Answer
parallel
Work Step by Step
Using the properties of equality, the first equation, $
-x+3y=2
,$ is equivalent to
\begin{array}{l}
3y=x+2
\\\\
y=\dfrac{1}{3}x+\dfrac{2}{3}
.\end{array}
Using $y=mx+b$, where $m$ is the slope, the slope is
\begin{array}{l}
m_1=\dfrac{1}{3}
.\end{array}
Using the properties of equality, the second equation, $
6x-18y=3
,$ is equivalent to
\begin{array}{l}
-18y=-6x+3
\\\\
y=\dfrac{-6}{-18}x+\dfrac{3}{-18}
\\\\
y=\dfrac{1}{3}x-\dfrac{1}{6}
.\end{array}
Using $y=mx+b$, where $m$ is the slope, the slope is
\begin{array}{l}
m_2=\dfrac{1}{3}
.\end{array}
Since $m_1=m_2,$ then the given lines are $\text{
parallel
.}$