The problem asks for the equation of line $L$ in point-slope form and slope-intercept form, and mentions that line $L$ is parallel to the line $y = 5x$. The graph shows that line $L$ passes through the point $(3, -2)$.

Step 1: Identify the Slope of Line $L$

Since line $L$ is parallel to the line $y = 5x$, it will have the same slope as that line. The slope of $y = 5x$ is 5. Therefore, the slope $m$ of line $L$ is also 5.

Step 2: Use the 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. Here, the slope $m = 5$ and the point is $(3, -2)$. Plugging these values into the equation, we get: $y - (-2) = 5(x - 3)$ Simplifying: $y + 2 = 5(x - 3)$ So, the point-slope form of the equation is: $y + 2 = 5(x - 3)$

Step 3: Convert to Slope-Intercept Form

To convert this into slope-intercept form $y = mx + b$, expand and simplify the equation: $y + 2 = 5(x - 3)$ $y + 2 = 5x - 15$ Now, subtract 2 from both sides: $y = 5x - 17$ So, the slope-intercept form of the equation is: $y = 5x - 17$

Final Answer:

• Point-Slope Form: $y + 2 = 5(x - 3)$
• Slope-Intercept Form: $y = 5x - 17$

Would you like further details or have any questions?

Here are some additional related questions for practice:

1. What is the slope of a line parallel to $y = 7x + 2$?
2. How would you find the equation of a line perpendicular to $y = 5x$ passing through the same point?
3. How can you derive the equation of a line given two points on the line?
4. What is the significance of the slope in real-life applications?
5. How does the graph of a line change if you only modify the y-intercept?

Tip: When a line is parallel to another line, their slopes are always equal.