Question

consider the line ( x - 9y = 9 ).
what is the slope of a line parallel to this line?
what is the slope of a line perpendicular to this line?
slope of a parallel line:
slope of a perpendicular line:

Explanation:

Step1: Rewrite the line equation in slope - intercept form ($y = mx + b$, where $m$ is the slope)

We are given the equation $x - 9y=9$. First, we want to solve for $y$.
Subtract $x$ from both sides: $- 9y=-x + 9$.
Then divide each term by $-9$: $y=\frac{-x}{-9}+\frac{9}{-9}$, which simplifies to $y=\frac{1}{9}x - 1$.
So the slope of the given line ($m$) is $\frac{1}{9}$.

Step2: Find the slope of a parallel line

Parallel lines have the same slope. So if a line is parallel to the line $x - 9y = 9$, its slope will be equal to the slope of the given line.
Since the slope of the given line is $\frac{1}{9}$, the slope of a parallel line is $\frac{1}{9}$.

Step3: Find the slope of a perpendicular line

The slope of a line perpendicular to a line with slope $m$ is the negative reciprocal of $m$. That is, if the slope of a line is $m$, the slope of the perpendicular line $m_{\perp}=-\frac{1}{m}$.
Here, $m = \frac{1}{9}$, so the negative reciprocal is $-\frac{1}{\frac{1}{9}}=-9$.

Answer:

Slope of a parallel line: $\frac{1}{9}$
Slope of a perpendicular line: $- 9$