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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

Krish
Krish

Haruki
Haruki

Tamara
Tamara
Payton

1 answer

To verify the claims made by Tamara, Payton, Krish, and Haruki, we need to solve each equation and check the solution sets.

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

Let's simplify: \[ 3(4) - 3(5h) - 7 = -12 + 2h \] \[ 12 - 15h - 7 = -12 + 2h \] \[ 5 - 15h = -12 + 2h \]

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

Add 12 to both sides: \[ 17 = 17h \]

Divide both sides by 17: \[ h = 1 \]

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

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

Let's simplify: \[ 2p - 6 - 1 = 6p + 142 \] \[ 2p - 7 = 6p + 142 \]

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

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

Divide both sides by 4: \[ p = -\frac{149}{4} \]

So the solution for Equation B is \( p = -\frac{149}{4} \).

Equation C: \[ 23v - 3 + 83 = -1 \]

Let's simplify: \[ 23v + 80 = -1 \]

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

Divide both sides by 23: \[ v = -\frac{81}{23} \]

So the solution for Equation C is \( v = -\frac{81}{23} \).

Now let's compare the solutions:

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

None of the equations have the same solutions. Therefore, Haruki is correct.

So the answer is: Haruki.