Question

Which of the following pairs of equations has exactly the same solution?(1 point)
Responses

−3/4x=5/2
and 5/2x=−3/4
negative Start Fraction 3 over 4 End Fraction x equals Start Fraction 5 over 2 End Fraction and Start Fraction 5 over 2 End Fraction x equals negative Start Fraction 3 over 4 End Fraction

−3.2x=0.64
and x4=−0.05
negative 3.2x equals 0.64 and Start Fraction x over 4 End Fraction equals negative 0.05

3/8x=1
and 1/3x=1/8
Start Fraction 3 over 8 End Fraction x equals 1 and Start Fraction 1 over 3 End Fraction x equals Start Fraction 1 over 8 End Fraction

x/3.2=1.8
and 1.8x=3.2

Answer 1

To determine which pairs of equations have exactly the same solution, we need to analyze each pair.

1. **Pair 1:**
\(-\frac{3}{4}x = \frac{5}{2}\) and \(\frac{5}{2}x = -\frac{3}{4}\)

- For the first equation, isolate \(x\):
\[
x = -\frac{5/2}{3/4} = -\frac{5}{2} \cdot \frac{4}{3} = -\frac{20}{6} = -\frac{10}{3}
\]
- For the second equation:
\[
x = -\frac{-3/4}{5/2} = -\frac{-3/4 \cdot 2/5} = \frac{6}{20} = \frac{3}{10}
\]
- The solutions are different.

2. **Pair 2:**
\(-3.2x = 0.64\) and \(\frac{x}{4} = -0.05\)

- For the first equation:
\[
x = \frac{0.64}{-3.2} = -0.2
\]
- For the second equation:
\[
x = -0.05 \cdot 4 = -0.2
\]
- The solutions are the same.

3. **Pair 3:**
\(\frac{3}{8}x = 1\) and \(\frac{1}{3}x = \frac{1}{8}\)

- For the first equation:
\[
x = \frac{1}{\frac{3}{8}} = \frac{8}{3}
\]
- For the second equation:
\[
x = \frac{1}{8} \cdot 3 = \frac{3}{8}
\]
- The solutions are different.

4. **Pair 4:**
\(\frac{x}{3.2} = 1.8\) and \(1.8x = 3.2\)

- For the first equation:
\[
x = 1.8 \cdot 3.2 = 5.76
\]
- For the second equation:
\[
x = \frac{3.2}{1.8} \approx 1.777... \text{ (which is also different)}
\]
- The solutions are different.

**Conclusion:** The pair that has exactly the same solution is **Pair 2: \(-3.2x = 0.64\) and \(\frac{x}{4} = -0.05\)**.