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The problem asks for the equation of a line that is **perpendicular** to a given line and passes through a specific point, \((-4, -6)\).

The equation of the given line is written as:
\[
y - 1 = -\frac{4}{3}(x + 1)
\]
This is in point-slope form, \(y - y_1 = m(x - x_1)\), where \(m\) is the slope of the line.

### Step 1: Find the slope of the given line
The slope \(m\) of the given line is \(-\frac{4}{3}\).

### Step 2: Slope of the perpendicular line
If two lines are perpendicular, their slopes are negative reciprocals of each other. So, if the slope of one line is \(-\frac{4}{3}\), the slope of the perpendicular line is:
\[
m_{\perpendicular} = \frac{3}{4}
\]

### Step 3: Use point-slope form
Now that we know the slope of the perpendicular line is \(\frac{3}{4}\), and it passes through the point \((-4, -6)\), we can use the point-slope form of the equation:
\[
y - y_1 = m(x - x_1)
\]
Substitute \(m = \frac{3}{4}\), \(x_1 = -4\), and \(y_1 = -6\):
\[
y + 6 = \frac{3}{4}(x + 4)
\]

### Step 4: Simplify to match the answer choices
Now, let's simplify this equation:
\[
y + 6 = \frac{3}{4}x + \frac{3}{4}(4)
\]
\[
y + 6 = \frac{3}{4}x + 3
\]
\[
y = \frac{3}{4}x + 3 - 6
\]
\[
y = \frac{3}{4}x - 3
\]

This matches option **B** in the answer choices.

### Final Answer: 
The equation of the line perpendicular to the given line and passing through the point \((-4, -6)\) is:
\[
\boxed{y = \frac{3}{4}x - 3}
\]

Let me know if you would like more details or have any other questions!

Here are 5 related questions to further your understanding:
1. How do you determine the slope of a line from an equation in point-slope form?
2. What is the relationship between the slopes of perpendicular lines?
3. How can you convert an equation from point-slope form to slope-intercept form?
4. How would you find the equation of a line parallel to a given line?
5. What is the general formula for the point-slope form of a line?

**Tip:** When finding perpendicular lines, always check if the slopes are negative reciprocals (flip the fraction and change the sign).

What equation is perpendicular to the line y - 1 = -4/3(x + 1) and also contains the point (-4, -6)?

Learn how to find the equation of a line that is perpendicular to y - 1 = -4/3(x + 1) and passes through the point (-4, -6). This problem uses point-slope form and the concept of negative reciprocal slopes. Suitable for high school students in Grades 9-11.

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