In Which I Connect Algebra to Schoenberg

Time to find equations based on perpendicular and parallel lines. It's easier than I thought but not as simple as I tried to make it.

The rules:

• Perpendicular lines have opposite reciprocal slopes. Idiot's Guide explains this as if line g is perpendicular to line h, m = a/b for g and m = -(b/a) for h. That wonked my head. I need an example I can process easier. Hello there, twelve tone row matrix! Opposite reciprocal = retrograde inversion. (Why, oh, why did I not have this connection to make the first time around?)

• Parallel lines have the same slope. Of course they do! They're parallel! Otherwise they'd intersect at some point. Maybe not on the part of the graph you're looking at but eventually.

My practice problem for perpendicular slopes is

Write the equation of line k in slope-intercept form if k passes through (2,-3) and is perpendicular to the line with the equation x - 5y = 7.

I can do this, right?

First solve for y.

x - x - 5y = -x + 7

-5y = -x + 7

(5y/5) = (-1x/-5) + (7/-5)

y = (1/5)x - (7/5) = slope-intercept of the line perpendicular to k

Now I need to take the retrograde inversion opposite reciprocal to get the slope of k.

(1/5)(-1) = -5 = m of k

On to point-slope form

y - (-3) = -5(x - 2)

y + 3 = -5x + 10

y + 3 - 3 = -5x + (10 - 3)

Which leaves me with slope-intercept

y = -5x + 7

Self-assigned extra credit -- write in standard form!

5x + y = 7

I am so proud of me. :)

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Then we come to parallel lines.

Write the equation of line j in standard form if j passes through (-6,1) and is parallel to the line with the equation -2x + 6y = 7

I can figure this out on my own. I don't need no stinkin' book!

...switch the signs? *check answer* I am un-proud of me.

Put in slope-intercept form to solve for y

-2x + 6y = 7

6y = 2x + 7

(6y/6) = (2/6) + (7/6)

y = (1/3)x  + (7/6)

Plug the point given in the problem into point-slope

y - 1 = (1/3)[x - (-6)]

y - 1 = (1/3)(x + 6)

y - 1 = (1/3)x + 2

y - 1 + 1 = (1/3)x + 2 + 1

y = (1/3)x + 3 = slope intercept of j

Convert to standard form

-(1/3)x + y = (1/3)x + -(1/3)x + 3

-(1/3)x + y = 3

[(3/1) * -(1/3)x] + [(3/1) * (y/1)] = 3(3)

-x + 3y = 9

(-1)(-x) + (-1)(3y) = (-1)(9)

x - 3y = -9

Note to self: always read the directions.

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I've already done the chapters on inequalities, but I will write about those over the weekend. Maybe I'll be able to finally sit down and write the 3 chapters I did pre-blog as well. Monday starts my descent into the matrix sans Neo. Wish me luck!