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To solve this problem, we need to write the equation of a line in standard form that:

1. Passes through the point \((-5, 9)\).
2. Is perpendicular to the line whose equation is \( y = \frac{5}{4}x + \frac{9}{4} \).

### Steps to find the equation:

1. **Find the slope of the given line**:
   - The equation is \( y = \frac{5}{4}x + \frac{9}{4} \), so the slope \(m_1\) of this line is \( \frac{5}{4} \).

2. **Find the slope of the perpendicular line**:
   - The slope of a line perpendicular to another is the negative reciprocal of the original slope. So, if the original slope is \(m_1 = \frac{5}{4}\), the slope \(m_2\) of the perpendicular line is:
     \[
     m_2 = -\frac{4}{5}
     \]

3. **Use the point-slope form to write the equation**:
   - The point-slope form of a line is:
     \[
     y - y_1 = m(x - x_1)
     \]
     Where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope.
   - Substituting \(x_1 = -5\), \(y_1 = 9\), and \(m = -\frac{4}{5}\), we get:
     \[
     y - 9 = -\frac{4}{5}(x + 5)
     \]

4. **Simplify the equation**:
   - Distribute the slope on the right-hand side:
     \[
     y - 9 = -\frac{4}{5}x - 4
     \]
   - Add 9 to both sides:
     \[
     y = -\frac{4}{5}x + 5
     \]

5. **Convert to standard form**:
   - Multiply through by 5 to eliminate the fraction:
     \[
     5y = -4x + 25
     \]
   - Rearrange to get the equation in standard form \(Ax + By = C\):
     \[
     4x + 5y = 25
     \]

Thus, the equation of the line in standard form is:
\[
4x + 5y = 25
\]

Would you like any further details or explanation on this solution?

Here are 5 related questions to explore:

1. How do you find the slope of a line from its equation in standard form?
2. What are the steps to convert a slope-intercept equation to standard form?
3. How do you determine if two lines are perpendicular or parallel?
4. Can you derive a line equation if you are given two points instead of a point and slope?
5. How does the slope-intercept form differ from the standard form of a line?

**Tip**: Always remember that the slope of a perpendicular line is the negative reciprocal of the original slope.

Use the given conditions to write an equation for the line passing through (-5, 9) and perpendicular to the line y = (5/4)x + (9/4).

Learn how to find the equation of a line that passes through (-5, 9) and is perpendicular to y = 5/4x + 9/4. This problem involves finding perpendicular slopes, using the point-slope form, and converting the equation to standard form. Ideal for students in Grades 9-10.

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