Let's solve each equation to determine their solution sets.
Equation A: \[ 3(4 - 5h) - 7 = -12 + 2h \]
Start by distributing on the left side: \[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \]
Now, move all terms involving \( h \) to one side and constant terms to the other: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \]
Dividing both sides by 17: \[ h = 1 \]
Thus, the solution to Equation A is \( h = 1 \).
Equation B: \[ 2(p - 3) - 1 = 6p + 142 \]
Distributing on the left side: \[ 2p - 6 - 1 = 6p + 142 \] \[ 2p - 7 = 6p + 142 \]
Now, move all terms involving \( p \) to one side: \[ -7 - 142 = 6p - 2p \] \[ -149 = 4p \]
Dividing both sides by 4: \[ p = -\frac{149}{4} \]
Thus, the solution to Equation B is \( p = -\frac{149}{4} \).
Equation C: \[ 23v - 3 + 83 = -1 \]
Simplifying the left side: \[ 23v + 80 = -1 \]
Subtract 80 from both sides: \[ 23v = -1 - 80 \] \[ 23v = -81 \]
Now divide by 23: \[ v = -\frac{81}{23} \]
Thus, the solution to Equation C is \( v = -\frac{81}{23} \).
Now, let's summarize the results:
- Equation A has the solution \( h = 1 \).
- Equation B has the solution \( p = -\frac{149}{4} \).
- Equation C has the solution \( v = -\frac{81}{23} \).
Comparing the solutions:
- The solutions for Equations A, B, and C are all different.
Conclusion: Haruki is correct; none of the equations have the same solution. The correct response is Haruki.
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