Answer:
[tex]\textsf{Perpendicular to given line}:y=-3x+14[/tex]
[tex]\textsf{Parallel to given line}: y=\dfrac13x-\dfrac83[/tex]
Step-by-step explanation:
Rewrite the given equation to make y the subject:
[tex]\begin{aligned}-3x+9y &=5 \\ \implies 9y &=3x+5 \\ \implies y &=\dfrac13x+\dfrac59\end{aligned}[/tex]
Therefore, the slope of the given equation is [tex]\frac13[/tex].
If two lines are perpendicular to each other, the product of their slopes will be -1. Therefore, the slope (m) of the line that is perpendicular to the given line is:
[tex]\begin{aligned}m \times \dfrac13 & =-1\\ \implies m & =-3\end{aligned}[/tex]
To find the equation of the line, substitute the found slope (-3) and the point (5, -1) into the point-slope form of a linear equation:
[tex]\begin{aligned}y-y_1 & =m(x-x_1)\\ \implies y-(-1) &=-3(x-5) \\ y+1 & =-3x+15 \\ y &=-3x+14\end{aligned}[/tex]
If two lines are parallel to each other, their slopes will be the same. Therefore, the slope (m) of the line that is parallel to the given line is [tex]\frac13[/tex]
To find the equation of the line, substitute the slope ([tex]\frac13[/tex]) and the point (5, -1) into the point-slope form of a linear equation:
[tex]\begin{aligned}y-y_1 & =m(x-x_1)\\\\ \implies y-(-1) &=\dfrac13(x-5) \\\\ y+1 & =\dfrac13x-\dfrac53 \\\\ y &=\dfrac13x-\dfrac83\end{aligned}[/tex]
Slope =m=1/3
Note that perpendicular lines have slopes negative reciprocal to each other .
Equation in point slope form
And
parallel lines have equal slope
Equation of parallel line
