Question

consider the line 8x + 7y = -9. 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 need to rewrite the equation \(8x + 7y=-9\) in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept.
Starting with \(8x+7y=-9\), we solve for \(y\):
Subtract \(8x\) from both sides: \(7y=-8x - 9\)
Then divide each term by \(7\): \(y=-\frac{8}{7}x-\frac{9}{7}\)
So the slope of the given line \(m =-\frac{8}{7}\)

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So if a line is parallel to the line \(8x + 7y=-9\), its slope \(m_{parallel}\) is equal to the slope of the given line.
Since the slope of the given line is \(-\frac{8}{7}\), the slope of the parallel line \(m_{parallel}=-\frac{8}{7}\)

Step3: Find the slope of the perpendicular line

The slopes of two perpendicular lines are negative reciprocals of each other. That is, if the slope of one line is \(m\), the slope of a line perpendicular to it, \(m_{perpendicular}\), is given by \(m_{perpendicular}=-\frac{1}{m}\) (when \(m
eq0\))
We know that \(m =-\frac{8}{7}\), so \(m_{perpendicular}=-\frac{1}{-\frac{8}{7}}\)
Simplifying \( -\frac{1}{-\frac{8}{7}}=\frac{7}{8}\)

Answer:

Slope of a perpendicular line: \(\frac{7}{8}\)
Slope of a parallel line: \(-\frac{8}{7}\)