Answer:
The two lines are neither parallel nor perpendicular to one another.
Step-by-step explanation:
The slope [tex]m[/tex] gives the orientation of a line.
Make sure that the equation of both lines are in the slope-intercept form [tex]y = m\, x + b[/tex] (where [tex]m\![/tex] is the slope and [tex]b[/tex] is the [tex]y[/tex]-intercept) before comparing their slopes.
The equation of the first line [tex]y = 3\, x - 5[/tex] is already in the slope-intercept form. Compare this equation with the standard [tex]y = m\, x + b[/tex]. The slope of this line would be [tex]m = 3[/tex].
Rewrite the equation of the second line [tex]9\, x + 3\, y = 1[/tex] to obtain the slope-intercept equation of that line:
[tex]3\, y = -9\, x + 1[/tex].
[tex]\displaystyle y = -3\, x + \frac{1}{3}[/tex].
Thus, the slope of this line would be [tex]m = (-3)[/tex].
Two lines are parallel to one another if and only if their slopes are equal. In this question, [tex]3 \ne (-3)[/tex]. Thus, the two lines are not parallel to one another.
On the other hand, two lines are perpendicular to one another if and only if the product of their slopes is [tex](-1)[/tex]. In this question, [tex]3\times (-3) = (-9)[/tex], which is not [tex](-1)\![/tex]. Thus, these two lines are not perpendicular to one another, either.
