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

Tamara
Tamara

Haruki
Haruki

Krish
Krish

Payton

Answer 1

Let's solve each of the equations to determine their solution sets and identify who is correct.

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

1. Distribute on the left side:
\[
3 \times 4 - 3 \times 5h - 7 = -12 + 2h
\]
\[
12 - 15h - 7 = -12 + 2h
\]
\[
5 - 15h = -12 + 2h
\]

2. Rearrange the equation:
\[
5 + 12 = 2h + 15h
\]
\[
17 = 17h
\]

3. Solve for \( h \):
\[
h = 1
\]

So the solution for Equation A is \( h = 1 \).

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

1. Simplify \( \frac{14}{2} \):
\[
2(p - 3) - 1 = 6p + 7
\]

2. Distribute on the left side:
\[
2p - 6 - 1 = 6p + 7
\]
\[
2p - 7 = 6p + 7
\]

3. Rearrange the equation:
\[
2p - 6p = 7 + 7
\]
\[
-4p = 14
\]

4. Solve for \( p \):
\[
p = -\frac{14}{4} = -\frac{7}{2}
\]

So the solution for Equation B is \( p = -\frac{7}{2} \).

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

1. Combine like terms:
\[
\frac{2}{3}v - 3 + \frac{8}{3} = -1
\]
\[
\frac{2}{3}v - 3 + \frac{8}{3} = -1
\]
\[
\frac{2}{3}v - 3 + 2.6667 = -1 \quad (as \, \frac{8}{3} = 2.6667)
\]
\[
\frac{2}{3}v - 0.3333 = -1
\]

2. Add 3:
\[
\frac{2}{3}v = -1 + 0.3333
\]
\[
\frac{2}{3}v = -0.6667
\]

3. Multiply both sides by \(\frac{3}{2}\):
\[
v = -0.6667 \frac{3}{2}
\]
\[
v = -1
\]

So the solution for Equation C is \( v = -1 \).

### Summary of Solutions
- Equation A: \( h = 1 \)
- Equation B: \( p = -\frac{7}{2} \)
- Equation C: \( v = -1 \)

Now we evaluate the claims:

- **Tamara**: Incorrect, because A and C have different solutions.
- **Payton**: Incorrect, because B and C have different solutions.
- **Krish**: Incorrect, because all equations have different solutions.
- **Haruki**: Correct, none of the equations have the same solution.

### Final Answer:
**Haruki** is correct.