To determine which pairs of equations have exactly the same solution, let's solve each pair step by step.
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First Pair:
- Equation 1: \( \frac{3}{8}x = 1 \)
- To isolate \( x \), multiply both sides by \( \frac{8}{3} \): \[ x = 1 \cdot \frac{8}{3} = \frac{8}{3} \]
- Equation 2: \( \frac{1}{3}x = \frac{1}{8} \)
- Multiply both sides by 3: \[ x = 3 \cdot \frac{1}{8} = \frac{3}{8} \]
- Conclusion: Solutions are \( \frac{8}{3} \) and \( \frac{3}{8} \); not the same.
- Equation 1: \( \frac{3}{8}x = 1 \)
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Second Pair:
- Equation 1: \( -3.2x = 0.64 \)
- Divide both sides by -3.2: \[ x = \frac{0.64}{-3.2} = -0.2 \]
- Equation 2: \( \frac{x}{4} = -0.05 \)
- Multiply both sides by 4: \[ x = -0.05 \cdot 4 = -0.2 \]
- Conclusion: Both have the solution \( x = -0.2 \); they are the same.
- Equation 1: \( -3.2x = 0.64 \)
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Third Pair:
- Equation 1: \( \frac{x}{3.2} = 1.8 \)
- Multiply both sides by 3.2: \[ x = 1.8 \cdot 3.2 = 5.76 \]
- Equation 2: \( 1.8x = 3.2 \)
- Divide both sides by 1.8: \[ x = \frac{3.2}{1.8} \approx 1.7777 \quad (\text{not equal to } 5.76) \]
- Conclusion: Solutions are not the same.
- Equation 1: \( \frac{x}{3.2} = 1.8 \)
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Fourth Pair:
- Equation 1: \( -\frac{3}{4}x = \frac{5}{2} \)
- Divide both sides by -3/4 (or multiply by -4/3): \[ x = \frac{5}{2} \cdot \frac{-4}{3} = -\frac{10}{3} \]
- Equation 2: \( \frac{5}{2}x = -\frac{3}{4} \)
- Divide both sides by \( \frac{5}{2} \) (or multiply by \( \frac{2}{5} \)): \[ x = -\frac{3/4} {5/2} = -\frac{3 \cdot 2}{4 \cdot 5} = -\frac{6}{20} = -\frac{3}{10} \]
- Conclusion: Solutions are not the same.
- Equation 1: \( -\frac{3}{4}x = \frac{5}{2} \)
By evaluating each pair, the only pair with the same solution is the Second Pair:
- \( -3.2x = 0.64 \) and \( \frac{x}{4} = -0.05 \), both yielding \( x = -0.2 \).