Step 1: Convert to slope-intercept form to find the slope of the line.
This isn't necessary if you know all the shortcuts, it just makes your life a bit easier.
[tex]3x+6y-1=0\\6y=-3x+1\\y=-\frac{3}{6}x+\frac{1}{6}\\-\frac{1}{2}x+\frac{1}{6}[/tex]
Now we want to find the equations for our new lines. It's easiest to do this in slope-intercept form, so let's start finding the slope and y-intercept.
So for the perpendicular line, the slope is going to flip and change signs, this is aka its opposite reciprocal.
And for the parallel line, it'll stay the same.
[tex]-\frac{1}{2}[/tex] ⇒ [tex]\frac{2}{1}=2[/tex]
[tex]-\frac{1}{2}[/tex] ⇒ [tex]-\frac{1}{2}[/tex]
As for the y-intercept, just apply that slope until you get a point where x=0.
(5, 1) with a rise of 2 and a run of 1...let's work backwards to (0, b)
(5, 1), (4, -1), (3, -3), (2, -5), (1, -7), (0, -9).
(5, 1) with a rise of -1 and a run of 2...let's work backwards to (0, b)
(5, 1), (3, 0), (1, -1)...gonna have to take half a step here...(0, -1.5).
Now let's construct our equations in slope-intercept form.
[tex]y = 2x-9\\y=-\frac{1}{2}x-\frac{3}{2}[/tex]
And now it's time to convert to general form!
Make sure we have common denominators...check.
Multiply by the denominator...
[tex]y=2x-9\\2y=1x-3[/tex]
...check.
Aaaand bring everything over!
[tex]\boxed{-2x+y-9=0}\\\boxed{-x+2y-3=0}[/tex]
