Question

Are the graphs of the equations parallel, perpendicular, or neither?x -3y = 6 and x - 3y = 9

Answer 1

The equation of a line in Slope-Intercept form, is:

$y=mx+b$

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:

$\begin{gathered} x-3y=6 \\ -3y=-x+6 \\ y=\frac{-x}{-3}+(\frac{6}{-3}) \\ \\ y=\frac{x}{3}-2 \end{gathered}$

You can identify that:

$\begin{gathered} m_1=\frac{1}{3} \\ \\ b_1=-2 \end{gathered}$

- Line #2:

$\begin{gathered} x-3y=9​ \\ -3y=-x+9 \\ y=\frac{-x}{-3}+(\frac{9}{-3}) \\ \\ y=\frac{x}{3}-3 \end{gathered}$

You can identify that:

$\begin{gathered} m_2=\frac{1}{3} \\ \\ b_2=-3_{}_{} \end{gathered}$

Therefore, since:

$\begin{gathered} m_1=m_2 \\ b_1\ne b_2 \end{gathered}$

You can conclude that: The graphs of the equation are parallel.