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+142

Equation C: 23v−3+83=−1

(1 point)
Responses

Krish
Krish

Payton
Payton

Tamara
Tamara

Haruki

Answer 1

To determine who is correct about the solution sets of the linear equations provided, let's solve each equation step-by-step.

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

1. Distribute the 3 on the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
2. Simplify:
\[ 5 - 15h = -12 + 2h \]
3. Add \(15h\) to both sides:
\[ 5 = -12 + 17h \]
4. Add 12 to both sides:
\[ 17 = 17h \]
5. Divide by 17:
\[ h = 1 \]

**Equation B:**
\[ 2(p - 3) - 1 = 6p + 142 \]

1. Distribute the 2 on the left side:
\[ 2p - 6 - 1 = 6p + 142 \]
2. Combine like terms:
\[ 2p - 7 = 6p + 142 \]
3. Subtract \(2p\) from both sides:
\[ -7 = 4p + 142 \]
4. Subtract 142 from both sides:
\[ -149 = 4p \]
5. Divide by 4:
\[ p = -\frac{149}{4} \]

**Equation C:**
\[ 23v - 3 + 83 = -1 \]

1. Simplify the left side:
\[ 23v + 80 = -1 \]
2. Subtract 80 from both sides:
\[ 23v = -81 \]
3. Divide by 23:
\[ v = -\frac{81}{23} \]

Now we have:

- \( h = 1 \) (from Equation A)
- \( p = -\frac{149}{4} \) (from Equation B)
- \( v = -\frac{81}{23} \) (from Equation C)

Now we can analyze the statements made by Tamara, Payton, Krish, and Haruki:

- **Tamara** says that Equation A and Equation C have the same solution. (Incorrect, \( h \neq v \))
- **Payton** says Equation B and Equation C have the same solution. (Incorrect, \( p \neq v \))
- **Krish** says all three have the same solution. (Incorrect)
- **Haruki** says that none of the equations have the same solution. (Correct)

Therefore, the correct answer is:

**Haruki**.