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Given that the augmented matrix in row-reduced form below is equivalent to the augmented matrix of a system of linear equations. Determine whether the system has a solution and find the solution(s) to the system, if they exist.

The given matrix is an augmented matrix in row-reduced form, meaning it's already in reduced row echelon form (RREF). The matrix is:

\[
\left[
\begin{array}{ccc|c}
1 & 0 & 0 & 5 \\
0 & 1 & 0 & 2 \\
0 & 0 & 1 & 4
\end{array}
\right]
\]

This represents a system of linear equations where each row corresponds to an equation. Let's interpret the matrix as follows:

1. The first row represents: \( x_1 = 5 \)
2. The second row represents: \( x_2 = 2 \)
3. The third row represents: \( x_3 = 4 \)

Since this is a row-reduced matrix and no row corresponds to an inconsistency (e.g., \( 0 = b \) where \( b \neq 0 \)), the system has a unique solution.

### Solution:
\[
x_1 = 5, \quad x_2 = 2, \quad x_3 = 4
\]

Thus, the solution to the system of equations is:
\[
\boxed{(5, 2, 4)}
\]

Would you like any further explanation or details?

### 5 Related Questions:
1. What is the process of reducing a matrix to row-reduced echelon form?
2. Can there be cases where an augmented matrix has no solution? How do we identify them?
3. How do you solve systems of linear equations using Gaussian elimination?
4. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
5. How can you tell if a system has infinitely many solutions based on the augmented matrix?

### Tip:
When working with augmented matrices, always check for rows like \( 0 = b \) (with \( b \neq 0 \)) in the reduced form, as these indicate that the system has no solution.

Discover how to determine the solution of a system of linear equations using the row-reduced echelon form of an augmented matrix. This guide explains key concepts in linear algebra, including matrix reduction and the interpretation of solutions.

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