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The given matrix is an augmented matrix, which represents a system of linear equations. To write the system of equations from the augmented matrix, we can interpret the entries of the matrix in the following way:

The matrix is:

\[
\begin{bmatrix}
1 & 2 & 3 & | & 10 \\
4 & 5 & 6 & | & 12 \\
7 & 8 & 9 & | & 14
\end{bmatrix}
\]

This corresponds to the following system of equations, where the variables are \( x \), \( y \), and \( z \):

1. \( 1x + 2y + 3z = 10 \)
2. \( 4x + 5y + 6z = 12 \)
3. \( 7x + 8y + 9z = 14 \)

Thus, the system of equations is:

\[
\begin{aligned}
x + 2y + 3z &= 10 \\
4x + 5y + 6z &= 12 \\
7x + 8y + 9z &= 14
\end{aligned}
\]

Would you like further explanation or details on solving this system of equations? 

Here are some related questions for further understanding:
1. How can you solve a system of equations using matrix operations?
2. What is the augmented matrix method for solving systems of linear equations?
3. How do you interpret the result if the system of equations has no solution?
4. What are some ways to solve a system of equations (e.g., substitution, elimination, or matrix methods)?
5. How does row reduction help in solving systems of equations?

**Tip:** When working with matrices, keep in mind that the augmented matrix form allows you to easily perform operations to find solutions using methods such as Gaussian elimination.

Write a systems of equations from the augmented matrix.

Learn how to write a system of linear equations from an augmented matrix with a step-by-step explanation. This problem demonstrates key linear algebra concepts, including interpreting matrices and solving systems of equations. Suitable for students in Grades 10-12.

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