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+14/2

Equation C: 2/3v−3+8/3=−1

(1 point)
Responses

Payton
Payton

Haruki
Haruki

Krish
Krish

Tamara

Answer 1

To determine who is correct about the solution sets of the equations, we need to solve each equation and see if they have the same solutions.

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

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

Bringing all terms involving \( h \) to one side:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]

Dividing by 17:
\[ h = 1 \]

**Solution A: \( h = 1 \)**

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**Equation B:**
\[ 2(p - 3) - 1 = \frac{6p + 14}{2} \]

Multiplying through by 2 to eliminate the fraction:
\[ 4(p - 3) - 2 = 6p + 14 \]
\[ 4p - 12 - 2 = 6p + 14 \]
\[ 4p - 14 = 6p + 14 \]

Bringing all terms involving \( p \) to one side:
\[ 4p - 6p = 14 + 14 \]
\[ -2p = 28 \]

Dividing by -2:
\[ p = -14 \]

**Solution B: \( p = -14 \)**

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**Equation C:**
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]

Adding 3 to both sides:
\[ \frac{2}{3}v + \frac{8}{3} = 2 \]

Multiplying through by 3 to eliminate the fraction:
\[ 2v + 8 = 6 \]
\[ 2v = 6 - 8 \]
\[ 2v = -2 \]

Dividing by 2:
\[ v = -1 \]

**Solution C: \( v = -1 \)**

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Now we compare the solutions:
- **Equation A:** \( h = 1 \)
- **Equation B:** \( p = -14 \)
- **Equation C:** \( v = -1 \)

No two equations have the same solution:

- Tamara (A and C have the same solution) is incorrect.
- Payton (B and C have the same solution) is incorrect.
- Krish (all three have the same solution) is incorrect.
- Haruki (none have the same solution) is correct.

Therefore, the correct response is:

**Haruki**