Question

Tamara, Payton, Krish, and Haruki are analyzing the solution sets to the following linear equations. Tamara says that Equation A and Equation C have the same solution. Payton says Equation B and Equation C have the same solution. Krish says all three have the same solution. Haruki says that none of the equations have the same solution. Identify who is correct about the linear equations.

Equation A: 3(4−5h)−7=−12+2h

Equation B: 2(p−3)−1=6p+142

Equation C: 23v−3+83=−1

Answer 1

To determine who is correct about the solutions of the equations, we'll first solve each equation step-by-step.

### Equation A: \( 3(4 - 5h) - 7 = -12 + 2h \)

1. Distribute \(3\):
\[
12 - 15h - 7 = -12 + 2h
\]
2. Combine like terms:
\[
5 - 15h = -12 + 2h
\]
3. Move \(2h\) to the left side and 5 to the right side:
\[
-15h - 2h = -12 - 5
\]
\[
-17h = -17
\]
4. Divide by -17:
\[
h = 1
\]

### Equation B: \( 2(p - 3) - 1 = 6p + 142 \)

1. Distribute \(2\):
\[
2p - 6 - 1 = 6p + 142
\]
2. Combine like terms:
\[
2p - 7 = 6p + 142
\]
3. Move \(6p\) to the left side and -7 to the right side:
\[
2p - 6p = 142 + 7
\]
\[
-4p = 149
\]
4. Divide by -4:
\[
p = -\frac{149}{4} \quad (\text{or } -37.25)
\]

### Equation C: \( 23v - 3 + 83 = -1 \)

1. Combine like terms:
\[
23v + 80 = -1
\]
2. Move 80 to the right side:
\[
23v = -1 - 80
\]
\[
23v = -81
\]
3. Divide by 23:
\[
v = -\frac{81}{23}
\]

### Summary of Solutions:
- Equation A: \( h = 1 \)
- Equation B: \( p = -\frac{149}{4} \)
- Equation C: \( v = -\frac{81}{23} \)

### Conclusion:
Now we examine the statements by Tamara, Payton, Krish, and Haruki:
- **Tamara**: Claims Equations A and C have the same solution. (Incorrect, \( h \neq v \))
- **Payton**: Claims Equations B and C have the same solution. (Incorrect, \( p \neq v \))
- **Krish**: Claims all three have the same solution. (Incorrect, all are different)
- **Haruki**: Claims none of the equations have the same solution. (Correct)

The correct conclusion is that **Haruki** is the only one who is correct.