To write the equation of a line that is parallel to the line given by the equation \( y = -3x - 2 \), we need to keep the same slope. The slope of the given line is -3.
Thus, a line that is parallel would also have a slope of -3. The general form of the equation for a line is:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept. Since we want the parallel line not to share any points with the original line, we need to choose a different y-intercept.
Among the options provided:
- \( y = 3x - 2 \) (not parallel, slope = 3)
- \( y = -3x - 2 \) (same line, slope = -3)
- \( y = 3x + 8 \) (not parallel, slope = 3)
- \( y = -3x + 8 \) (parallel, slope = -3)
The correct choice for a parallel line that does not share any points with the original line \( y = -3x - 2 \) is:
\( y = -3x + 8 \).