Question

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

Answer 1

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`.