https://file.aiquickdraw.com/mathbot/file/1726830076191trw6asgb.jpg

The image appears to show a system of linear equations presented in an augmented matrix format. To check for consistency and solve the system if possible, we first need to determine if the matrix has any inconsistencies such as contradictory equations or dependencies that might indicate no solution or infinitely many solutions.

The augmented matrix provided is:
\[
\begin{array}{ccc|c}
5 & 3 & 14 & 4 \\
0 & 1 & 2 & 1 \\
1 & 1 & 2 & 0 \\
2 & 1 & 6 & 2 \\
\end{array}
\]

To analyze this, we will perform row operations to attempt to reach reduced row echelon form (RREF), which simplifies checking for consistency. Let's do this now.The reduced row echelon form (RREF) of the augmented matrix is:

\[
\begin{array}{ccc|c}
1 & 0.6 & 2.8 & 0.8 \\
0 & 1 & 2 & 1 \\
0 & 0 & 1 & 0.75 \\
0 & 0 & 0 & 1 \\
\end{array}
\]

Analyzing the RREF, the last row of the matrix, `[0, 0, 0, 1]`, indicates an inconsistency because it represents the equation \(0x + 0y + 0z = 1\), which is impossible (i.e., \(0 = 1\)). Thus, this system of equations is **inconsistent** and has **no solution**.

Would you like to explore any specific details or further clarify any concepts related to this analysis? Let me know if you have any questions!

### Possible next questions:
1. How does the presence of a row `[0, 0, 0, 1]` in the RREF indicate that the system is inconsistent?
2. What would indicate that a system has infinitely many solutions?
3. How do we perform row operations to achieve RREF?
4. What other methods can be used to solve systems of linear equations?
5. Could errors in the original matrix transcription affect the result?

### Tip:
Always double-check the matrix entries when copying them for calculations to avoid any computational errors based on incorrect inputs.

Check for consistency of the given augmented matrix.

Explore how to check for the consistency of a system of linear equations using an augmented matrix. Learn the steps of row operations and Gaussian elimination to determine if the system has no solution, one solution, or infinitely many solutions. Suitable for advanced high school or college students.

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