Question

consider the line $4x + 2y = 5$. 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: slope of a parallel line:

Explanation:

Step1: Rewrite the line in slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.
We start with the equation $4x+2y = 5$.
Subtract $4x$ from both sides: $2y=-4x + 5$.
Divide each term by 2: $y=\frac{-4x + 5}{2}=-2x+\frac{5}{2}$.
So the slope of the given line, $m=-2$.

Step2: Find the slope of the parallel line

Parallel lines have the same slope. So if a line is parallel to the line $y = - 2x+\frac{5}{2}$, its slope will be equal to the slope of the given line.
So the slope of the parallel line, $m_{parallel}=-2$.

Step3: Find the slope of the perpendicular line

The slopes of two perpendicular lines are negative reciprocals of each other. That is, if the slope of one line is $m$, the slope of the line perpendicular to it, $m_{perpendicular}=-\frac{1}{m}$.
Given $m = - 2$, then $m_{perpendicular}=-\frac{1}{-2}=\frac{1}{2}$.

Answer:

Slope of a perpendicular line: $\frac{1}{2}$
Slope of a parallel line: $-2$