Question

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

Tamara
Tamara

Krish
Krish

Payton
Payton

Haruki

Answer 1

To determine who is correct about the solution sets of the equations, we must solve each equation and see if they yield the same solution.

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

First, simplify the left side:
\[ 3(4) - 3(5h) - 7 = 12 - 15h - 7 = 5 - 15h \]

Now the equation looks like this:
\[ 5 - 15h = -12 + 2h \]

Now, move the terms involving \(h\) to one side:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]

Now, divide by 17:
\[ h = 1 \]

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

Simplify the left side:
\[ 2p - 6 - 1 = 6p + 142 \]
\[ 2p - 7 = 6p + 142 \]

Now, move the terms involving \(p\) to one side:
\[ -7 - 142 = 6p - 2p \]
\[ -149 = 4p \]

Now, divide by 4:
\[ p = -\frac{149}{4} \]

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

First, simplify:
\[ 23v + 80 = -1 \]

Now, move constant terms:
\[ 23v = -1 - 80 \]
\[ 23v = -81 \]

Now, divide by 23:
\[ v = -\frac{81}{23} \]

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

Since the solutions for all three equations are different, Haruki is correct that none of the equations have the same solution.

Final answer: **Haruki**