To determine who among Tamara, Payton, Krish, and Haruki is correct about the linear equations, let's solve each of the equations one by one.
### Equation A:
\[ 3(4 - 5h) - 7 = -12 + 2h \]
1. Distribute:
\[ 12 - 15h - 7 = -12 + 2h \]
2. Simplify:
\[ 5 - 15h = -12 + 2h \]
3. Rearranging gives:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]
4. Solving for h:
\[ h = 1 \]
### Equation B:
\[ 2(p - 3) - 1 = \frac{6p + 14}{2} \]
1. Distribute:
\[ 2p - 6 - 1 = 3p + 7 \]
2. Simplifying gives:
\[ 2p - 7 = 3p + 7 \]
3. Rearranging gives:
\[ -7 - 7 = 3p - 2p \]
\[ -14 = p \]
4. So, \( p = -14 \).
### Equation C:
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]
1. Combine constants:
\[ \frac{2}{3}v + \frac{8}{3} - 3 = -1 \]
\[ \frac{2}{3}v + \frac{8 - 9}{3} = -1 \]
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]
2. Adding \( \frac{1}{3} \):
\[ \frac{2}{3}v = -1 + \frac{1}{3} \]
\[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \]
\[ \frac{2}{3}v = -\frac{2}{3} \]
3. Multiplying by \( \frac{3}{2} \):
\[ v = -1 \]
### Summary of Solutions:
- Equation A: \( h = 1 \)
- Equation B: \( p = -14 \)
- Equation C: \( v = -1 \)
### Analyzing the Claims:
1. Tamara (Equation A and Equation C have the same solution): **Incorrect**
- A: \( h = 1 \), C: \( v = -1 \) (different solutions)
2. Payton (Equation B and Equation C have the same solution): **Incorrect**
- B: \( p = -14 \), C: \( v = -1 \) (different solutions)
3. Krish (All three have the same solution): **Incorrect**
- All solutions are different.
4. Haruki (None of the equations have the same solution): **Correct**
- All three equations have different solutions.
**Final Answer:** Haruki is correct.