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The question asks to find the equation of a line that is **perpendicular** to the line \( y = -\frac{1}{2}x - 3 \) and passes through the point \( (3, 6) \). We can solve this by following these steps:

### Step 1: Find the slope of the given line.
The equation of the given line is in slope-intercept form \( y = mx + b \), where \( m \) is the slope. For the given line:
\[
y = -\frac{1}{2}x - 3
\]
The slope of the line is \( m = -\frac{1}{2} \).

### Step 2: Find the slope of the perpendicular line.
Two lines are **perpendicular** if the product of their slopes is \( -1 \). Let the slope of the perpendicular line be \( m_p \). Thus:
\[
m \times m_p = -1
\]
Substituting \( m = -\frac{1}{2} \):
\[
-\frac{1}{2} \times m_p = -1
\]
Solving for \( m_p \):
\[
m_p = 2
\]
So, the slope of the perpendicular line is \( 2 \).

### Step 3: Use the point-slope form of the equation of a line.
We know the slope \( m_p = 2 \) and the line passes through the point \( (3, 6) \). The point-slope form is:
\[
y - y_1 = m(x - x_1)
\]
where \( (x_1, y_1) \) is the point \( (3, 6) \) and \( m = 2 \). Substituting:
\[
y - 6 = 2(x - 3)
\]

### Step 4: Simplify to slope-intercept form.
Distribute the \( 2 \):
\[
y - 6 = 2x - 6
\]
Add \( 6 \) to both sides:
\[
y = 2x
\]

### Final Answer:
The equation of the line that is perpendicular to \( y = -\frac{1}{2}x - 3 \) and passes through the point \( (3, 6) \) is:
\[
y = 2x
\]

Would you like further clarification or more details? Here are 5 related questions for further exploration:

1. How do you find the equation of a line parallel to a given line?
2. What is the general form of a line equation?
3. How do you determine if two lines are perpendicular just by looking at their slopes?
4. How can we use the point-slope form to derive equations of lines with different conditions?
5. What happens to the slopes of two lines when they are parallel?

**Tip:** Always remember, the slope of a line perpendicular to another is the negative reciprocal of the original slope!

Find an equation for the line with the given properties. Express your answer using either the slope-intercept or the slope-point form of the equation of a line. Find the equation for the line with the given properties: the line is perpendicular to the line y = -1/2x - 2 and passes through the point (3,6).

This tutorial explains how to find the equation of a line that is perpendicular to another given line and passes through a specific point, utilizing the point-slope and slope-intercept forms of linear equations. Suitable for high school level students learning geometry or algebra.

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