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

The constant of variation is ##-2##.

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 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##.