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To solve this problem, let's go through the following steps:

### Step 1: Identify the slope of the given line

The line equation given is:
\[
x + 3y = 6
\]
To find the slope, we should convert it to slope-intercept form \(y = mx + b\), where \(m\) is the slope.

1. Rearrange for \(y\):
   \[
   3y = -x + 6
   \]
2. Divide by 3:
   \[
   y = -\frac{1}{3}x + 2
   \]

The slope of this line is \(m = -\frac{1}{3}\).

### Step 2: Find the equation of a line parallel to this one

Since parallel lines have the same slope, the line we are looking for will also have a slope of \(m = -\frac{1}{3}\).

### Step 3: Use the point-slope form to find the equation of the new line

We are given a point \((-3, 2)\) through which the line passes. Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \(m = -\frac{1}{3}\), \(x_1 = -3\), and \(y_1 = 2\):
\[
y - 2 = -\frac{1}{3}(x + 3)
\]
Simplify this equation:
1. Distribute \(-\frac{1}{3}\):
   \[
   y - 2 = -\frac{1}{3}x - 1
   \]
2. Add 2 to both sides:
   \[
   y = -\frac{1}{3}x + 1
   \]

### Step 4: Identify the correct answer

The equation we derived is:
\[
y = -\frac{1}{3}x + 1
\]

Looking at the options, the correct answer is:
\[
\boxed{y = -\frac{1}{3}x + 1}
\]

Would you like further details on this solution, or do you have any additional questions?

Here are five related questions to explore further:
1. What is the formula for determining if two lines are parallel?
2. How would you find the perpendicular line to \(x + 3y = 6\) through a given point?
3. How do you convert from point-slope form to slope-intercept form?
4. What is the significance of the slope in different forms of line equations?
5. How can you verify if two lines are identical or just parallel?

**Tip:** When dealing with line equations, always rearrange them into slope-intercept form (\(y = mx + b\)) to easily identify the slope.

Write an equation of the line passing through the point (-3, 2) and parallel to the line defined by x + 3y = 6.

Learn how to find the equation of a line passing through a given point and parallel to another line. This example involves identifying the slope, using point-slope form, and converting to slope-intercept form. Suitable for Grades 8-10.

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