Question

consider the line $6x + 8y = -6$,
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

The equation of a line in slope - intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We start with the given equation \(6x+8y=-6\).
Subtract \(6x\) from both sides: \(8y=-6x - 6\).
Then divide each term by \(8\): \(y=\frac{-6x}{8}-\frac{6}{8}\).
Simplify the fractions: \(y =-\frac{3}{4}x-\frac{3}{4}\). So the slope \(m\) of the given line is \(-\frac{3}{4}\).

Step2: Slope of parallel line

Parallel lines have the same slope. So the slope of a line parallel to the given line is also \(-\frac{3}{4}\).

Step3: Slope of perpendicular line

The slope of a line perpendicular to a line with slope \(m\) is the negative reciprocal of \(m\). If \(m =-\frac{3}{4}\), the negative reciprocal \(m_{\perp}\) is given by \(m_{\perp}=-\frac{1}{m}\).
Substitute \(m =-\frac{3}{4}\) into the formula: \(m_{\perp}=-\frac{1}{-\frac{3}{4}}=\frac{4}{3}\).

Answer:

Slope of a parallel line: \(-\frac{3}{4}\)
Slope of a perpendicular line: \(\frac{4}{3}\)