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The problem asks for the equation of line \( L \), which is parallel to the line \( y = 3x \) and passes through the point \( (1, -2) \). We are required to express this equation in both point-slope form and slope-intercept form.

### 1. **Point-Slope Form:**

The point-slope form of a line 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.

- Since line \( L \) is parallel to \( y = 3x \), it has the same slope, which is \( m = 3 \).
- The given point on the line is \( (1, -2) \).

Substituting these values into the point-slope form:

\[
y - (-2) = 3(x - 1)
\]

Simplifying:

\[
y + 2 = 3(x - 1)
\]

This is the equation of line \( L \) in point-slope form.

### 2. **Slope-Intercept Form:**

To convert this to slope-intercept form, we'll expand and simplify the equation. Start from the point-slope form:

\[
y + 2 = 3(x - 1)
\]

First, distribute the 3 on the right-hand side:

\[
y + 2 = 3x - 3
\]

Now subtract 2 from both sides:

\[
y = 3x - 3 - 2
\]

Simplifying:

\[
y = 3x - 5
\]

Thus, the equation of line \( L \) in slope-intercept form is:

\[
y = 3x - 5
\]

### Final Answer:
- **Point-slope form:** \( y + 2 = 3(x - 1) \)
- **Slope-intercept form:** \( y = 3x - 5 \)

Would you like further details on the process or clarification on any step?

Here are some related questions to deepen your understanding:
1. What is the relationship between slopes of parallel lines?
2. How would the equation change if the line were perpendicular to \( y = 3x \)?
3. Can you derive the equation of a line if only two points are given instead of a point and a slope?
4. How does the slope-intercept form help in graphing a line?
5. What is the general method for converting from point-slope form to slope-intercept form?

**Tip:** When dealing with parallel lines, remember that they always have the same slope!

Write an equation for line L in point-slope form and slope-intercept form. Line L is parallel to y = 3x and passes through the point (1, -2).

Learn how to write the equation of a line parallel to y = 3x that passes through the point (1, -2). This problem covers key linear algebra concepts such as slope and point-slope form. Suitable for students in Grades 8-10.

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