The problem asks to find the equation of a line that is parallel to the line $y = \frac{2}{3}x - 3$ and passes through the point $(-3, 1)$.

Step 1: Identify the slope of the given line.

The given line is in the slope-intercept form $y = mx + b$, where $m$ is the slope. From the equation $y = \frac{2}{3}x - 3$, the slope $m = \frac{2}{3}$.

Since parallel lines have the same slope, the slope of the new line will also be $m = \frac{2}{3}$.

Step 2: Use the point-slope form of a line equation.

The point-slope form is given by: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line, and $m$ is the slope.

We know the slope $m = \frac{2}{3}$ and the line passes through the point $(-3, 1)$. Substituting these values into the point-slope form:

$y - 1 = \frac{2}{3}(x + 3)$

Step 3: Simplify the equation.

Distribute the slope $\frac{2}{3}$:

$y - 1 = \frac{2}{3}x + 2$

Now, add 1 to both sides to isolate $y$:

$y = \frac{2}{3}x + 3$

Final Equation:

The equation of the line parallel to $y = \frac{2}{3}x - 3$ and passing through the point $(-3, 1)$ is: $y = \frac{2}{3}x + 3$

Would you like more details or further clarification on this? Here are some related questions:

1. How do you find the slope of a line from its equation?
2. What is the difference between the point-slope form and the slope-intercept form of a line?
3. How can you find the equation of a line perpendicular to another line?
4. What is the geometric interpretation of parallel and perpendicular lines?
5. How do you graph a line using its slope and a point?

Tip: When two lines are parallel, they share the same slope but have different $y$-intercepts.