Intermediate Algebra, Chapter 3, Test, Section Test, Problem 8

Determine whether the pair of lines $2y = 3x + 12$ and $3y = 2x - 5$ is parallel, perpendicular, or neither.

We write each equation in slope-intercept form.

Equation 1


$
\begin{equation}
\begin{aligned}

2y =& 3x + 12
&& \text{Given equation}
\\
\\
y =& \frac{3}{2}x + \frac{12}{2}
&& \text{Divide each side by $2$}
\\
\\
y =& \frac{3}{2}x + 6
&& \text{Slope Intercept Form}

\end{aligned}
\end{equation}
$


Equation 2


$
\begin{equation}
\begin{aligned}

3y =& 2x - 5
&& \text{Given equation}
\\
\\
y =& \frac{2}{3}x - \frac{5}{3}
&& \text{Slope Intercept Form}

\end{aligned}
\end{equation}
$


Since the slopes are not equal and the product of their slopes is $\displaystyle \frac{3}{2} \left( \frac{2}{3} \right) = 1$, not $-1$, the two lines are neither parallel nor perpendicular.