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.

Question

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

Tamara
Tamara

Krish
Krish

Payton
Payton

Haruki
Haruki
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Answer 1

To determine who is correct about the solution sets of the equations, we need to solve each equation one by one.

Equation A:

\[ 3(4 - 5h) - 7 = -12 + 2h \]

First, distribute on the left side:

\[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \]

Now, let's collect the terms involving \( h \):

\[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \]

Now divide both sides by 17:

\[ h = 1 \]

Equation B:

\[ 2(p - 3) - 1 = 6p + 142 \]

First, distribute on the left side:

\[ 2p - 6 - 1 = 6p + 142 \] \[ 2p - 7 = 6p + 142 \]

Now, let's collect the terms involving \( p \):

\[ -7 - 142 = 6p - 2p \] \[ -149 = 4p \]

Now divide both sides by 4:

\[ p = -\frac{149}{4} \]

Equation C:

\[ 23v - 3 + 83 = -1 \]

Combine like terms on the left side:

\[ 23v + 80 = -1 \]

Subtract 80 from both sides:

\[ 23v = -1 - 80 \] \[ 23v = -81 \]

Now divide both sides by 23:

\[ v = -\frac{81}{23} \]

Summary of Solutions:

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

Conclusion:

• Tamara said A and C are the same (False).
• Payton said B and C are the same (False).
• Krish said all three are the same (False).
• Haruki said none of the equations have the same solution (True).

The correct answer is that Haruki is correct.