Question
Thomas, Petra, Holland, and Dayton are analyzing solution sets for linear equations.
Thomas says that Equation A and Equation C have the same solution. Petra says that Equation B and Equation C have the same solution, Holland says that all three have the same solution. Dayton says that none of the equations have the same solution.
Identify who is correct about linear equations. Then, explain how you determined who was correct below
Equation A: -12 + 3x + 2 = x + 2
Equation B: 8 - 3 (x - 1) = 2
Equation C. -2 + 4х = 10
(1 point)
• Thomas
© Petra
• Holland
© Dayton
Which student was correct? How did you determine which student was correct?
Thomas says that Equation A and Equation C have the same solution. Petra says that Equation B and Equation C have the same solution, Holland says that all three have the same solution. Dayton says that none of the equations have the same solution.
Identify who is correct about linear equations. Then, explain how you determined who was correct below
Equation A: -12 + 3x + 2 = x + 2
Equation B: 8 - 3 (x - 1) = 2
Equation C. -2 + 4х = 10
(1 point)
• Thomas
© Petra
• Holland
© Dayton
Which student was correct? How did you determine which student was correct?
Answers
GPT-4o mini
To determine who is correct among Thomas, Petra, Holland, and Dayton, we need to solve each equation and compare their solution sets.
### Equation A: \(-12 + 3x + 2 = x + 2\)
Simplifying:
\[
-12 + 3x + 2 = x + 2 \\
-10 + 3x = x + 2 \\
3x - x = 2 + 10 \\
2x = 12 \\
x = 6
\]
Thus, the solution for Equation A is \(x = 6\).
### Equation B: \(8 - 3(x - 1) = 2\)
Simplifying:
\[
8 - 3(x - 1) = 2 \\
8 - 3x + 3 = 2 \\
11 - 3x = 2 \\
-3x = 2 - 11 \\
-3x = -9 \\
x = 3
\]
The solution for Equation B is \(x = 3\).
### Equation C: \(-2 + 4x = 10\)
Simplifying:
\[
-2 + 4x = 10 \\
4x = 10 + 2 \\
4x = 12 \\
x = 3
\]
The solution for Equation C is also \(x = 3\).
### Summary of Solutions:
- **Equation A:** \(x = 6\)
- **Equation B:** \(x = 3\)
- **Equation C:** \(x = 3\)
### Comparing the Solutions:
- Thomas claims that Equation A and C have the same solution (False: A has \(x = 6\), C has \(x = 3\)).
- Petra claims that Equation B and C have the same solution (True: both have \(x = 3\)).
- Holland claims that all three equations have the same solution (False: each has different solutions).
- Dayton claims that none of the equations have the same solution (True).
Based on this analysis, **Petra and Dayton are correct.**
However, since the question asks for a single student who is correct regarding the solution sets being the same, Petra is the only one who correctly identified at least one pair of equations with the same solution (B and C). Thus, Petra is the student most appropriately recognized for correctness, while Dayton also holds a true statement about the unique solutions across all equations.
In conclusion, the answer is:
- **Correct Student:** Petra
### Equation A: \(-12 + 3x + 2 = x + 2\)
Simplifying:
\[
-12 + 3x + 2 = x + 2 \\
-10 + 3x = x + 2 \\
3x - x = 2 + 10 \\
2x = 12 \\
x = 6
\]
Thus, the solution for Equation A is \(x = 6\).
### Equation B: \(8 - 3(x - 1) = 2\)
Simplifying:
\[
8 - 3(x - 1) = 2 \\
8 - 3x + 3 = 2 \\
11 - 3x = 2 \\
-3x = 2 - 11 \\
-3x = -9 \\
x = 3
\]
The solution for Equation B is \(x = 3\).
### Equation C: \(-2 + 4x = 10\)
Simplifying:
\[
-2 + 4x = 10 \\
4x = 10 + 2 \\
4x = 12 \\
x = 3
\]
The solution for Equation C is also \(x = 3\).
### Summary of Solutions:
- **Equation A:** \(x = 6\)
- **Equation B:** \(x = 3\)
- **Equation C:** \(x = 3\)
### Comparing the Solutions:
- Thomas claims that Equation A and C have the same solution (False: A has \(x = 6\), C has \(x = 3\)).
- Petra claims that Equation B and C have the same solution (True: both have \(x = 3\)).
- Holland claims that all three equations have the same solution (False: each has different solutions).
- Dayton claims that none of the equations have the same solution (True).
Based on this analysis, **Petra and Dayton are correct.**
However, since the question asks for a single student who is correct regarding the solution sets being the same, Petra is the only one who correctly identified at least one pair of equations with the same solution (B and C). Thus, Petra is the student most appropriately recognized for correctness, while Dayton also holds a true statement about the unique solutions across all equations.
In conclusion, the answer is:
- **Correct Student:** Petra