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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.

Which of the following linear equations represents a line that passes through the point (-3, 3) and is parallel to the given line?

Learn how to solve for a linear equation that is parallel to a given line and passes through the point (-3, 3). This problem covers key concepts such as the slope-intercept form and the point-slope form, and is suitable for students in Grades 8-10.

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