Question

consider the line $y = \frac{5}{2} - \frac{1}{5}x$. what is the slope of a line parallel to this line? what is the slope of a line perpendicular to this line? your answer slope of a parallel line: \boxed{} slope of a perpendicular line: \boxed{}

Explanation:

Step1: Identify the slope of the given line

The equation of the line is given in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. The given line is \(y=\frac{5}{2}-\frac{1}{5}x\), which can be rewritten as \(y =-\frac{1}{5}x+\frac{5}{2}\). So, the slope (\(m\)) of the given line is \(-\frac{1}{5}\).

Step2: Find the slope of a parallel line

Parallel lines have the same slope. So, if a line is parallel to the given line, its slope will be equal to the slope of the given line. Therefore, the slope of a line parallel to the given line is \(-\frac{1}{5}\).

Step3: Find the slope of a 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 \(m
eq0\)). Here, \(m =-\frac{1}{5}\), so the negative reciprocal is \(-\frac{1}{-\frac{1}{5}} = 5\).

Answer:

Slope of a parallel line: \(-\frac{1}{5}\)
Slope of a perpendicular line: \(5\)