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!