To determine which pairs of equations have exactly the same solution, we can solve each one.
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For the first pair: \(-3.2x = 0.64\)
To solve for \(x\): \[ x = \frac{0.64}{-3.2} = -0.2 \]
Now for the second equation: \[ \frac{x}{4} = -0.05 \]
Multiplying both sides by 4: \[ x = -0.2 \]
Both equations have the solution \(x = -0.2\).
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For the second pair: \(\frac{3}{8}x = 1\)
Multiplying both sides by \(\frac{8}{3}\): \[ x = \frac{8}{3} \]
Now for the second equation: \(\frac{1}{3}x = \frac{1}{8}\)
Multiplying both sides by 3: \[ x = \frac{3}{8} \]
These equations do not have the same solution.
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For the third pair: \(\frac{x}{3.2} = 1.8\)
Multiplying both sides by 3.2: \[ x = 1.8 \times 3.2 = 5.76 \]
Now for the second equation: \(1.8x = 3.2\)
Dividing both sides by 1.8: \[ x = \frac{3.2}{1.8} = \frac{16}{9} \approx 1.78 \]
These equations do not have the same solution.
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For the fourth pair: \(-\frac{3}{4}x = \frac{5}{2}\)
Multiplying both sides by -\(\frac{4}{3}\): \[ x = -\frac{4}{3} \times \frac{5}{2} = -\frac{10}{3} \]
Now for the second equation: \(\frac{5}{2}x = -\frac{3}{4}\)
Dividing both sides by \(\frac{5}{2}\): \[ x = -\frac{3}{4} \times \frac{2}{5} = -\frac{3}{10} \]
These equations do not have the same solution.
From our work, the first pair of equations: \(-3.2x = 0.64\) and \(\frac{x}{4} = -0.05\)
are the only ones that have exactly the same solution, which is \(x = -0.2\).