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

(1 point)
Responses

Haruki

Payton

Tamara

Krish

Answer 1

To identify who is correct about the solution sets of the linear equations, we will solve each equation one by one.

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

Expanding the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
\[ 5 - 15h = -12 + 2h \]

Now, rearranging gives:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]

Solving for \( h \):
\[ h = 1 \]

**Equation B:**
\[ 2(p-3) - 1 = 6p + 142 \]

Expanding the left side:
\[ 2p - 6 - 1 = 6p + 142 \]
\[ 2p - 7 = 6p + 142 \]

Now, rearranging:
\[ -7 - 142 = 6p - 2p \]
\[ -149 = 4p \]

Solving for \( p \):
\[ p = -\frac{149}{4} \]

**Equation C:**
\[ 23v - 3 + 83 = -1 \]
\[ 23v + 80 = -1 \]

Now, rearranging gives:
\[ 23v = -1 - 80 \]
\[ 23v = -81 \]

Solving for \( v \):
\[ v = -\frac{81}{23} \]

Now we can summarize the solutions:
- **Equation A** has the solution \( h = 1 \).
- **Equation B** has the solution \( p = -\frac{149}{4} \).
- **Equation C** has the solution \( v = -\frac{81}{23} \).

Now let's analyze the claims:
- **Tamara** says Equation A and Equation C have the same solution. (False)
- **Payton** says Equation B and Equation C have the same solution. (False)
- **Krish** says all three have the same solution. (False)
- **Haruki** says that none of the equations have the same solution. (True)

In conclusion, the correct answer is **Haruki**.