To find the equation of the line passing through $(3, 2)$ and parallel to the line $3x + y = -3$, we can follow these steps:

Step 1: Find the slope of the given line.

We start by converting the equation $3x + y = -3$ into slope-intercept form $(y = mx + b)$, where $m$ is the slope.

$3x + y = -3$ Subtract $3x$ from both sides: $y = -3x - 3$ Now, the equation is in slope-intercept form, and the slope $m = -3$.

Step 2: Use the point-slope form.

Since the new line is parallel to the given line, it will have the same slope, $m = -3$. The equation of a line in point-slope form is:

$y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point the line passes through and $m$ is the slope. Using the point $(3, 2)$ and the slope $m = -3$:

$y - 2 = -3(x - 3)$

Step 3: Simplify to slope-intercept form.

Now expand and simplify:

$y - 2 = -3x + 9$ Add 2 to both sides: $y = -3x + 11$

Final Answer:

The equation of the line is: $y = -3x + 11$

Would you like a more detailed breakdown of any steps, or do you have any further questions?

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Here are five related questions to expand on this topic:

1. How would you find the equation of a line perpendicular to $3x + y = -3$?
2. Can you derive the equation of a line given its slope and any other point on the line?
3. How does the slope of parallel and perpendicular lines relate?
4. How would you determine the x- and y-intercepts of the line $y = -3x + 11$?
5. How do you graph the line $y = -3x + 11$ on a coordinate plane?

Tip: When two lines are parallel, their slopes are always equal!