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Let's break down the problem to find the equation of a line perpendicular to the given line \( 9x - 5y = -8 \), and passing through the point \( (-9, 4) \).

### Step 1: Convert the given equation to slope-intercept form
We need to express \( 9x - 5y = -8 \) in the form \( y = mx + b \) where \( m \) is the slope.

Starting with the equation:
\[
9x - 5y = -8
\]
Solve for \( y \):
\[
-5y = -9x - 8
\]
\[
y = \frac{9}{5}x + \frac{8}{5}
\]

Thus, the slope of the given line is \( m_1 = \frac{9}{5} \).

### Step 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 perpendicular line will be:
\[
m_2 = -\frac{5}{9}
\]

### Step 3: Use point-slope form to find the equation
Now, we use the point-slope form \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) = (-9, 4) \) and \( m = -\frac{5}{9} \).

Substitute these values:
\[
y - 4 = -\frac{5}{9}(x + 9)
\]

### Step 4: Simplify to slope-intercept form
Distribute the slope:
\[
y - 4 = -\frac{5}{9}x - 5
\]
Now add 4 to both sides:
\[
y = -\frac{5}{9}x - 1
\]

### Final Answer:
The equation of the line is:
\[
y = -\frac{5}{9}x - 1
\]

Would you like further details or explanations on any of the steps? Here are some additional questions to deepen your understanding:

1. How do we find the slope of a line given in standard form?
2. Why do perpendicular slopes have a negative reciprocal relationship?
3. What is the point-slope form of a linear equation?
4. How do we convert a linear equation from point-slope form to slope-intercept form?
5. What are the key differences between the slope-intercept form and standard form of a line?

**Tip:** When finding a perpendicular line, always remember that the slopes of two perpendicular lines multiply to give \(-1\).

Write the equation of a line perpendicular to 9x - 5y = -8 that passes through the point (-9, 4).

Learn how to find the equation of a line that is perpendicular to 9x - 5y = -8 and passes through the point (-9, 4). This problem involves key algebraic concepts like finding the slope, negative reciprocals for perpendicular lines, and using the point-slope form to find the equation of a line.

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