To solve this problem, we need to find an equation of a line that is parallel to $y = 5x + 2$ and passes through the point $(-6, -1)$.

Step-by-Step Solution:

1. Identify the slope of the given line: The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. From the given equation $y = 5x + 2$, the slope $m$ is 5.

2. Use the fact that parallel lines have the same slope: Since we are looking for a line that is parallel to $y = 5x + 2$, the slope of the new line must also be $m = 5$.

3. Substitute the slope into the slope-intercept form: We now have the equation in the form $y = 5x + b$. The only unknown left is the y-intercept $b$.

4. Use the point (-6, -1) to solve for $b$:

   • Substitute $x = -6$ and $y = -1$ into the equation: $-1 = 5(-6) + b$ $-1 = -30 + b$ $b = 29$

5. Write the equation of the line: Now that we have $b = 29$, the equation of the line is: $y = 5x + 29$

Conclusion:

The correct answer is C. $y = 5x + 29$.

Would you like more details on any step? Here are 5 questions to expand your understanding:

1. Why do parallel lines have the same slope?
2. How would you find the equation if the line were perpendicular instead of parallel?
3. How can we use point-slope form to find the equation of a line?
4. What happens if the line passed through a different point, like $(-2, 3)$?
5. How would you graph this equation to verify its correctness?

Tip: Always double-check your work by substituting the point back into the final equation to ensure it's correct!