The question asks to find the equation of a line perpendicular to the given line $-5x + y = -2$ and passing through the point $(5, 4)$.

Step-by-step solution:

1. Rewrite the given equation:
   The line equation is $-5x + y = -2$. Let's rewrite it in slope-intercept form $y = mx + b$ to identify its slope: $y = 5x - 2$ So, the slope of this line is $m_1 = 5$.

2. Find the slope of the perpendicular line:
   The slope of a line perpendicular to another is the negative reciprocal of the original line's slope.
   Therefore, the slope of the new line is: $m_2 = -\frac{1}{5}$

3. 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. Using the point $(5, 4)$ and the slope $m_2 = -\frac{1}{5}$, we have: $y - 4 = -\frac{1}{5}(x - 5)$

4. Simplify the equation:
   Distribute and simplify the equation: $y - 4 = -\frac{1}{5}x + 1$ $y = -\frac{1}{5}x + 5$

So, the equation of the line perpendicular to $-5x + y = -2$ and passing through $(5, 4)$ is: $y = -\frac{1}{5}x + 5$

Would you like more details or explanations? Here are a few related questions to consider:

1. How do you convert any linear equation to slope-intercept form?
2. Why is the slope of a perpendicular line the negative reciprocal?
3. What are other ways to find the equation of a line besides point-slope form?
4. How would you graph two perpendicular lines on the coordinate plane?
5. What happens if the line is parallel instead of perpendicular?

Tip: To verify your work, always check that the slopes of perpendicular lines multiply to $-1$.