Let's review the steps to determine if there are any mistakes in the student's work.
Step 1: The student begins with the point-slope form of a line, which is: \[ y - y_1 = m(x - x_1) \] Where \( m \) is the slope and \((x_1, y_1)\) is a point on the line. For this problem, the given slope \( m \) is \(\frac{3}{4}\) and the point is \((3, -2)\). The point-slope equation would therefore be: \[ y - (-2) = \frac{3}{4}(x - 3) \] This simplifies to: \[ y + 2 = \frac{3}{4}(x - 3) \] This matches with Step 2 of the student's work, so Step 1 and Step 2 are correct.
Step 2: The student correctly expands the right side of the equation: \[ y + 2 = \frac{3}{4}x - \frac{3 \cdot 3}{4} \] This simplifies to: \[ y + 2 = \frac{3}{4}x - \frac{9}{4} \] So Step 2 is also correct.
Step 3: The student then tries to isolate \( y \) by subtracting 2 from both sides. However, it seems that the student attempts to add \(\frac{8}{4}\) (which equals 2) to both sides: \[ y + 2 + 2 = \frac{3}{4}x - \frac{9}{4} + \frac{8}{4} \] Separating the terms on the left gives us \(y + 4\). Now, the right side combines as follows: \[ y + 4 = \frac{3}{4}x - \frac{1}{4} \] This is where the student makes an error. They should have subtracted 2 directly from both sides instead of adding \(\frac{8}{4}\).
When doing this correctly, it would look like: \[ y + 2 - 2 = \frac{3}{4}x - \frac{9}{4} - \frac{8}{4} \] This simplifies to: \[ y = \frac{3}{4}x - \frac{17}{4} \]
Step 4: The final equation in the student's work states: \[ y = \frac{3}{4}x - \frac{1}{4} \] This is incorrect. The correct result should be: \[ y = \frac{3}{4}x - \frac{17}{4} \]
Conclusion: The student made an error in Step 3 when trying to isolate \( y \). They should have subtracted 2 from both sides, but instead added \(\frac{8}{4}\), leading to an incorrect final answer. The correct final equation is \( y = \frac{3}{4}x - \frac{17}{4} \).
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