An augmented matrix for a linear system is given below. {(1,0,0,9),(0,0,1,-4)} Identify all of the free variables. Note that more variables may be listed than are actually in the system! Check all that apply from x1, x2, x3, x4, x5

The given augmented matrix is:

\[
\begin{pmatrix}
1 & 0 & 0 & | & 9 \\
0 & 0 & 1 & | & -4
\end{pmatrix}
\]

This matrix corresponds to the following system of equations:

1. \( x_1 = 9 \)
2. \( x_3 = -4 \)

From this, we can see:

- \( x_1 \) and \( x_3 \) are **bound** variables, as they are explicitly solved in terms of constants.
- The matrix provides no information about \( x_2 \), \( x_4 \), and \( x_5 \), so they are **free variables**.

### Conclusion:
The free variables are \( x_2 \), \( x_4 \), and \( x_5 \).

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Do you have any questions or would you like more details?  
Here are some related questions to consider:

1. What is the rank of the matrix?
2. How can you identify free variables in general for any augmented matrix?
3. What is the meaning of a "free variable" in the context of linear systems?
4. How can you express the general solution for this system?
5. What is the significance of the number of free variables relative to the number of equations?

**Tip**: When solving linear systems, always check for consistency of the system (i.e., whether the system has one solution, infinitely many solutions, or no solution).

Learn how to identify free variables in an augmented matrix for a linear system. This problem demonstrates key linear algebra concepts such as row-reduced echelon form and matrix representation of systems. Suitable for students in advanced high school mathematics or undergraduate linear algebra courses.

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