Question

consider the line $y = -\frac{5}{2} - \frac{2}{3}x$. what is the slope of a line perpendicular to this line? what is the slope of a line parallel to this line? slope of a perpendicular line: \boxed{} slope of a parallel line: \boxed{}

Explanation:

Step1: Identify the slope of the given line

The equation of the line is \( y = -\frac{5}{2} - \frac{2}{3}x \), which can be rewritten in slope - intercept form \( y=mx + b \) (where \( m \) is the slope and \( b \) is the y - intercept) as \( y=-\frac{2}{3}x-\frac{5}{2} \). So the slope of the given line, \( m_{given}=-\frac{2}{3} \).

Step2: Find the slope of a parallel line

For two parallel lines, their slopes are equal. So if a line is parallel to the given line, its slope \( m_{parallel}=m_{given}=-\frac{2}{3} \).

Step3: Find the slope of a perpendicular line

If two lines are perpendicular, the product of their slopes is - 1. Let the slope of the perpendicular line be \( m_{perpendicular} \). Then we have the equation \( m_{given}\times m_{perpendicular}=- 1 \). Substituting \( m_{given}=-\frac{2}{3} \) into the equation, we get \( -\frac{2}{3}\times m_{perpendicular}=-1 \). To solve for \( m_{perpendicular} \), we can multiply both sides of the equation by \( -\frac{3}{2} \). So \( m_{perpendicular}=(-1)\times(-\frac{3}{2})=\frac{3}{2} \).

Answer:

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