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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

Payton
Payton

Tamara
Tamara

Haruki
Haruki

Krish

1 answer

To determine who among Tamara, Payton, Krish, and Haruki is correct about the solution sets of the linear equations, we need to solve each equation.

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

  1. Distribute on the left: \[ 12 - 15h - 7 = -12 + 2h \] Simplifying gives: \[ 5 - 15h = -12 + 2h \]

  2. Add \( 15h \) and \( 12 \) to both sides: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \]

  3. Solve for \( h \): \[ h = 1 \]

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

  1. Distributing on the left: \[ 2p - 6 - 1 = 6p + 142 \] Simplifying gives: \[ 2p - 7 = 6p + 142 \]

  2. Subtract \( 2p \) from both sides: \[ -7 = 4p + 142 \]

  3. Subtract 142 from both sides: \[ -149 = 4p \]

  4. Solve for \( p \): \[ p = -\frac{149}{4} \]

Equation C: \( 23v - 3 + 83 = -1 \)

  1. Combine like terms on the left: \[ 23v + 80 = -1 \]

  2. Subtract 80 from both sides: \[ 23v = -81 \]

  3. Solve for \( v \): \[ v = -\frac{81}{23} \]

Summary of Solutions:

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

Evaluating Statements:

  • Tamara: Claims A and C have the same solution. (Incorrect).
  • Payton: Claims B and C have the same solution. (Incorrect).
  • Krish: Claims A, B, and C all have the same solution. (Incorrect).
  • Haruki: Claims that none of the equations have the same solution. (Correct).

Thus, the correct answer is Haruki.