To determine who is correct about the solution sets of the equations, we must solve each equation and see if they yield the same solution.
Equation A: \[ 3(4-5h) - 7 = -12 + 2h \]
First, simplify the left side: \[ 3(4) - 3(5h) - 7 = 12 - 15h - 7 = 5 - 15h \]
Now the equation looks like this: \[ 5 - 15h = -12 + 2h \]
Now, move the terms involving \(h\) to one side: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \]
Now, divide by 17: \[ h = 1 \]
Equation B: \[ 2(p-3) - 1 = 6p + 142 \]
Simplify the left side: \[ 2p - 6 - 1 = 6p + 142 \] \[ 2p - 7 = 6p + 142 \]
Now, move the terms involving \(p\) to one side: \[ -7 - 142 = 6p - 2p \] \[ -149 = 4p \]
Now, divide by 4: \[ p = -\frac{149}{4} \]
Equation C: \[ 23v - 3 + 83 = -1 \]
First, simplify: \[ 23v + 80 = -1 \]
Now, move constant terms: \[ 23v = -1 - 80 \] \[ 23v = -81 \]
Now, divide by 23: \[ v = -\frac{81}{23} \]
Now we have:
- For Equation A: \( h = 1 \)
- For Equation B: \( p = -\frac{149}{4} \)
- For Equation C: \( v = -\frac{81}{23} \)
Since the solutions for all three equations are different, Haruki is correct that none of the equations have the same solution.
Final answer: Haruki