Question

consider the line $2x + 3y = -4$. 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 of the line \(2x + 3y=-4\). We want to rewrite 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 of the equation: \(3y=-2x - 4\).
Then divide each term by 3: \(y=-\frac{2}{3}x-\frac{4}{3}\).
So the slope (\(m\)) of the given line is \(-\frac{2}{3}\).

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So if a line is parallel to the line \(2x + 3y=-4\) (with slope \(-\frac{2}{3}\)), its slope will also be \(-\frac{2}{3}\).

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 (\(m_{\perp}\)) is \(m_{\perp}=-\frac{1}{m}\) (when \(m
eq0\)).
For \(m =-\frac{2}{3}\), the negative reciprocal is \(-\frac{1}{-\frac{2}{3}}=\frac{3}{2}\).

Answer:

Slope of a parallel line: \(-\frac{2}{3}\)
Slope of a perpendicular line: \(\frac{3}{2}\)