Question

Is `3y + 2= 2x` a direct variation equation and if so, what is the constant of variation?

Answer 1

`3y+2=2x` is not a direct variation equation

Explanation:

Method 1 For Seeing This
A direct variation equation will always pass through the origin
i.e. `(x,y)=(0,0)` will always be a solution to the equation.
`color(white)("XXX")`for `3y+2=2x`
`color(white)("XXX")3(0)+2 != 2(0)` so this condition is not satisfied.

Method 2 For Seeing This
For a direct variation equation, if `(a,b)` is a solution, then `(cxxa,cxxb)` is also a solution (for any constant `c`).
`color(white)("XXX")`Noting that `(4,3)` is a solution to `3y+2=2x`
`color(white)("XXX")`We can check using `c=5`
`color(white)("XXX")`to se if `(x,y)=(5xx4,5xx3)=(20,15)` is a solution.
`color(white)("XXX")3(15)+2 = 47 != 40 = 2(20)`
`color(white)("XXXXXX")`So, once again, we see that this is not a direct variation.

Method 3 For Seeing This
Any direct variation can be transformed into the form:
`color(white)("XXX")y=m*x` for some constant `m`
`color(white)("XXX")3y+2=2x` can be transformed into
`color(white)("XXXXXX")y= 2/3x-2/3`
`color(white)("XXX")`but there is no way to dispose of the `(-2/3)` to get it into the above form.