https://file.aiquickdraw.com/mathbot/file/1730495502215sq19zbdj.jpg

Let's check the correctness of the solutions:

### Given Line:
The line provided is:
\[
y = \frac{4}{3}x - 4
\]

### Step 1: Finding the Slope of the Given Line
The equation is in slope-intercept form \( y = mx + b \), where \( m \) represents the slope. Here, the slope of the given line is:
\[
m = \frac{4}{3}
\]

### Step 2: Perpendicular Line
1. **Finding the Perpendicular Slope:** The slope of a line perpendicular to the given line will be the negative reciprocal of \( \frac{4}{3} \). So, the perpendicular slope is:
   \[
   m_{\text{perpendicular}} = -\frac{3}{4}
   \]

2. **Equation of the Perpendicular Line Passing Through (3, -2):**
   Using the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (3, -2) \) and \( m = -\frac{3}{4} \), we get:
   \[
   y + 2 = -\frac{3}{4}(x - 3)
   \]
   Expanding this:
   \[
   y + 2 = -\frac{3}{4}x + \frac{9}{4}
   \]
   Subtracting 2 (or \(\frac{8}{4}\)) from both sides:
   \[
   y = -\frac{3}{4}x + \frac{1}{4}
   \]
   
   The correct equation for the perpendicular line should be:
   \[
   y = -\frac{3}{4}x + \frac{1}{4}
   \]

   However, the image solution shows:
   \[
   y = -\frac{3}{4}x - 4
   \]
   **This is incorrect.** The correct y-intercept should be \(\frac{1}{4}\).

### Step 3: Parallel Line
1. **Finding the Parallel Slope:** Since the parallel line has the same slope as the given line, it will also have \( m = \frac{4}{3} \).

2. **Equation of the Parallel Line Passing Through (3, -2):**
   Using the point-slope form with \( (x_1, y_1) = (3, -2) \) and \( m = \frac{4}{3} \), we get:
   \[
   y + 2 = \frac{4}{3}(x - 3)
   \]
   Expanding this:
   \[
   y + 2 = \frac{4}{3}x - 4
   \]
   Subtracting 2 from both sides:
   \[
   y = \frac{4}{3}x - 6
   \]

   The correct equation for the parallel line should be:
   \[
   y = \frac{4}{3}x - 6
   \]

   However, the image solution shows:
   \[
   y = \frac{4}{3}x - 4
   \]
   **This is also incorrect.** The correct y-intercept should be \(-6\).

### Summary
The correct answers are:
- **Equation of the perpendicular line:** \( y = -\frac{3}{4}x + \frac{1}{4} \)
- **Equation of the parallel line:** \( y = \frac{4}{3}x - 6 \)

Both equations in the image are incorrect. Let me know if you'd like further details on any of these steps.

---

Here are some related questions you might find helpful:

1. How do you determine the slope of a line from its equation?
2. Why is the perpendicular slope the negative reciprocal of the original slope?
3. What is the point-slope form of a line, and when is it useful?
4. How can you check if two lines are parallel or perpendicular using their slopes?
5. Why is expanding and simplifying important when working with point-slope form?

**Tip:** Always double-check calculations by substituting the given point into the final equation to verify if it satisfies it.

Consider the line y = (4/3)x - 4. Find the equation of the line that is perpendicular to this line and passes through the point (3, -2). Find the equation of the line that is parallel to this line and passes through the point (3, -2).

Learn how to find the equation of a line perpendicular or parallel to y = (4/3)x - 4 that passes through a specific point. This problem includes key algebra concepts like slopes of lines and point-slope form, suitable for students in Grades 8-10.

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