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Let's solve the problem step-by-step.

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### Step 1: Analyze the given line equation
The given line equation is:
\[
6x + 3y = 12
\]

We convert this into slope-intercept form \(y = mx + b\) to find the slope \(m\):
\[
3y = -6x + 12
\]
\[
y = -2x + 4
\]
Thus, the slope of the given line is \(m = -2\).

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### Step 2: Equation of the line parallel to the given line
A line parallel to the given line has the same slope \(m = -2\). We use the point-slope form of a line equation:
\[
y - y_1 = m(x - x_1)
\]
Here, \(m = -2\) and the point is \((-3, 9)\). Substituting:
\[
y - 9 = -2(x + 3)
\]
\[
y - 9 = -2x - 6
\]
\[
y = -2x + 3
\]
Thus, the equation of the parallel line is:
\[
y = -2x + 3
\]

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### Step 3: Equation of the line perpendicular to the given line
For a line perpendicular to the given line, the slope is the negative reciprocal of \(-2\), which is:
\[
m = \frac{1}{2}
\]
Using the point-slope form again with the point \((-3, 9)\):
\[
y - 9 = \frac{1}{2}(x + 3)
\]
\[
y - 9 = \frac{1}{2}x + \frac{3}{2}
\]
\[
y = \frac{1}{2}x + \frac{3}{2} + 9
\]
\[
y = \frac{1}{2}x + \frac{21}{2}
\]
Thus, the equation of the perpendicular line is:
\[
y = \frac{1}{2}x + \frac{21}{2}
\]

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### Final Answers:
1. **Parallel line:** \(y = -2x + 3\)
2. **Perpendicular line:** \(y = \frac{1}{2}x + \frac{21}{2}\)

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Would you like a more detailed explanation of any step?  
Here are 5 related questions to explore further:  

1. How do you determine the slope of a line from its equation?  
2. Why do parallel lines have the same slope?  
3. What is the geometric interpretation of the negative reciprocal for perpendicular lines?  
4. How do you rewrite equations in slope-intercept form?  
5. How can the point-slope form be used for other types of problems?  

**Tip:** Always convert equations to slope-intercept form \(y = mx + b\) for an easier analysis of slope and intercept.

Consider the line with the equation: 6x + 3y = 12. Give the equation of the line parallel to this line which passes through (-3, 9). Give the equation of the line perpendicular to this line which passes through (-3, 9).

Solve for the equations of lines parallel and perpendicular to 6x + 3y = 12 passing through (-3, 9). This problem covers slope-intercept form, point-slope form, and the properties of parallel and perpendicular lines.

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