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

Krish
Krish

Tamara
Tamara

Haruki
Haruki

Payton

Answer 1

To determine who is correct about the linear equations, we will solve each equation and compare their solution sets.

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

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

Next, add \(15h\) and \(12\) to both sides:
\[ 5 + 12 = 2h + 15h \]
\[ 17 = 17h \]

Now, divide by 17:
\[ h = 1 \]

### Equation B:
\[ 2(p - 3) - 1 = 6p + \frac{14}{2} \]
First, simplify the right side:
\[ \frac{14}{2} = 7 \]

Now we can rewrite the equation:
\[ 2(p - 3) - 1 = 6p + 7 \]

Distribute:
\[ 2p - 6 - 1 = 6p + 7 \]
\[ 2p - 7 = 6p + 7 \]

Next, subtract \(2p\) from both sides:
\[ -7 = 4p + 7 \]

Now, subtract 7 from both sides:
\[ -14 = 4p \]

Finally, divide by 4:
\[ p = -\frac{7}{2} \]

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

First, simplify the left side:
\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \]

Convert \(-3\) into a fraction with a denominator of 3:
\[ -3 = -\frac{9}{3} \]

Now combine:
\[ \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = -1 \]
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]

Add \( \frac{1}{3} \) to both sides:
\[ \frac{2}{3}v = -1 + \frac{1}{3} \]

Convert \(-1\) into a fraction:
\[ -1 = -\frac{3}{3} \]
\[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \]
\[ \frac{2}{3}v = -\frac{2}{3} \]

Now multiply by \(\frac{3}{2}\) to solve for \(v\):
\[ v = -1 \]

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

Now we can evaluate the claims:

- **Tamara** states that Equation A and Equation C have the same solution. This is **false**.
- **Payton** states Equation B and Equation C have the same solution. This is **false**.
- **Krish** states all three have the same solution. This is **false**.
- **Haruki** states that none of the equations have the same solution. This is **true**.

Thus, the correct answer is **Haruki**.