Question

How do you determine the constant of variation for the direct variation given by `2x=4y`?

Answer 1

Constant of variation `= 1/2` (assuming the normal situation where `y` is considered the dependent variable).

Explanation:

Normally when defining a direct variation between the variables `x` and `y`, the variable `y` is considered dependent upon `x` (i.e. `y` is thought of as being equivalent to `f(x)`)
and
the direct variation equation, in this case, is of the form:
`color(white)("XXX")y=cx` where `c` is the constant of variation.

`color(white)("XXXXXX")2x=4y`
`color(white)("XXXXXX")iff 4y=2x`
`color(white)("XXXXXX")iff y = (1/2)x`
`color(white)("XXX")rarr c= (1/2)`

It is possible (although unlikely) that the intent was to specify `x` as the dependent variable (i.e. `x _= g(y)`)
In this case the direct variation equation would be of the form:
`color(white)("XXX")x=cy`

`color(white)("XXXXXX")2x=4y`
`color(white)("XXXXXX")iff x=2y`
`color(white)("XXX")rarr c=2`