Bot GPT 4-o mini To determine which pair of equations has exactly the same solution, let's analyze each pair of equations one by one.
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Pair 1: \[ \frac{x}{3.2} = 1.8 \quad \text{and} \quad 1.8x = 3.2 \]
- From the first equation: \( x = 1.8 \cdot 3.2 = 5.76 \)
- From the second equation: \( x = \frac{3.2}{1.8} \) \[ x = \frac{32}{18} = \frac{16}{9} \approx 1.78 \]
- Not the same solution.
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Pair 2: \[ -3.2x = 0.64 \quad \text{and} \quad \frac{x}{4} = -0.05 \]
- From the first equation: \[ x = \frac{0.64}{-3.2} = -0.2 \]
- From the second equation: \[ x = -0.05 \cdot 4 = -0.2 \]
- Same solution: both equations give \( x = -0.2 \).
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Pair 3: \[ \frac{3}{8}x = 1 \quad \text{and} \quad \frac{1}{3}x = \frac{1}{8} \]
- From the first equation: \[ x = \frac{1 \cdot 8}{3} = \frac{8}{3} \approx 2.67 \]
- From the second equation: \[ x = \frac{1 \cdot 8}{1} = 8 \cdot \frac{1}{3} = \frac{8}{3} \approx 2.67 \]
- Same solution: both equations yield \( x = \frac{8}{3} \).
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Pair 4: \[ -\frac{3}{4}x = \frac{5}{2} \quad \text{and} \quad \frac{5}{2}x = -\frac{3}{4} \]
- From the first equation: \[ x = -\frac{5}{2} \cdot \frac{4}{3} = -\frac{10}{3} \]
- From the second equation: \[ x = -\frac{3}{4} \cdot \frac{2}{5} = -\frac{3}{10} \]
- Not the same solution.
So, the pairs that have the same solution are:
- Pair 2: \( x = -0.2 \)
- Pair 3: \( x = \frac{8}{3} \)
Both pairs have consistent solutions. However, if only one answer is sought from the list provided, the strongest confirmation is Pair 2, as it is further confirmed with simpler operations.
The correct response is:
- −3.2x=0.64 and x/4=−0.05
1 answer