Question

consider the line $-x - 6y = -7$. 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 in slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope. We start with the equation $-x - 6y=-7$.
First, we isolate $y$. Add $x$ to both sides of the equation: $-6y=x - 7$.
Then, divide each term by $-6$: $y=\frac{x}{-6}+\frac{-7}{-6}$, which simplifies to $y =-\frac{1}{6}x+\frac{7}{6}$.
So the slope of the given line, $m$, is $-\frac{1}{6}$.

Step2: Find slope of parallel line

Parallel lines have the same slope. So if a line is parallel to the line $y =-\frac{1}{6}x+\frac{7}{6}$, its slope will be equal to the slope of the given line.
So the slope of a line parallel to the given line is $-\frac{1}{6}$.

Step3: Find 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 $m_{\perp}=-\frac{1}{m}$.
We know that $m =-\frac{1}{6}$, so $m_{\perp}=-\frac{1}{-\frac{1}{6}} = 6$.

Answer:

Slope of a parallel line: $-\frac{1}{6}$
Slope of a perpendicular line: $6$