To solve this problem, we need to determine which linear equation represents a line that is parallel to the given line and passes through the point $(-3, 3)$.

Step 1: Understanding Parallel Lines

Parallel lines have the same slope. Therefore, the slope of the new line must match the slope of the given line in the choices.

Step 2: Slope of the Line in the Equations

Each of the choices is in slope-intercept form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• 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 $m = -\frac{2}{3}$

Step 3: Equation of the New Line

The line passing through $(-3, 3)$ is parallel to the given line, so it must have the same slope. Thus, we need to find the equation of the line that passes through $(-3, 3)$ and has the correct slope.

We will use the point-slope form of the equation of a line: $y - y_1 = m(x - x_1)$ Where $(x_1, y_1)$ is the point on the line, and $m$ is the slope.

Case 1: Slope $\frac{3}{2}$ (from options A and B)

Using the point $(-3, 3)$ and slope $m = \frac{3}{2}$: $y - 3 = \frac{3}{2}(x + 3)$ Simplifying: $y - 3 = \frac{3}{2}x + \frac{9}{2}$ $y = \frac{3}{2}x + \frac{9}{2} + 3 = \frac{3}{2}x + \frac{9}{2} + \frac{6}{2} = \frac{3}{2}x + \frac{15}{2}$ This does not match either option A or B, so the line is not one of these.

Case 2: Slope $-\frac{2}{3}$ (from options C and D)

Now, using the point $(-3, 3)$ and slope $m = -\frac{2}{3}$: $y - 3 = -\frac{2}{3}(x + 3)$ Simplifying: $y - 3 = -\frac{2}{3}x - 2$ $y = -\frac{2}{3}x - 2 + 3 = -\frac{2}{3}x + 1$ This matches option D.

Final Answer:

The correct answer is D: $y = -\frac{2}{3}x + 1$.

Would you like more details or have any questions?

Here are five related questions to expand your understanding:

1. How do you find the slope of a line between two points?
2. What is the point-slope form of a line, and how is it used?
3. Can two lines be parallel if they have different slopes?
4. What is the relationship between the slope and y-intercept in the slope-intercept form of a line?
5. How do you find the equation of a line perpendicular to a given line?

Tip: Parallel lines always have the same slope, so checking the slope first can simplify finding the right equation.