The slope of a line that is parallel to another line is the same as the slope of the given line.

1. The equation of the given lines in each option is in slope-intercept form: $y = mx + b$, where $m$ represents the slope.

2. Let's determine the slope of each equation:

   • Option A: $y = \frac{3}{2}x + 1$, slope $m = \frac{3}{2}$
   • Option B: $y = \frac{3}{2}x - 1$, slope $m = \frac{3}{2}$
   • Option C: $y = -\frac{2}{3}x - 1$, slope $m = -\frac{2}{3}$
   • Option D: $y = -\frac{2}{3}x + 1$, slope $m = -\frac{2}{3}$

3. The slope of the line passing through the point $(-3, 3)$ must be the same as the slope of the parallel line. Since the point is on the new line and the lines are parallel, we need to find a line with the same slope as one of the given options.

4. To be parallel, the correct slope must match the slope of the given line. In this case, the slopes of the parallel lines are $\frac{3}{2}$. So, the correct equation must have a slope of $\frac{3}{2}$, which corresponds to either option A or B.

5. Now, substitute $(-3, 3)$ into the equations to see which one holds:

• For Option A: $y = \frac{3}{2}x + 1 \Rightarrow 3 = \frac{3}{2}(-3) + 1 = -\frac{9}{2} + 1 = -\frac{7}{2} \quad (\text{False})$

• For Option B: $y = \frac{3}{2}x - 1 \Rightarrow 3 = \frac{3}{2}(-3) - 1 = -\frac{9}{2} - 1 = -\frac{11}{2} \quad (\text{False})$

Since both calculations do not work out for options A and B, this suggests I made an earlier mistake. Would you like to clarify? or