Question

Which of the following pairs of equations has exactly the same solution?
3/8x=1and 1/3x=1/8, -3.2x=0.64 and x/4=-0.05, -3/4x=5/2x=-3/4, 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 solve each equation or pair of equations.

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

- Solve \( \frac{3}{8}x = 1 \):
\[
x = 1 \cdot \frac{8}{3} = \frac{8}{3}
\]

- Solve \( \frac{1}{3}x = \frac{1}{8} \):
\[
x = \frac{1}{8} \cdot 3 = \frac{3}{8}
\]

**These equations do not have the same solution.**

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

- Solve \( -3.2x = 0.64 \):
\[
x = \frac{0.64}{-3.2} = -0.2
\]

- Solve \( \frac{x}{4} = -0.05 \):
\[
x = -0.05 \cdot 4 = -0.2
\]

**These equations have the same solution: \( x = -0.2 \).**

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

- Solve \( -\frac{3}{4}x = 5 \):
\[
x = 5 \cdot -\frac{4}{3} = -\frac{20}{3}
\]

- Solve \( 2x = -\frac{3}{4} \):
\[
x = -\frac{3}{4} \cdot \frac{1}{2} = -\frac{3}{8}
\]

**These equations do not have the same solution.**

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

- Solve \( \frac{x}{3.2} = 1.8 \):
\[
x = 1.8 \cdot 3.2 = 5.76
\]

- Solve \( 1.8x = 3.2 \):
\[
x = \frac{3.2}{1.8} = \frac{32}{18} = \frac{16}{9} \approx 1.7777
\]

**These equations do not have the same solution.**

Based on the analysis, the only pair of equations that has exactly the same solution is the second pair:
**\( -3.2x = 0.64 \) and \( \frac{x}{4} = -0.05 \)**.