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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 \(-1\).

Find the equation of a line perpendicular to -5x + y = -2 that passes through the point (5,4).

This problem explores finding the equation of a line perpendicular to the line -5x + y = -2 that passes through the point (5,4). It covers concepts such as slope-intercept form, perpendicular slopes, and point-slope form. Suitable for students in Grades 9-10.

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