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

Payton
Payton

Tamara
Tamara

Haruki
Haruki

Krish

Answer 1

To determine who is correct regarding the solutions to the equations, we'll first solve each equation separately.

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

First, distribute:
\[ 12 - 15h - 7 = -12 + 2h \]

Combine like terms:
\[ 5 - 15h = -12 + 2h \]

Rearranging gives:
\[ 5 + 12 = 15h + 2h \]
\[ 17 = 17h \]

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

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

Distribute:
\[ 2p - 6 - 1 = 6p + 142 \]

Combine like terms:
\[ 2p - 7 = 6p + 142 \]

Rearranging gives:
\[ -7 - 142 = 6p - 2p \]
\[ -149 = 4p \]

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

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

Combine like terms:
\[ 23v + 80 = -1 \]

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

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

### **Review of Solutions:**
- **Equation A:** \( h = 1 \)
- **Equation B:** \( p = -\frac{149}{4} \)
- **Equation C:** \( v = -\frac{81}{23} \)

### **Analyzing the Claims:**
1. **Tamara:** Claims A and C have the same solution. (False: \( h \neq v \))
2. **Payton:** Claims B and C have the same solution. (False: \( p \neq v \))
3. **Krish:** Claims A, B, and C all have the same solution. (False: All values are different)
4. **Haruki:** Claims none of the equations have the same solution. (True: \( h \), \( p \), and \( v \) are all unique)

### **Conclusion:**
Haruki is correct; none of the equations have the same solution.