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

Krish
Krish

Tamara
Tamara

Haruki
Haruki

Payton

1 answer

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

Equation A: \[ 3(4 - 5h) - 7 = -12 + 2h \] Expanding and simplifying: \[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \] Now, let's get all terms involving \( h \) on one side and constants on the other: \[ 5 + 12 = 15h + 2h \] \[ 17 = 17h \] Dividing by 17: \[ h = 1 \]

Equation B: \[ 2(p - 3) - 1 = 6p + 142 \] Expanding: \[ 2p - 6 - 1 = 6p + 142 \] \[ 2p - 7 = 6p + 142 \] Rearranging: \[ -7 - 142 = 6p - 2p \] \[ -149 = 4p \] Dividing by -4: \[ p = -\frac{149}{4} \]

Equation C: \[ 23v - 3 + 83 = -1 \] Combining like terms: \[ 23v + 80 = -1 \] Rearranging: \[ 23v = -1 - 80 \] \[ 23v = -81 \] Dividing by 23: \[ v = -\frac{81}{23} \]

Now we have the solutions:

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

Evaluating the claims:

  • Tamara claims Equations A and C have the same solution. (This is false, since \( h \neq v \)).
  • Payton claims Equations B and C have the same solution. (This is false, since \( p \neq v \)).
  • Krish claims all three equations have the same solution. (This is false since none of the equations yield the same solution).
  • Haruki claims none of the equations have the same solution. (This is true).

Thus, Haruki is correct.