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Bot GPT 4-o mini To solve the equations and evaluate the claims made by Tamara, Payton, Krish, and Haruki, let’s simplify each equation one by one.
### Equation A:
\[ 3(4 - 5h) - 7 = -12 + 2h \]
Step 1: Distribute on the left side.
\[ 12 - 15h - 7 = -12 + 2h \]
Step 2: Combine like terms.
\[ 5 - 15h = -12 + 2h \]
Step 3: Rearrange to isolate \(h\).
\[ 5 + 12 = 2h + 15h \]
\[ 17 = 17h \]
Step 4: Divide by 17.
\[ h = 1 \]
### Equation B:
\[ 2(p - 3) - 1 = \frac{6p}{2} + \frac{14}{2} \]
Step 1: Simplify the right side.
\[ \frac{6p}{2} = 3p \quad \text{and} \quad \frac{14}{2} = 7 \]
So, simplifying the equation we get:
\[ 2(p - 3) - 1 = 3p + 7 \]
Step 2: Distribute on the left side.
\[ 2p - 6 - 1 = 3p + 7 \]
\[ 2p - 7 = 3p + 7 \]
Step 3: Rearrange to isolate \(p\).
\[ 2p - 3p = 7 + 7 \]
\[ -p = 14 \]
Step 4: Divide by -1.
\[ p = -14 \]
### Equation C:
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]
Step 1: Combine the constant terms on the left.
\[ -3 + \frac{8}{3} = -\frac{9}{3} + \frac{8}{3} = -\frac{1}{3} \]
So we rewrite the equation as:
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]
Step 2: Add \(\frac{1}{3}\) to both sides.
\[ \frac{2}{3}v = -1 + \frac{1}{3} = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3} \]
Step 3: Multiply both sides by \(\frac{3}{2}\) to isolate \(v\).
\[ v = -1 \]
### Summary of Solutions:
- \(h = 1\) (from Equation A)
- \(p = -14\) (from Equation B)
- \(v = -1\) (from Equation C)
### Claims Analysis:
1. Tamara says Equation A and Equation C have the same solution. **(False)**
- \(h = 1\), \(v = -1\)
2. Payton says Equation B and Equation C have the same solution. **(False)**
- \(p = -14\), \(v = -1\)
3. Krish says all three have the same solution. **(False)**
4. Haruki says that none of the equations have the same solution. **(True)**
### Conclusion:
Haruki is correct, as none of the equations share the same solution.