Intermediate Algebra, Chapter 3, 3.2, Section 3.2, Problem 72

State whether the lines with equation $2x + 5y = -7$ and $5x - 2y = 1$ 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 =& -7
&& \text{Given equation}
\\
\\
5y =& -2x - 7
&& \text{Subtract each side by $2x$}
\\
\\
y =& - \frac{2}{5}x - \frac{7}{5}
&& \text{Divide each side by $5$}

\end{aligned}
\end{equation}
$



Equation 2



$
\begin{equation}
\begin{aligned}

5x - 2y =& 1
&& \text{Given equation}
\\
\\
-2y =& -5x + 1
&& \text{Subtract each side by $5x$}
\\
\\
y =& \frac{5}{2}x - \frac{1}{2}
&& \text{Divide each side by $-2$}

\end{aligned}
\end{equation}
$


We know that the slope is given by the coefficient of $x$ and since the product of the slopes is $\displaystyle - \frac{2}{5} \left( \frac{5}{2} \right) = -1$, the lines are perpendicular.