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

Haruki
Haruki

Tamara
Tamara

Payton

Answer 1

To determine who is correct about the solution sets of the equations, we need to solve each equation and compare their solutions.

**Equation A:**
\[ 3(4-5h) - 7 = -12 + 2h \]
Distributing on the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
Combine like terms:
\[ 5 - 15h = -12 + 2h \]
Add \(15h\) to both sides and move constants:
\[ 5 + 12 = 2h + 15h \]
\[ 17 = 17h \]
Dividing both sides by \(17\):
\[ h = 1 \]

**Equation B:**
\[ 2(p-3) - 1 = 6p + 142 \]
Distributing on the left side:
\[ 2p - 6 - 1 = 6p + 142 \]
Combine like terms:
\[ 2p - 7 = 6p + 142 \]
Subtract \(6p\) from both sides:
\[ 2p - 6p - 7 = 142 \]
\[ -4p - 7 = 142 \]
Adding 7 to both sides:
\[ -4p = 149 \]
Dividing both sides by \(-4\):
\[ p = -\frac{149}{4} \]

**Equation C:**
\[ 23v - 3 + 83 = -1 \]
Combine constants on the left side:
\[ 23v + 80 = -1 \]
Subtract \(80\) from both sides:
\[ 23v = -81 \]
Dividing both sides by \(23\):
\[ v = -\frac{81}{23} \]

Now we have the solutions:
- From Equation A: \( h = 1 \)
- From Equation B: \( p = -\frac{149}{4} \)
- From Equation C: \( v = -\frac{81}{23} \)

Since all three equations have different solutions:
- Tamara is incorrect (Equation A and C do not have the same solution).
- Payton is incorrect (Equation B and C do not have the same solution).
- Krish is incorrect (not all three have the same solution).
- Haruki is correct (none of the equations have the same solution).

Therefore, the correct answer is: **Haruki**.