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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 + 14/2
Equation C: 2/3v - 3 + 8/3 = -1

1 answer

Let's analyze each of the equations to determine their solution sets.

Equation A:

\[ 3(4 - 5h) - 7 = -12 + 2h \] First, we'll simplify the left-hand side:

  1. Distribute: \[ 3(4) - 3(5h) - 7 = -12 + 2h \] \[ 12 - 15h - 7 = -12 + 2h \]
  2. Combine like terms: \[ 5 - 15h = -12 + 2h \]
  3. Add \(15h\) to both sides: \[ 5 = -12 + 17h \]
  4. Add \(12\) to both sides: \[ 17 = 17h \]
  5. Divide by \(17\): \[ h = 1 \]

So, the solution for Equation A is \( h = 1 \).

Equation B:

\[ 2(p - 3) - 1 = 6p + \frac{14}{2} \] We'll simplify the left-hand side as follows:

  1. Distribute: \[ 2p - 6 - 1 = 6p + 7 \]
  2. Combine like terms: \[ 2p - 7 = 6p + 7 \]
  3. Subtract \(2p\) from both sides: \[ -7 = 4p + 7 \]
  4. Subtract \(7\) from both sides: \[ -14 = 4p \]
  5. Divide by \(4\): \[ p = -\frac{14}{4} \] \[ p = -\frac{7}{2} \]

So, the solution for Equation B is \( p = -\frac{7}{2} \).

Equation C:

\[ \frac{2}{3}v - 3 + \frac{8}{3} = -1 \] Combine like terms on the left-hand side:

  1. Convert \(-3\) to a fraction: \[ -3 = -\frac{9}{3} \] So the equation is: \[ \frac{2}{3}v - \frac{9}{3} + \frac{8}{3} = -1 \]
  2. Combine the constants: \[ \frac{2}{3}v - \frac{1}{3} = -1 \]
  3. Add \(\frac{1}{3}\) to both sides: \[ \frac{2}{3}v = -1 + \frac{1}{3} \] Convert \(-1\) into thirds: \[ -1 = -\frac{3}{3} \] Therefore: \[ \frac{2}{3}v = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3} \]
  4. Multiply by \(\frac{3}{2}\): \[ v = -1 \]

So, the solution for Equation C is \( v = -1 \).

Summary of Solutions:

  • Equation A: \( h = 1 \)
  • Equation B: \( p = -\frac{7}{2} \)
  • Equation C: \( v = -1 \)

Conclusion:

  • Tamara: Incorrect (A and C have different solutions)
  • Payton: Incorrect (B and C have different solutions)
  • Krish: Incorrect (All three have different solutions)
  • Haruki: Correct (None of the equations have the same solution)

Thus, Haruki is the one who is correct.