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To solve this problem, we need to find the equation of a line that is **perpendicular** to the line \( y = \frac{1}{2}x - 6 \) and passes through the point \( (-3, 0) \).

### Step 1: Identify the slope of the given line
The slope of the line \( y = \frac{1}{2}x - 6 \) is \( m = \frac{1}{2} \).

### Step 2: Determine the slope of the perpendicular line
The slope of a line that is perpendicular to another line is the **negative reciprocal** of the slope of the original line. Thus, the slope \( m' \) of the perpendicular line is:
\[
m' = -\frac{1}{\frac{1}{2}} = -2
\]

### Step 3: Use the point-slope form of the line equation
We use the point-slope form again, where \( (x_1, y_1) \) is the point \( (-3, 0) \) and \( m' = -2 \):
\[
y - 0 = -2(x + 3)
\]

### Step 4: Simplify the equation
Expand and simplify the equation:
\[
y = -2(x + 3)
\]
\[
y = -2x - 6
\]

### Step 5: Identify the correct answer
The equation of the line is \( y = -2x - 6 \). This equation, however, does not directly match any of the provided options.

Let's compare with the closest options:
- The closest answer is **A: \( y = -3x + 9 \)**, but this is not the same as \( y = -2x - 6 \).

However, it's possible there is a mistake in the question or the provided options do not exactly match the required form. The correct approach and method give \( y = -2x - 6 \) as the solution, which should be checked against similar forms.

Would you like to discuss this further or clarify the options provided?

Here are some related questions for practice:
1. How do you find the equation of a line that passes through two given points?
2. What is the slope of a line perpendicular to \( y = -\frac{3}{4}x + 2 \)?
3. How do you verify if a point lies on a given line?
4. What are the properties of perpendicular lines in terms of slopes?
5. How can the equation of a line be rewritten in standard form?

**Tip:** When determining perpendicularity, always remember that the product of the slopes of two perpendicular lines is \(-1\).

Learn how to solve a problem involving perpendicular lines in geometry. Find the equation of a line perpendicular to y = (1/2)x - 6 passing through point (-3, 0). Understand the point-slope form and the concept of perpendicular slopes. Suitable for Grades 9-12.

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