The student's work contains an error in the step where they distribute the slope and rearrange the equation. Let's go through it step by step to identify the mistake:
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The student correctly starts with the point-slope form of a line: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope. Here, \( m = -6 \) and the point is \( (2, -8) \).
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Substituting the values into the equation gives: \[ y - (-8) = -6(x - 2) \] This step is correct.
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The mistake occurs in the next step where they try to simplify: \[ y - (-8) = -6x + 12 \] The left-hand side simplifies to: \[ y + 8 = -6x + 12 \] The error is in the following attempt to manipulate the equation. They didn't correctly manage the left side of the equation to isolate \( y \).
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Next, the student wrote: \[ y - (-8) + 8 = -6x + 12 + 8 \] This contains an additional \( + 8 \) on the left-hand side which is incorrect. Instead, it should have been: \[ y + 8 = -6x + 12 \] To isolate \( y \), you would subtract \( 8 \) from both sides: \[ y = -6x + 12 - 8 \]
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The final simplified equation should be: \[ y = -6x + 4 \]
Therefore, the correct equation for the line with a slope of -6 passing through the point (2, -8) is: \[ y = -6x + 4 \] The student's final result of \( y = -6x + 20 \) is incorrect due to the miscalculation during the simplification steps.