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

Haruki
Haruki
Tamara
Tamara
Payton
Payton
Krish

Answer 1

To determine whose assertion about the equations A, B, and C is correct, we need to solve each equation and find their respective solution sets.

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

1. Distribute: \( 12 - 15h - 7 = -12 + 2h \)
2. Combine like terms: \( 5 - 15h = -12 + 2h \)
3. Add \( 15h \) and \( 12 \) to both sides: \( 5 + 12 = 15h + 2h \)
4. Simplify: \( 17 = 17h \)
5. Divide by 17: \( h = 1 \)

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

1. Distribute: \( 2p - 6 - 1 = \frac{6p + 14}{2} \)
2. Combine like terms: \( 2p - 7 = \frac{6p + 14}{2} \)
3. Multiply both sides by 2 to eliminate the fraction: \( 2(2p - 7) = 6p + 14 \)
4. Distribute: \( 4p - 14 = 6p + 14 \)
5. Rearranging gives: \( 4p - 6p = 14 + 14 \)
6. Combine like terms: \( -2p = 28 \)
7. Divide by -2: \( p = -14 \)

### Equation C:
\( \frac{2}{3}v - 3 + \frac{8}{3} = -1 \)

1. Combine like terms: \( \frac{2}{3}v - 3 + \frac{8}{3} = -1 \)
2. Convert -3 to thirds: \( -3 = -\frac{9}{3} \), so the equation becomes:
\[ \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = -1 \]
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]
3. Add \(\frac{1}{3}\) to both sides:
\[ \frac{2}{3}v = -1 + \frac{1}{3} \]
\[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \]
\[ \frac{2}{3}v = -\frac{2}{3} \]
4. Multiply both sides by \(\frac{3}{2}\):
\[ v = -1 \]

### Summary of Solutions:
- Equation A: \( h = 1 \)
- Equation B: \( p = -14 \)
- Equation C: \( v = -1 \)

### Conclusion:
- Equation A, B, and C all have different solutions (h=1, p=-14, and v=-1 respectively).
- **Haruki is correct:** none of the equations have the same solution.

So the answer is **Haruki**.