Question

consider the line -4x - 5y = 3. what is the slope of a line perpendicular to this line? what is the slope of a line parallel to this line? slope of a perpendicular line: slope of a parallel line:

Explanation:

Step1: Find the slope of the given line

First, we rewrite the equation \(-4x - 5y = 3\) in slope - intercept form \(y=mx + b\) (where \(m\) is the slope and \(b\) is the y - intercept).
Start with \(-4x-5y = 3\).
Add \(4x\) to both sides: \(-5y=4x + 3\).
Divide each term by \(-5\): \(y=-\frac{4}{5}x-\frac{3}{5}\).
So the slope of the given line \(m =-\frac{4}{5}\).

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So if a line is parallel to the line \(-4x - 5y=3\), its slope \(m_{parallel}=-\frac{4}{5}\).

Step3: Find the slope of the 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 \(m_{perpendicular}=-\frac{1}{m}\) (for \(m
eq0\)).
Here, \(m =-\frac{4}{5}\), so \(m_{perpendicular}=-\frac{1}{-\frac{4}{5}}=\frac{5}{4}\).

Answer:

Slope of a perpendicular line: \(\frac{5}{4}\)
Slope of a parallel line: \(-\frac{4}{5}\)