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The system of equations provided is inconsistent. This means that the equations contradict each other, and no solution exists that satisfies all three equations simultaneously. 

To summarize the key issue:
- The first two equations are \(x - y + z = 6\) and \(x - y + z = 2\). These two equations imply a contradiction, as \(x - y + z\) cannot equal both 6 and 2 at the same time. This inconsistency leads to the conclusion that the system has no solutions.

Would you like further details or explanations? Here are 5 related questions for further exploration:
1. How can you tell if a system of equations is inconsistent?
2. What is the difference between consistent and inconsistent systems?
3. Can a system of linear equations have infinitely many solutions?
4. How would the solution change if the second equation were different?
5. What methods can be used to solve consistent systems of equations?

**Tip:** When you spot identical expressions with different results, that's a quick sign of inconsistency in a system.

Check consistency for the following system of equations:
1) x - y + z = 6
2) x - y + z = 2
3) 2x + y - z = 1

Analyze the consistency of the system of equations: x - y + z = 6, x - y + z = 2, and 2x + y - z = 1. This problem introduces key concepts from linear algebra, such as the consistency of systems and how contradictions arise when solving equations. Suitable for students in Grades 10-12.

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