Question

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

x3.2=1.8
and 1.8x=3.2

3/8x=1
and 1/3x=1/8

−3.2x=0.64
and x/4=−0.05


−3/4x=5/2
and 5/2x=−3/4

Answer 1

To determine which pairs of equations have exactly the same solution, let's solve each pair separately.

1. **For the equations \( x^{3.2} = 1.8 \) and \( 1.8x = 3.2 \)**:
- The first equation can be rewritten as \( x = 1.8^{1/3.2} \).
- The second equation can be solved for \( x \):
\[ x = \frac{3.2}{1.8} \approx 1.777... \]

2. **For the equations \( \frac{3}{8}x = 1 \) and \( \frac{1}{3}x = \frac{1}{8} \)**:
- The first equation gives:
\[ x = \frac{1 \cdot 8}{3} = \frac{8}{3} \approx 2.666... \]
- The second equation gives:
\[ x = \frac{1/8}{1/3} = \frac{1/8 \cdot 3}{1} = \frac{3}{8} = 0.375 \]

3. **For the equations \( -3.2x = 0.64 \) and \( \frac{x}{4} = -0.05 \)**:
- The first equation gives:
\[ x = \frac{0.64}{-3.2} = -0.2 \]
- The second equation gives:
\[ x = -0.05 \cdot 4 = -0.2 \]

4. **For the equations \( -\frac{3}{4}x = \frac{5}{2} \) and \( \frac{5}{2}x = -\frac{3}{4} \)**:
- The first equation gives:
\[ x = \frac{5/2}{-3/4} = \frac{5 \cdot 4}{2 \cdot -3} = \frac{20}{-6} = -\frac{10}{3} \approx -3.33... \]
- The second equation gives:
\[ x = \frac{-3/4}{5/2} = \frac{-3/4 \cdot 2/5}{1} = \frac{-6}{20} = -\frac{3}{10} = -0.3 \]

Now let's summarize the results:
- Pair 1: No common solution (different values).
- Pair 2: No common solution (different values).
- Pair 3: Same solution (\( x = -0.2 \)).
- Pair 4: No common solution (different values).

Thus, the pair of equations that has exactly the same solution is:

**−3.2x=0.64 and x/4=−0.05.**