Answer
$L_{1}$ and $L_{2}$ are perpendicular lines.
Work Step by Step
Line $L_{1}$ has the equation $2x-5y=8$. Line $L_{2}$ passes through the points $(1,4)$ and $(-1,-1)$. Determine whether these lines are parallel lines, perpendicular lines or neither.
Solve the equation of $L_{1}$ for $y$, to express it in slope-intercept form:
$2x-5y=8$
$-5y=-2x+8$
$y=\dfrac{-2}{-5}x+\dfrac{8}{-5}$
$y=\dfrac{2}{5}x-\dfrac{8}{5}$
Since the slope-intercept form of the equation of a line is $y=mx+b$, where $m$ is the slope of the line and $b$ is its $y$-intercept, it can be seen from the equation obtained that the slope of $L_{1}$ is $m_{1}=\dfrac{2}{5}$.
Use the formula for the slope of a line, which is $m=\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}$, where $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are points through which it passes, to obtain the slope of $L_{2}$:
$m_{2}=\dfrac{-1-4}{-1-1}=\dfrac{-5}{-2}=\dfrac{5}{2}$
It can be seen that $m_{2}$ is equal to $-\dfrac{1}{m_{1}}$. Because of this, it can be concluded that $L_{1}$ and $L_{2}$ are perpendicular lines.
