Question
Which of the following pairs of equations has exactly the same solution?
Responses
x/3.2=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
Responses
x/3.2=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
Answers
GPT-4o mini
To determine which pairs of equations have exactly the same solution, we can solve each pair separately.
1. **Equations: \( \frac{x}{3.2} = 1.8 \) and \( 1.8x = 3.2 \)**
- For the first equation, multiply both sides by \( 3.2 \):
\[ x = 1.8 \times 3.2 = 5.76 \]
- For the second equation, divide both sides by \( 1.8 \):
\[ x = \frac{3.2}{1.8} \approx 1.777 \]
- **These equations do not have the same solution.**
2. **Equations: \( \frac{3}{8}x = 1 \) and \( \frac{1}{3}x = \frac{1}{8} \)**
- For the first equation, multiply both sides by \( \frac{8}{3} \):
\[ x = \frac{8}{3} \]
- For the second equation, multiply both sides by \( 8 \) and then by \( 3 \):
\[ x = \frac{1}{8} \times 3 \]
\[ x = \frac{3}{8} \]
- **These equations do not have the same solution.**
3. **Equations: \( -3.2x = 0.64 \) and \( \frac{x}{4} = -0.05 \)**
- For the first equation, divide both sides by \( -3.2 \):
\[ x = \frac{0.64}{-3.2} = -0.2 \]
- For the second equation, multiply both sides by 4:
\[ x = -0.05 \times 4 = -0.2 \]
- **These equations have the same solution: \( x = -0.2 \).**
4. **Equations: \( -\frac{3}{4}x = \frac{5}{2} \) and \( \frac{5}{2}x = -\frac{3}{4} \)**
- For the first equation, multiply both sides by \( -\frac{4}{3} \):
\[ x = \frac{5}{2} \times (-\frac{4}{3}) = -\frac{10}{3} \]
- For the second equation, multiply both sides by \( \frac{2}{5} \):
\[ x = -\frac{3}{4} \times \frac{2}{5} = -\frac{3}{10} \]
- **These equations do not have the same solution.**
Based on this analysis, the pair of equations that has exactly the same solution is:
**−3.2x=0.64 and x/4=−0.05**.
1. **Equations: \( \frac{x}{3.2} = 1.8 \) and \( 1.8x = 3.2 \)**
- For the first equation, multiply both sides by \( 3.2 \):
\[ x = 1.8 \times 3.2 = 5.76 \]
- For the second equation, divide both sides by \( 1.8 \):
\[ x = \frac{3.2}{1.8} \approx 1.777 \]
- **These equations do not have the same solution.**
2. **Equations: \( \frac{3}{8}x = 1 \) and \( \frac{1}{3}x = \frac{1}{8} \)**
- For the first equation, multiply both sides by \( \frac{8}{3} \):
\[ x = \frac{8}{3} \]
- For the second equation, multiply both sides by \( 8 \) and then by \( 3 \):
\[ x = \frac{1}{8} \times 3 \]
\[ x = \frac{3}{8} \]
- **These equations do not have the same solution.**
3. **Equations: \( -3.2x = 0.64 \) and \( \frac{x}{4} = -0.05 \)**
- For the first equation, divide both sides by \( -3.2 \):
\[ x = \frac{0.64}{-3.2} = -0.2 \]
- For the second equation, multiply both sides by 4:
\[ x = -0.05 \times 4 = -0.2 \]
- **These equations have the same solution: \( x = -0.2 \).**
4. **Equations: \( -\frac{3}{4}x = \frac{5}{2} \) and \( \frac{5}{2}x = -\frac{3}{4} \)**
- For the first equation, multiply both sides by \( -\frac{4}{3} \):
\[ x = \frac{5}{2} \times (-\frac{4}{3}) = -\frac{10}{3} \]
- For the second equation, multiply both sides by \( \frac{2}{5} \):
\[ x = -\frac{3}{4} \times \frac{2}{5} = -\frac{3}{10} \]
- **These equations do not have the same solution.**
Based on this analysis, the pair of equations that has exactly the same solution is:
**−3.2x=0.64 and x/4=−0.05**.