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.
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.
⇒ 
⇒
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.
And now it's time to convert to general form!
Make sure we have common denominators...check.
Multiply by the denominator...
...check.
Aaaand bring everything over!
This isn't necessary if you know all the shortcuts, it just makes your life a bit easier.
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.
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.
And now it's time to convert to general form!
Make sure we have common denominators...check.
Multiply by the denominator...
...check.
Aaaand bring everything over!