Let's analyze the equation and the work of both students step-by-step.
Given the equation: \[ 2w + 4w + 3 = 8 - 5w + 6 \]
First, we can combine like terms on both sides.
Left Side:
- Combine \(2w\) and \(4w\): \[ 2w + 4w = 6w \]
- So, the left side becomes: \[ 6w + 3 \]
Right Side:
- Combine \(8\) and \(6\): \[ 8 + 6 = 14 \]
- So, the right side becomes: \[ 14 - 5w \]
Now the equation simplifies to: \[ 6w + 3 = 14 - 5w \]
Now let's evaluate each student's steps.
Student A:
- Step 1: \(6w + 3 = 14 - 5w\) (This is correct.)
- Step 2: \(6w + 5w = 14 - 3\) (This is incorrect. The left side should still be \(6w + 3\), and the right side should be \(14 - 5w\) at this step, not \(14 - 3\)).
Student B:
- Step 1: \(2w + 4w + 5w = 8 + 6 - 3\) (This is incorrect. It should read \(2w + 4w + 3 = 8 + 6 - 5w\). They should have kept the -5w on the right.)
- Step 2: \(11w = 11\) (This cannot be affirmed as correct since their previous step was invalid).
Conclusion:
- Student A correctly started but made an error in their algebra in Step 2.
- Student B did not manipulate the equation correctly from the beginning.
Therefore, neither student's work is entirely correct. The correct approach after Student A's correct first step should have been to isolate \(w\) by adding \(5w\) to both sides: \[ 6w + 5w + 3 = 14 \] This leads to: \[ 11w + 3 = 14 \] Then, subtract 3 from both sides: \[ 11w = 11 \] Finally, dividing both sides by 11 gives: \[ w = 1 \]
In summary, Student A's starting equation is correct, but their second step has errors, while Student B's expressions are incorrect from the beginning.
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