Question

consider the line $6x + 8y = -1$. 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 of the line \(6x + 8y=-1\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope.
Subtract \(6x\) from both sides: \(8y=-6x - 1\).
Then divide each term by \(8\): \(y=\frac{-6}{8}x-\frac{1}{8}\).
Simplify \(\frac{-6}{8}\) to \(\frac{-3}{4}\), so \(y =-\frac{3}{4}x-\frac{1}{8}\). The slope \(m\) of the given line is \(-\frac{3}{4}\).

Step2: Slope of parallel line

Parallel lines have the same slope. So if the slope of the given line is \(m =-\frac{3}{4}\), the slope of a line parallel to it is also \(-\frac{3}{4}\).

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 \(m_{\perp}=-\frac{1}{m}\) (when \(m
eq0\)).
Given \(m =-\frac{3}{4}\), then \(m_{\perp}=-\frac{1}{-\frac{3}{4}}=\frac{4}{3}\).

Answer:

Slope of a parallel line: \(-\frac{3}{4}\)
Slope of a perpendicular line: \(\frac{4}{3}\)