Question

consider the line $2x + 9y = -1$. 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: Convert to slope - intercept form

We start with the equation of the line \(2x + 9y=-1\). We want to write it in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
First, we isolate \(y\):
Subtract \(2x\) from both sides: \(9y=-2x - 1\)
Then divide each term by \(9\): \(y=-\frac{2}{9}x-\frac{1}{9}\)
So the slope of the given line (\(m\)) is \(-\frac{2}{9}\).

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So if a line is parallel to the line \(y =-\frac{2}{9}x-\frac{1}{9}\), its slope will be equal to the slope of the given line.
So the slope of the parallel line \(m_{parallel}=-\frac{2}{9}\)

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 \(m\). The formula for the slope of a perpendicular line \(m_{perpendicular}\) is \(m_{perpendicular}=-\frac{1}{m}\) (when \(m
eq0\))
We know that \(m =-\frac{2}{9}\), so \(m_{perpendicular}=-\frac{1}{-\frac{2}{9}}=\frac{9}{2}\)

Answer:

Slope of a perpendicular line: \(\frac{9}{2}\)
Slope of a parallel line: \(-\frac{2}{9}\)