To determine which of the given pairs of equations has exactly the same solution, we need to simplify each pair and check if they yield the same result.
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First pair:
- \(38x = 1\)
- \(13x = 18\)
Solve for \(x\):
- \(x = \frac{1}{38}\)
- \(x = \frac{18}{13}\)
These are not equal, so this pair does not have the same solution.
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Second pair:
- \(\frac{3}{8}x = 1\)
- \(\frac{1}{3}x = \frac{1}{8}\)
Solve for \(x\):
- \(x = \frac{8}{3}\)
- Multiplying both sides of \(\frac{1}{3}x = \frac{1}{8}\) by 3 gives \(x = \frac{3}{8}\).
These are not equal, so this pair does not have the same solution.
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Third pair:
- \(-3.2x = 0.64\)
- \(x/4 = -0.05\)
Solve for \(x\):
- \(x = \frac{0.64}{-3.2} = -0.2\)
- \(x = -0.05 \cdot 4 = -0.2\)
These are equal, so this pair has the same solution.
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Fourth pair:
- \(-\frac{3}{4}x = \frac{5}{2}\)
- \(\frac{5}{2}x = -\frac{3}{4}\)
Solve for \(x\):
- \(x = -\frac{5}{2} \cdot \left(-\frac{4}{3}\right) = \frac{10}{3}\)
- \(x = -\frac{3}{4} \cdot \left(\frac{2}{5}\right) = -\frac{3}{10}\)
These are not equal, so this pair does not have the same solution.
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Fifth pair:
- \(x \cdot 3.2 = 1.8\)
- \(1.8x = 3.2\)
Solve for \(x\):
- \(x = \frac{1.8}{3.2}\)
- \(x = \frac{3.2}{1.8}\)
These are not equal, so this pair does not have the same solution.
Conclusion:
The pair of equations that has exactly the same solution is: −3.2x = 0.64 and x/4 = −0.05
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