Intermediate Algebra, Chapter 3, 3.2, Section 3.2, Problem 78

State whether the lines with equation $2x + 5y = -8$ and $6 + 2x = 5y$ is parallel, perpendicular, or neither.

We find the slope of each line by solving each equation for $y$

Equation 1


$
\begin{equation}
\begin{aligned}

2x + 5y =& -8
&& \text{Given equation}
\\
\\
5y =& -2x - 8
&& \text{Subtract each side by $2x$}
\\
\\
y =& - \frac{2}{5}x - \frac{8}{5}
&& \text{Divide each side by $5$}


\end{aligned}
\end{equation}
$


Equation 2


$
\begin{equation}
\begin{aligned}

6 + 2x =& 5y
&& \text{Given equation}
\\
\\
y =& \frac{2}{5}x + \frac{6}{5}
&& \text{Divide each side by $5$}

\end{aligned}
\end{equation}
$


We know that the slope is given by the coefficient of $x$ and since the slopes $\displaystyle - \frac{2}{5}$ and $\displaystyle \frac{2}{5}$ are not equal and they are not negative reciprocals because their product is $\displaystyle - \frac{4}{25}$, not $-1$. Thus, the two lines are neither parallel nor perpendicular.