To determine the number of solutions for the given system of equations:
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Write down the equations: \[ -x + y = 3 \quad \text{(1)} \] \[ -2x + 2y = 3 \quad \text{(2)} \]
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Simplify the second equation (if possible): Divide the second equation by 2: \[ -x + y = \frac{3}{2} \quad \text{(3)} \]
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Compare equations (1) and (3): \[ -x + y = 3 \quad \text{(1)} \] \[ -x + y = \frac{3}{2} \quad \text{(3)} \]
From equations (1) and (3), we can see that both have the same left-hand side, but different right-hand sides: \(-x + y = 3\) and \(-x + y = \frac{3}{2}\).
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Check for consistency: Since \(3\) is not equal to \(\frac{3}{2}\), these two equations are inconsistent.
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Conclusion: Because the two equations represent parallel lines (having the same slope but different y-intercepts), there are no solutions to the system of equations.
Thus, the number of solutions is: \[ \boxed{0} \]