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: 23v−3+83=−1

(1 point)
Responses

Tamara
Tamara

Haruki
Haruki

Krish
Krish

Payton

Answer 1

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

**Equation A:**
\[ 3(4-5h) - 7 = -12 + 2h \]
Expanding the left side:
\[ 12 - 15h - 7 = -12 + 2h \]
Combining like terms:
\[ 5 - 15h = -12 + 2h \]
Bringing all terms involving \(h\) to one side and constant terms to the other side:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]
Dividing both sides by 17:
\[ h = 1 \]

**Equation B:**
\[ 2(p-3) - 1 = \frac{6p + 14}{2} \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 2(2(p-3) - 1) = 6p + 14 \]
Expanding:
\[ 4p - 6 - 2 = 6p + 14 \]
Combining like terms:
\[ 4p - 8 = 6p + 14 \]
Bringing all terms involving \(p\) to one side:
\[ 4p - 6p = 14 + 8 \]
\[ -2p = 22 \]
Dividing both sides by -2:
\[ p = -11 \]

**Equation C:**
\[ 23v - 3 + 83 = -1 \]
Combining like terms:
\[ 23v + 80 = -1 \]
Bringing 80 to the other side:
\[ 23v = -1 - 80 \]
\[ 23v = -81 \]
Dividing both sides by 23:
\[ v = -\frac{81}{23} \]

Now we have:
- Equation A has solution \(h = 1\)
- Equation B has solution \(p = -11\)
- Equation C has solution \(v = -\frac{81}{23}\)

Based on these results:
- Tamara is incorrect (A and C do not have the same solution).
- Payton is incorrect (B and C do not have the same solution).
- Krish is incorrect (A, B, and C do not have the same solution).
- Haruki is correct (none of the equations have the same solution).

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