For the equation #-4y=8x#, what is the constant of variation?

1 Answer
Jan 9, 2017

The constant of variation is #-2#.

Explanation:

We can solve this equation for #y# in terms of #x#, by dividing both sides by #-4#:

#-4y=8x#

#color(white)(-4)y=(8x)/-4#

#color(white)(-4y)=-2x#

Now we have an equation that says, "#y# is always #-2# times as much as #x# is". It is this #-2# that is our constant of variation, because every time #x# goes up by #1#, #y# will go "up" by #-2# (i.e. down by #2#).

Can we show this?

Let #x^star=x+1# (i.e. #x^star# is one more than #x#). If #y^star# is in direct variation with #x^star#, with a constant of variation of #-2#, then

#y^star = -2x^star#

Which means

#y^star = -2(x+1)" "#(since #x^star = x+1#)
#color(white)(y^star) = -2x-2#

But wait, #y=-2x#, so we have

#y^star = y - 2#

And there we go! When #x^star# is 1 more than #x#, we see #y^star# is #2# less than #y#. In other words: when #x# goes up by #1#, #y# goes down by #2#.