Intermediate Algebra, Chapter 3, 3.2, Section 3.2, Problem 76

State whether the lines with equation $3x = y$ and $2y - 6x = 5$ is parallel, perpendicular, or neither.

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

Equation 1


$
\begin{equation}
\begin{aligned}

y =& 3x
&& \text{Given equation}

\end{aligned}
\end{equation}
$


Equation 2


$
\begin{equation}
\begin{aligned}

2y -6x =& 5
&& \text{Given equation}
\\
\\
2y =& 6x + 5
&& \text{Add each side by $6x$}
\\
\\
y =& 3x + \frac{5}{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 slopes are equal, the two lines are parallel.