To solve the given system of equations:

1. $4x + 2y - 2z = 1$
2. $2x + y - z = 1$
3. $-3x - 2y = 1$

Step 1: Express $z$ in terms of $x$ and $y$

From the second equation: $z = 2x + y - 1$

Step 2: Substitute $z = 2x + y - 1$ into the first equation

Substitute into $4x + 2y - 2z = 1$: $4x + 2y - 2(2x + y - 1) = 1$ Simplify: $4x + 2y - 4x - 2y + 2 = 1$ $2 = 1$

Step 3: Analyze inconsistency

The result $2 = 1$ is a contradiction, meaning the system of equations has no solution.

Conclusion

The given system of equations is inconsistent and therefore has no solution.

Would you like to delve deeper into this analysis or try similar systems?

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Relative Questions:

1. What does it mean for a system of equations to be inconsistent?
2. Can inconsistent systems have approximate solutions under certain conditions?
3. How can we determine consistency of equations without solving them?
4. What methods (substitution, elimination, matrix) are best for solving systems of equations?
5. How can graphing help in understanding systems with no solutions?

Tip:

Always check for contradictions early when solving systems of equations—it can save significant time and effort!