To determine who is correct regarding the solutions of the equations, let's solve each equation step by step.
Equation A:
Equation A: \[ 3 (4 - 5h) - 7 = -12 + 2h \]
- Distribute the 3: \[ 12 - 15h - 7 = -12 + 2h \]
- Combine constants on the left side: \[ 5 - 15h = -12 + 2h \]
- Add \(15h\) to both sides: \[ 5 = -12 + 17h \]
- Add 12 to both sides: \[ 17 = 17h \]
- Divide by 17: \[ h = 1 \]
Equation B:
Equation B: \[ 2 (p - 3) - 1 = \frac{6p + 14}{2} \]
- Distribute the 2 on the left side: \[ 2p - 6 - 1 = \frac{6p + 14}{2} \]
- Combine constants: \[ 2p - 7 = \frac{6p + 14}{2} \]
- To eliminate the fraction, multiply the entire equation by 2: \[ 2(2p - 7) = 6p + 14 \] \[ 4p - 14 = 6p + 14 \]
- Rearranging gives: \[ 4p - 6p = 14 + 14 \] \[ -2p = 28 \]
- Divide by -2: \[ p = -14 \]
Equation C:
Equation C: \[ \frac{2}{3} v - 3 + \frac{8}{3} = -1 \]
- Combine constants on the left side: \[ \frac{2}{3} v + \frac{8}{3} - 3 = -1 \]
- Rewrite 3 as \(\frac{9}{3}\) for easier calculation: \[ \frac{2}{3} v + \frac{8}{3} - \frac{9}{3} = -1 \] \[ \frac{2}{3} v - \frac{1}{3} = -1 \]
- Add \(\frac{1}{3}\) to both sides: \[ \frac{2}{3} v = -1 + \frac{1}{3} \]
- Rewrite -1 as \(-\frac{3}{3}\): \[ \frac{2}{3} v = -\frac{3}{3} + \frac{1}{3} \]
- Combine: \[ \frac{2}{3} v = -\frac{2}{3} \]
- Multiply both sides by \(\frac{3}{2}\): \[ v = -1 \]
Summary of Solutions:
- Equation A: \(h = 1\)
- Equation B: \(p = -14\)
- Equation C: \(v = -1\)
Analyzing the Claims:
- Tamara: claims A and C have the same solution → Incorrect (values differ).
- Payton: claims B and C have the same solution → Incorrect (values differ).
- Krish: claims all three have the same solution → Incorrect (values differ).
- Haruki: claims none of the equations have the same solution → Correct (all values differ).
Conclusion:
Haruki is correct; none of the equations have the same solution.