Let's break down the steps provided by the student to check for consistency and any errors:
Starting with the initial point-slope form of the equation, which is:
Step 1: \( y - (-2) = \frac{3}{4}(x - 3) \)
This step accurately represents the point-slope form, where \(-2\) is the \(y\)-coordinate and represents the line's slope \( \frac{3}{4} \).
Step 2: \[ y + 2 = \frac{3}{4}(x) - \frac{9}{4} \]
This expansion looks correct, as multiplying out the right side yields: \[ \frac{3}{4}x - \frac{9}{4} \]
Step 3: This step aims to isolate \(y\): \[ y + 2 + 2 = \frac{3}{4}x - \frac{9}{4} + \frac{8}{4} \] So, on the left side, we add \(2\) correctly to both sides, but let’s check the right side: \[ -\frac{9}{4} + \frac{8}{4} = -\frac{1}{4} \]
So the adjusted equation should be: \[ y + 4 = \frac{3}{4}x - \frac{1}{4} \] However, the student has an issue in the last step.
Step 4: \[ y = \frac{3}{4}x - \frac{1}{4} \]
The issue is in Step 3; when adding \(2\) (which is \( \frac{8}{4} \)), the student correctly added \(8/4\) to \(-9/4\), but the left side states that \(y + 2 + 2 = y + 4\). Instead, it should be:
So the correct equation should have transformed to: \[ y = \frac{3}{4}x - \frac{1}{4} - 2 \] To clarify, adding \(2\) is also \(\frac{8}{4}\). Thus the student has two errors in that sum where they believed they needed to add \(2\) derived from 0.
In summary, the student made a miscalculation, incorrectly manipulating the addition of \(2\) alongside the transformed equation of \(y\).
Therefore, the student made a calculation error in combining terms within Step 3 and didn't accurately isolate \(y\).
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