Ask a New Question
Menu

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 point)
Responses

A) Tamara

B) Haruki

C) Payton

D) Krish

1 answer

To determine who is correct about the solution sets of the equations, we will solve each equation step-by-step.

Equation A:

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

Expanding and simplifying the left side:

  1. Distributing: \( 34 - 35h - 7 = 12 - 15h - 7 = 5 - 15h \)
  2. Setting the equation: \( 5 - 15h = -12 + 2h \)

Bringing all terms involving \( h \) to one side and constants to the other:

  1. \( 5 + 12 = 15h + 2h \)
  2. \( 17 = 17h \)
  3. \( h = 1 \)

Equation B:

\( 2(p - 3) - 1 = \frac{6p + 14}{2} \)

Start by simplifying the left side:

  1. \( 2p - 6 - 1 = 2p - 7 \)
  2. Setting the equation: \( 2p - 7 = \frac{6p + 14}{2} \)

Multiply both sides by 2 to eliminate the fraction:

  1. \( 2(2p - 7) = 6p + 14 \)
  2. \( 4p - 14 = 6p + 14 \)

Rearranging gives:

  1. \( 4p - 6p = 14 + 14 \)
  2. \( -2p = 28 \)
  3. \( p = -14 \)

Equation C:

\( \frac{2}{3}v - 3 + \frac{8}{3} = -1 \)

First, combine constants on the left side:

  1. \( -3 + \frac{8}{3} = -\frac{9}{3} + \frac{8}{3} = -\frac{1}{3} \)
  2. So the equation is: \( \frac{2}{3}v - \frac{1}{3} = -1 \)

Adding \( \frac{1}{3} \) to both sides:

  1. \( \frac{2}{3}v = -1 + \frac{1}{3} = -\frac{3}{3} + \frac{1}{3} = -\frac{2}{3} \)

Solving for \( v \):

  1. \( v = -\frac{2}{3} \div \frac{2}{3} = -1 \)

Summary of Solutions:

  • Equation A: \( h = 1 \)
  • Equation B: \( p = -14 \)
  • Equation C: \( v = -1 \)

Conclusion:

  • Tamara claims Equations A and C have the same solution: False (1 vs -1).
  • Payton claims Equations B and C have the same solution: False (-14 vs -1).
  • Krish claims all three equations have the same solution: False.
  • Haruki claims none of the equations have the same solution: True.

Thus, the correct answer is:

B) Haruki.