Find the equation of a line perpendicular tominus, 5, x, plus, y, equals, minus, 2−5x+y=−2that passes through the point left parenthesis, 5, comma, 4, right parenthesis(5,4).

To find the equation of a line perpendicular to the line given by the equation \(-5x + y = -2\) and passing through the point \((5, 4)\), follow these steps:

1. **Convert the line to slope-intercept form**:
   \[
   y = 5x - 2
   \]
   The slope of this line is \(m = 5\).

2. **Find the slope of the perpendicular line**:
   The slope of a line perpendicular to another is the negative reciprocal of the original slope. Thus:
   \[
   m_{\text{perpendicular}} = -\frac{1}{5}
   \]

3. **Use the point-slope form of the equation**:
   The point-slope form is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting the point \((5, 4)\) and the slope \(-\frac{1}{5}\):
   \[
   y - 4 = -\frac{1}{5}(x - 5)
   \]

4. **Simplify the equation**:
   \[
   y - 4 = -\frac{1}{5}x + 1
   \]
   \[
   y = -\frac{1}{5}x + 5
   \]

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

### Questions for Further Exploration:
1. How can we graph this line on a coordinate plane?
2. What are the coordinates of another point on this line?
3. How would the equation change if the point was different?
4. Can you find the intersection of this line with the original line?
5. How do you derive the slope-intercept form from a general linear equation?

### Tip:
When finding perpendicular lines, always remember to use the negative reciprocal of the slope for accurate results.

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

Discover how to find the equation of a line perpendicular to -5x + y = -2 that passes through (5, 4). This problem illustrates key concepts in linear equations and the relationship between slopes. Ideal for students in Grades 8-10.

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