Let's solve the problem in detail.

We need to write the equation of a line that is parallel to $y = 3x - 2$ and passes through the point $(3, 2)$.

Step 1: Determine the slope

Since the line is parallel to $y = 3x - 2$, it has the same slope. The slope-intercept form of a line is:

$y = mx + b$

where $m$ is the slope. From the given equation $y = 3x - 2$, the slope is $m = 3$.

Step 2: Use Point-Slope Form

The point-slope form of a line is:

$y - y_1 = m(x - x_1)$

Here, $(x_1, y_1) = (3, 2)$ is the point the line passes through, and the slope $m = 3$. Plugging in these values:

$y - 2 = 3(x - 3)$

This is the point-slope form of the equation.

Step 3: Convert to Slope-Intercept Form

Now, let's simplify the point-slope form into slope-intercept form $y = mx + b$.

$y - 2 = 3(x - 3)$ $y - 2 = 3x - 9$ $y = 3x - 9 + 2$ $y = 3x - 7$

This is the equation of the line in slope-intercept form.

Final Answer:

1. Point-Slope Form: $y - 2 = 3(x - 3)$
2. Slope-Intercept Form: $y = 3x - 7$

Would you like any additional details or clarifications?

Here are 5 related questions you might consider:

1. How do we find the slope of a line given two points?
2. What is the general process for converting from point-slope form to slope-intercept form?
3. How do we find the equation of a line that is perpendicular to another line?
4. What happens if two lines have the same slope but different intercepts?
5. How do we determine the distance between a point and a line?

Tip: Remember, parallel lines always have the same slope but different y-intercepts!