Bot GPT 4-o mini To analyze both systems of equations given, we'll identify whether they have a unique solution, infinitely many solutions, or no solution.
System A:
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Equations: \[ 3x - y = -8 \quad (1) \] \[ -3x + y = 8 \quad (2) \]
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Rearranging the equations for comparison:
- From equation (1): \( y = 3x + 8 \)
- From equation (2): \( y = 3x + 8 \)
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Analysis: The two equations are identical, which means they represent the same line. Thus, there are infinitely many solutions.
Conclusion for System A: The system has infinitely many solutions.
System B:
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Equations: \[ x + 3y = 11 \quad (3) \] \[ -x + 3y = 7 \quad (4) \]
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Rearranging the equations:
- From equation (3): \( 3y = 11 - x \) or \( y = \frac{11 - x}{3} \)
- From equation (4): \( 3y = 7 + x \) or \( y = \frac{7 + x}{3} \)
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Setting the equations for \(y\) equal to find \(x\): \[ \frac{11 - x}{3} = \frac{7 + x}{3} \]
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Multiplying both sides by 3 to eliminate the denominator: \[ 11 - x = 7 + x \]
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Solving for \(x\): \[ 11 - 7 = 2x \ 4 = 2x \ x = 2 \]
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Substituting \(x = 2\) back into one of the original equations to find \(y\): Using equation (3): \[ 2 + 3y = 11 \quad \Rightarrow \quad 3y = 9 \quad \Rightarrow \quad y = 3 \]
Conclusion for System B: The system has a unique solution: \((x, y) = (2, 3)\).
Final Descriptions:
- System A: The system has infinitely many solutions.
- System B: The system has a unique solution: \((x, y) = (2, 3)\).
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