To determine which pairs of equations have exactly the same solution, we can solve each pair of equations.
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First pair of equations:
- \( x/3.2 = 1.8 \)
- \( 1.8x = 3.2 \)
For the first equation: \[ x = 1.8 \times 3.2 = 5.76 \]
For the second equation: \[ x = \frac{3.2}{1.8} \approx 1.78 \]
So the first pair does not have the same solution.
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Second pair of equations:
- \( -3.2x = 0.64 \)
- \( x/4 = -0.05 \)
For the first equation: \[ x = \frac{0.64}{-3.2} = -0.2 \]
For the second equation: \[ x = -0.05 \times 4 = -0.2 \]
So the second pair has the same solution.
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Third pair of equations:
- \( \frac{3}{8}x = 1 \)
- \( \frac{1}{3}x = \frac{1}{8} \)
For the first equation: \[ x = 1 \div \frac{3}{8} = \frac{8}{3} \]
For the second equation: \[ x = \frac{1}{8} \times 3 = \frac{3}{8} \]
So the third pair does not have the same solution.
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Fourth pair of equations:
- \( -\frac{3}{4}x = \frac{5}{2} \)
- \( \frac{5}{2}x = -\frac{3}{4} \)
For the first equation: \[ x = \frac{5/2}{-3/4} = \frac{5/2} \times \left(-\frac{4}{3}\right) = -\frac{10}{3} \]
For the second equation: \[ x = \frac{-3/4}{5/2} = -\frac{3/4} \times \left(\frac{2}{5}\right) = -\frac{3}{10} \]
So the fourth pair does not have the same solution.
From the checks, the only pair of equations that has exactly the same solution is the second pair: \[ -3.2x = 0.64 \text{ and } x/4 = -0.05 \]