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To solve this problem, we need to determine the equation of a line that is perpendicular to the given line and passes through the point (-3, -5).

### Step 1: Analyze the given line
The given line equation is:

\[
y - 4 = 2(x - 6)
\]

This is in **point-slope form**: \( y - y_1 = m(x - x_1) \), where \( m \) is the slope, and \( (x_1, y_1) \) is a point on the line. From this, we can extract the slope \( m \).

The slope of the given line is **2** (the coefficient of \( x \)).

### Step 2: Find the slope of the perpendicular line
If two lines are perpendicular, their slopes are negative reciprocals of each other. So, the slope \( m_2 \) of the line perpendicular to the given line is:

\[
m_2 = -\frac{1}{2}
\]

### Step 3: Write the equation of the perpendicular line
Now, we know the slope \( m_2 = -\frac{1}{2} \), and the line passes through the point (-3, -5). We can use the **point-slope form** again to write the equation of the line:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( y_1 = -5 \), \( m = -\frac{1}{2} \), and \( x_1 = -3 \):

\[
y - (-5) = -\frac{1}{2}(x - (-3))
\]

Simplifying:

\[
y + 5 = -\frac{1}{2}(x + 3)
\]

### Step 4: Match with the options
Looking at the options provided:

- **Option B**: \( y + 5 = -\frac{1}{2}(x + 3) \)

This matches our equation.

Thus, the correct answer is **B**.

---

Would you like any further explanation or details?

Here are some related questions for further practice:

1. What is the equation of a line parallel to the given line passing through (-3, -5)?
2. How do you convert a point-slope form equation into slope-intercept form?
3. How do you find the equation of a line given two points?
4. What is the slope of a vertical line and how does it affect perpendicular lines?
5. How do you find the distance between a point and a line?

**Tip:** When dealing with perpendicular lines, always remember that their slopes are negative reciprocals!

What is an equation of the line that is perpendicular to y - 4 = 2(x - 6) and passes through the point (-3, -5)?

Learn how to find the equation of a line that is perpendicular to y - 4 = 2(x - 6) and passes through the point (-3, -5). This problem focuses on using the point-slope form and understanding the relationship between slopes of perpendicular lines. Suitable for students in Grades 9-11.

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