Question

consider the line 3x + 9y = 8.
what is the slope of a line parallel to this line?
what is the slope of a line perpendicular to this line?
your answer
slope of a parallel line:
slope of a perpendicular line:

Explanation:

Step1: Convert to slope - intercept form

The equation of a line in slope - intercept form is $y = mx + b$, where $m$ is the slope. We start with the given equation $3x+9y = 8$.
Subtract $3x$ from both sides: $9y=-3x + 8$.
Then divide each term by $9$: $y=\frac{-3x}{9}+\frac{8}{9}$.
Simplify the fraction: $y =-\frac{1}{3}x+\frac{8}{9}$. So the slope of the given line $m =-\frac{1}{3}$.

Step2: Slope of parallel line

Parallel lines have the same slope. So if a line is parallel to the line $3x + 9y=8$, its slope is equal to the slope of the given line. So the slope of the parallel line is $-\frac{1}{3}$.

Step3: Slope of perpendicular line

The slope of a line perpendicular to a line with slope $m$ is the negative reciprocal of $m$. The formula for the slope of a perpendicular line $m_{\perp}$ is $m_{\perp}=-\frac{1}{m}$.
Given $m =-\frac{1}{3}$, then $m_{\perp}=-\frac{1}{-\frac{1}{3}} = 3$.

Answer:

Slope of a parallel line: $-\frac{1}{3}$
Slope of a perpendicular line: $3$