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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
−
12
+
3
𝑥
+
2
=
𝑥
+
2

Equation B: 8−3(x−1)=2
8
−
3
(
𝑥
−
1
)
=
2

Equation C: −2+4x=10
−
2
+
4
𝑥
=
10

(1 point)
Responses

Thomas
Thomas

Petra
Petra

Holland
Holland

Dayton
Dayton
Question 2
Which student was correct? How did you determine which student was correct?(2 points)

1 answer

To determine who among Thomas, Petra, Holland, and Dayton is correct about the solution sets of the equations, we need to solve each equation and check their solutions.

  1. Equation A: \[ -12 + 3x + 2 = x + 2 \] Simplifying: \[ 3x - 10 = x + 2 \] Subtract \(x\) from both sides: \[ 2x - 10 = 2 \] Adding 10 to both sides: \[ 2x = 12 \] Dividing by 2: \[ x = 6 \]

  2. Equation B: \[ 8 - 3(x - 1) = 2 \] Distributing \(-3\): \[ 8 - 3x + 3 = 2 \] Combine like terms: \[ 11 - 3x = 2 \] Subtract 11 from both sides: \[ -3x = -9 \] Dividing by \(-3\): \[ x = 3 \]

  3. Equation C: \[ -2 + 4x = 10 \] Adding 2 to both sides: \[ 4x = 12 \] Dividing by 4: \[ x = 3 \]

Now let's summarize the solutions:

  • Equation A has the solution \(x = 6\).
  • Equation B has the solution \(x = 3\).
  • Equation C has the solution \(x = 3\).

From this analysis:

  • Thomas is incorrect because Equation A and Equation C do not have the same solution.
  • Petra is correct because Equation B and Equation C have the same solution, both being \(x = 3\).
  • Holland is incorrect because not all three equations have the same solution.
  • Dayton is incorrect because Equation B and Equation C share a solution.

Conclusion

Petra is correct. I determined this by solving each equation and comparing their solutions, which showed that only Equations B and C have the same solution.