Let's solve each of the equations to determine their solution sets and identify who is correct.
Equation A: \( 3(4−5h)−7=−12+2h \)
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Distribute on the left side: \[ 3 \times 4 - 3 \times 5h - 7 = -12 + 2h \] \[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \]
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Rearrange the equation: \[ 5 + 12 = 2h + 15h \] \[ 17 = 17h \]
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Solve for \( h \): \[ h = 1 \]
So the solution for Equation A is \( h = 1 \).
Equation B: \( 2(p−3)−1=6p+\frac{14}{2} \)
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Simplify \( \frac{14}{2} \): \[ 2(p - 3) - 1 = 6p + 7 \]
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Distribute on the left side: \[ 2p - 6 - 1 = 6p + 7 \] \[ 2p - 7 = 6p + 7 \]
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Rearrange the equation: \[ 2p - 6p = 7 + 7 \] \[ -4p = 14 \]
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Solve for \( p \): \[ p = -\frac{14}{4} = -\frac{7}{2} \]
So the solution for Equation B is \( p = -\frac{7}{2} \).
Equation C: \( \frac{2}{3}v−3+\frac{8}{3}=−1 \)
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Combine like terms: \[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \] \[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \] \[ \frac{2}{3}v - 3 + 2.6667 = -1 \quad (as , \frac{8}{3} = 2.6667) \] \[ \frac{2}{3}v - 0.3333 = -1 \]
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Add 3: \[ \frac{2}{3}v = -1 + 0.3333 \] \[ \frac{2}{3}v = -0.6667 \]
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Multiply both sides by \(\frac{3}{2}\): \[ v = -0.6667 \frac{3}{2} \] \[ v = -1 \]
So the solution for Equation C is \( v = -1 \).
Summary of Solutions
- Equation A: \( h = 1 \)
- Equation B: \( p = -\frac{7}{2} \)
- Equation C: \( v = -1 \)
Now we evaluate the claims:
- Tamara: Incorrect, because A and C have different solutions.
- Payton: Incorrect, because B and C have different solutions.
- Krish: Incorrect, because all equations have different solutions.
- Haruki: Correct, none of the equations have the same solution.
Final Answer:
Haruki is correct.