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/2+14/2 Equation C: 2/3v−3+8/3=−1

Answer 1

To solve the equations and evaluate the claims made by Tamara, Payton, Krish, and Haruki, let’s simplify each equation one by one.

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

Step 1: Distribute on the left side.
\[ 12 - 15h - 7 = -12 + 2h \]

Step 2: Combine like terms.
\[ 5 - 15h = -12 + 2h \]

Step 3: Rearrange to isolate \(h\).
\[ 5 + 12 = 2h + 15h \]
\[ 17 = 17h \]

Step 4: Divide by 17.
\[ h = 1 \]

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

Step 1: Simplify the right side.
\[ \frac{6p}{2} = 3p \quad \text{and} \quad \frac{14}{2} = 7 \]

So, simplifying the equation we get:
\[ 2(p - 3) - 1 = 3p + 7 \]

Step 2: Distribute on the left side.
\[ 2p - 6 - 1 = 3p + 7 \]
\[ 2p - 7 = 3p + 7 \]

Step 3: Rearrange to isolate \(p\).
\[ 2p - 3p = 7 + 7 \]
\[ -p = 14 \]

Step 4: Divide by -1.
\[ p = -14 \]

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

Step 1: Combine the constant terms on the left.
\[ -3 + \frac{8}{3} = -\frac{9}{3} + \frac{8}{3} = -\frac{1}{3} \]

So we rewrite the equation as:
\[ \frac{2}{3}v - \frac{1}{3} = -1 \]

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

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

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

### Claims Analysis:
1. Tamara says Equation A and Equation C have the same solution. **(False)**
- \(h = 1\), \(v = -1\)
2. Payton says Equation B and Equation C have the same solution. **(False)**
- \(p = -14\), \(v = -1\)
3. Krish says all three have the same solution. **(False)**
4. Haruki says that none of the equations have the same solution. **(True)**

### Conclusion:
Haruki is correct, as none of the equations share the same solution.