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)

Answer 1

Let's analyze each equation to determine the solution set for each one.

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

First, simplify the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
\[ 5 - 15h = -12 + 2h \]
Now, combine like terms:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]
Divide both sides by 17:
\[ h = 1 \]

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

First, simplify the left side:
\[ 2p - 6 - 1 = \frac{6p + 14}{2} \]
\[ 2p - 7 = 3p + 7 \]
Now, combine like terms:
\[ 2p - 3p = 7 + 7 \]
\[ -p = 14 \]
Multiply both sides by -1:
\[ p = -14 \]

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

First, combine the constants:
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]
Convert -3 to a fraction with a denominator of 3:
\[ \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = -1 \]
Combine the constants:
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]

Now, add \(\frac{1}{3}\) to both sides:
\[ \frac{2}{3}v = -1 + \frac{1}{3} \]
Convert -1 to a fraction with a denominator of 3:
\[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \]
Combine the constants:
\[ \frac{2}{3}v = -\frac{2}{3} \]
Multiply both sides by \(\frac{3}{2}\):
\[ v = -1 \]

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

Now let's evaluate the claims:
- **Tamara** claims that Equation A and Equation C have the same solution. Incorrect.
- **Payton** claims that Equation B and Equation C have the same solution. Incorrect.
- **Krish** claims that all three have the same solution. Incorrect.
- **Haruki** claims that none of the equations have the same solution. Correct.

Thus, **Haruki is correct**: none of the equations have the same solution.