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: 2/3 v−3+8/3=−1

(1 point)
Responses

Haruki
Haruki

Tamara
Tamara

Payton
Payton

Krish

1 answer

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

Equation A: \[ 3(4 - 5h) - 7 = -12 + 2h \] Distributing the 3: \[ 12 - 15h - 7 = -12 + 2h \] Simplifying: \[ 5 - 15h = -12 + 2h \] Adding \(15h\) and \(12\) to both sides: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \] Dividing both sides by 17: \[ h = 1 \]

Equation B: \[ 2(p - 3) - 1 = \frac{6p + 14}{2} \] Distributing the 2: \[ 2p - 6 - 1 = 3p + 7 \] Simplifying: \[ 2p - 7 = 3p + 7 \] Subtracting \(2p\) from both sides: \[ -7 = p + 7 \] Subtracting \(7\) from both sides: \[ -14 = p \]

Equation C: \[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \] Combining constants on the left: \[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \] Changing -3 to have a denominator of 3: \[ \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = -1 \] Combining like terms: \[ \frac{2}{3}v - \frac{1}{3} = -1 \] Adding \(\frac{1}{3}\) to both sides: \[ \frac{2}{3}v = -1 + \frac{1}{3} \] Changing -1 to have a denominator of 3: \[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} \] Thus: \[ \frac{2}{3}v = -\frac{2}{3} \] Multiplying both sides by \(\frac{3}{2}\): \[ v = -1 \]

Summarizing the results:

  • Equation A has the solution \(h = 1\).
  • Equation B has the solution \(p = -14\).
  • Equation C has the solution \(v = -1\).

From this information:

  • The solutions are different for all three equations: \(h = 1\), \(p = -14\), \(v = -1\).

Therefore, Haruki is correct in saying that none of the equations have the same solution.

Final answer: Haruki