6. the equation of line a is 3x + 6y - 1 = 0. give the equation of a line that passes through the
point (5,1) that is
/1 a. perpendicular to line a

/1 b. parallel to line a

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!