First, put the original equation into slope-intercept form, or y=mx+ b, where m is the slope and b is the y-intercept. If you don't know about slope-intercept form, read about it on Purple Math: https://www.purplemath.com/modules/strtlneq.htm, or watch a video on Khan Academy:
Starting with -4x-9y=2, add 4x to both sides to get
Divide every term by (-9) to get
This is slope intercept form, so we see that the slope, or m, of the given line is -4/9.
Parallel lines never run into each other, so they have to have the same slope, right? So m of the parallel equation, let's call it m2, is also -4/9. Our equation so far for the second line is
(5, -1) is a point on the line, so plug these in for x and y to find b: -1= (-4/9)5 +b. Converting -1 to -9/9 and multiplying (-4/9) by 5, we get
-9/9 = -20/9 +b.
Add 45/9 to both sides to get b =11/9. Plug in m and b to get the line of the parallel equation:
For the Perpendicular line:
The slope of lines that are perpendicular are negative reciprocals of each other. So, change the sign, and flip the fraction. If the first slope is 4, the perpendicular slope is -1/4. Our original slope is -4/9, so the slope of any line perpendicular to that is positive 9/4.
For the third (perpendicular) equation, we start with y3=(9/4)x+b.
Again, plug (5, -1) in for x and y: -1=(9/4)5 +b.
Multiply on the right side: -1=45/4 +b, and change -1 to -4/4:
-4/4=45/4 +b, and subtract to get -49/4=b.
Plug in the new m and b to get the third equation:
Perpendicular: y=(9/4)x - 49/4
I hope that helps! Slope-intercept form is the best! It's super simple to work with!
Good luck in your math class!