Question

consider the line 4x - 6y = -9.
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 \(4x - 6y=-9\). We want to solve for \(y\) to get it in the form \(y = mx + b\), where \(m\) is the slope.
First, subtract \(4x\) from both sides: \(-6y=-4x - 9\).
Then, divide every term by \(-6\): \(y=\frac{-4x}{-6}+\frac{-9}{-6}\).
Simplify the fractions: \(y = \frac{2}{3}x+\frac{3}{2}\). So the slope \(m\) of the given line is \(\frac{2}{3}\).

Step2: Slope of parallel line

Parallel lines have the same slope. So if the slope of the given line is \(\frac{2}{3}\), the slope of a line parallel to it is also \(\frac{2}{3}\).

Step3: Slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is the negative reciprocal of \(m\). The negative reciprocal of \(\frac{2}{3}\) is \(-\frac{3}{2}\) (because if \(m_1\) and \(m_2\) are slopes of perpendicular lines, then \(m_1\times m_2=- 1\), so \(m_2=-\frac{1}{m_1}\), here \(m_1 = \frac{2}{3}\), so \(m_2=-\frac{3}{2}\)).

Answer:

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