As a recap, standard form is writing an equation as Ax+By = C, with the stipulation that A, B, and C must be integers. I prefer the form Ax-By=C instead of + because then the A and the B values end up being the rise and the run of the line. For example, 2x-5y=20 has a rise of 2 units for every run of 5 units, for a slope of 2/5. Additionally, the example x+3y=15 rises 1 unit for every run of 3 units backward -- backwards because it was written with + instead of -.
Here's some reasons I think standard form is useful:
- It's neat and orderly, with no fractions or decimals
- It's easy to plug in points (because x and y are in order and the multipliers are always integers) which is how you check if a point is on a line
- It still reveals rise and run, with some understanding that the terms must be subtracted
- It is easier to answer questions like finding equation of lines through (3,7) and (8, 2) because you don't have to find the y-intercept
- It can describe vertical lines (x=___) that have undefined slope
- It can be scaled by multiplying. (by the way, dividing everything by C gives you an interesting form, x/dx-y/dy=1....)
- It is the form of choice for combining equations when systems of equations rolls around... my favorite topic in the algebra curriculum.
- It can easily be expanded into 3 (or more) dimensions, where y=mx+b has no easy expansion
To celebrate standard form, and praise some of its merits, I wrote a fight song for it, which is to the tune of the greatest college fight song out there, Michigan State. The lyrics are posted below, as well as a link to a tune on Youtube so you can sing along.
A x + b Y equals C
is known as standard form by all
It always graphs a straight line
The only form that does them ALL
Other forms can't do straight up lines
'cause their slopes are undefined
We simply let x equals ___
we'll be fine!
A's the rise and B's the run
As long as we subtract 'em!
Plug in a point to find the C
And make sure they are nice IN-TE-GERS!
Multi-pl'ing by de -nom'-na-tors
Makes fractions disappear! (poof!)
X's! Y's! Equals C!
Standard form's the one for me!
(Dance during musical interlude, then sing chorus again)