Question

consider the line $-x - 6y = -4$. 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: Convert to slope - intercept form

We start with the equation \(-x - 6y=-4\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
First, add \(x\) to both sides of the equation: \(-6y=x - 4\).
Then, divide each term by \(-6\): \(y=\frac{x}{-6}+\frac{-4}{-6}\), which simplifies to \(y =-\frac{1}{6}x+\frac{2}{3}\).

Step2: Find the slope of a parallel line

Parallel lines have the same slope. From the equation \(y =-\frac{1}{6}x+\frac{2}{3}\), the slope \(m\) of the given line is \(-\frac{1}{6}\). So the slope of a line parallel to this line is also \(-\frac{1}{6}\).

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 \(m\). If \(m =-\frac{1}{6}\), the negative reciprocal is \(-\frac{1}{m}=-\frac{1}{-\frac{1}{6}} = 6\).

Answer:

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