The equation of a line in Slope-Intercept form, is:
[tex]y=mx+b[/tex]Where "m" is the slope of the line and "b" is the y-intercept.
By definition:
- The slopes of parallel lines are equal and the y-intercepts are different.
- The slopes of perpendicular lines are opposite reciprocals.
For this case you need to rewrite the equations given in the exercise in Slope-Intercept form by solving for "y".
- Line #1:
[tex]\begin{gathered} x-3y=6 \\ -3y=-x+6 \\ y=\frac{-x}{-3}+(\frac{6}{-3}) \\ \\ y=\frac{x}{3}-2 \end{gathered}[/tex]You can identify that:
[tex]\begin{gathered} m_1=\frac{1}{3} \\ \\ b_1=-2 \end{gathered}[/tex]- Line #2:
[tex]\begin{gathered} x-3y=9 \\ -3y=-x+9 \\ y=\frac{-x}{-3}+(\frac{9}{-3}) \\ \\ y=\frac{x}{3}-3 \end{gathered}[/tex]You can identify that:
[tex]\begin{gathered} m_2=\frac{1}{3} \\ \\ b_2=-3_{}_{} \end{gathered}[/tex]Therefore, since:
[tex]\begin{gathered} m_1=m_2 \\ b_1\ne b_2 \end{gathered}[/tex]You can conclude that: The graphs of the equation are parallel.
