begin{pmatrix} 7 & 22 & | & 371 \\ 4 & -2 & | & 8 \end{pmatrix}
\]

### Part b: Use the RREF Function to Write the Matrix in Row-Reduced Echelon Form

To reduce the matrix to Row-Reduced Echelon Form (RREF), we perform row operations. The RREF function on a calculator or software will transform the augmented matrix into this form.

Applying the RREF:

\[
\begin{pmatrix} 1 & 0 & | & 38 \\ 0 & 1 & | & 11 \end{pmatrix}
\]

### Part c: State the Solution for the System

From the RREF, we can directly read off the solutions for \( x \) and \( y \):

\[
\begin{aligned}
x & = 38, \\
y & = 11.
\end{aligned}
\]

### Solution:
The solution to the system is \( x = 38 \) and \( y = 11 \).

---

Would you like any further details or explanations on these steps? Here are some related questions:

1. What are the steps involved in performing Gaussian elimination manually?
2. How does the RREF function work on a calculator?
3. Can a system of linear equations have no solution? If so, how does that appear in RREF?
4. What is the difference between Row Echelon Form (REF) and Row-Reduced Echelon Form (RREF)?
5. How can we check the consistency of a system of equations?

**Tip:** When using RREF on a calculator, make sure to enter the augmented matrix correctly, including all coefficients and constants.

Learn how to solve the system of linear equations 7x + 22y = 371 and 4x - 2y = 8 using matrix representation and row-reduced echelon form. Detailed step-by-step solution provided. Suitable for high school students studying linear algebra.

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