Question

consider the line ( 9x + 6y = -3 ).
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 \(9x + 6y=-3\). We want to solve for \(y\) to get it in the form \(y = mx + b\) (where \(m\) is the slope).
Subtract \(9x\) from both sides: \(6y=-9x - 3\).
Then divide each term by \(6\): \(y=\frac{-9x}{6}-\frac{3}{6}\).
Simplify the fractions: \(y =-\frac{3}{2}x-\frac{1}{2}\). So the slope \(m\) of the given line is \(-\frac{3}{2}\).

Step2: Slope of parallel line

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

Step3: Slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is the negative reciprocal of \(m\). If \(m =-\frac{3}{2}\), the negative reciprocal \(m_{\perp}\) is given by \(m_{\perp}=-\frac{1}{m}\).
Substitute \(m =-\frac{3}{2}\) into the formula: \(m_{\perp}=-\frac{1}{-\frac{3}{2}}=\frac{2}{3}\).

Answer:

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