To solve this problem, let's work step-by-step:

Problem:

We need to find the equation of a line perpendicular to the given line:

$y = -\frac{1}{9}x - 28$

and passing through the point $P(9, -3)$.

Step 1: Determine the slope of the given line

The equation of the line is in slope-intercept form $y = mx + b$, where $m$ is the slope. From the given equation:

$m = -\frac{1}{9}.$

Step 2: Find the slope of the perpendicular line

For two lines to be perpendicular, the product of their slopes must equal $-1$. If $m_1$ is the slope of the given line, and $m_2$ is the slope of the perpendicular line, then:

$m_1 \cdot m_2 = -1.$

Substitute $m_1 = -\frac{1}{9}$:

$-\frac{1}{9} \cdot m_2 = -1.$

Solve for $m_2$:

$m_2 = 9.$

Thus, the slope of the perpendicular line is $m_2 = 9$.

Step 3: Write the equation of the perpendicular line

The perpendicular line passes through the point $P(9, -3)$ and has slope $m = 9$. The equation of a line can be written as:

$y - y_1 = m(x - x_1),$

where $m$ is the slope, and $(x_1, y_1)$ is the given point. Substitute $m = 9$, $x_1 = 9$, and $y_1 = -3$:

$y - (-3) = 9(x - 9).$

Simplify:

$y + 3 = 9x - 81.$

$y = 9x - 84.$

Final Answer:

The equation of the perpendicular line is:

$y = 9x - 84.$

Do you have any questions, or would you like me to explain a specific step in more detail?

Related Questions:

1. How do you find the slope of a line given two points instead of an equation?
2. What is the general form of a linear equation, and how does it relate to the slope-intercept form?
3. How do you verify that two lines are perpendicular?
4. What is the geometric interpretation of a perpendicular line on a graph?
5. How does changing the slope of a line affect its steepness?

Tip:

Always double-check the sign of the slope when working with perpendicular lines—it’s easy to miss the negative reciprocal!