To determine who is correct about the solution sets of the three equations, we need to solve each equation.
Equation A: \[ 3(4 - 5h) - 7 = -12 + 2h \]
First, distribute the 3: \[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \]
Now, combine like terms: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \]
Dividing by 17 gives: \[ h = 1 \]
Equation B: \[ 2(p - 3) - 1 = \frac{6p + 14}{2} \]
Distribute the 2: \[ 2p - 6 - 1 = \frac{6p + 14}{2} \] \[ 2p - 7 = 3p + 7 \] (Multiplying both sides of the right equation by 2)
Now, combine like terms: \[ -7 - 7 = 3p - 2p \] \[ -14 = p \]
Equation C: \[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]
Firstly, combine like terms on the left: \[ \frac{2}{3}v - 3 + \frac{8}{3} = \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = \frac{2}{3}v - \frac{1}{3} \]
Next, set the equation equal to \(-1\): \[ \frac{2}{3}v - \frac{1}{3} = -1 \]
Add \(\frac{1}{3}\) to both sides: \[ \frac{2}{3}v = -1 + \frac{1}{3} \] \[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3} \]
Multiply both sides by \(\frac{3}{2}\): \[ v = -1 \]
Solutions Summary:
- Equation A has a solution: \( h = 1 \)
- Equation B has a solution: \( p = -14 \)
- Equation C has a solution: \( v = -1 \)
Verification of Statements:
- Tamara believes A and C have the same solution (incorrect).
- Payton believes B and C have the same solution (incorrect).
- Krish believes all three have the same solution (incorrect).
- Haruki believes none of the equations have the same solution (correct).
Conclusively, Haruki is correct.
1 answer