Why does $3x^2+y^3+9=-2xy$ cannot be written in explicit form?
By Andrew Adams •
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For this equation
$$3x^2 + y^3 + 9 = -2xy,$$
I checked on wolfram alpha and it doesn't seem to have an explicit form where $y$ is isolated and equals a function of $x$'s only.
The problem is that I don't understand why it can't be written explicitly, as the graph of the relation is a [non injective] function : it doesn't have more than one value of $y$ for each value of $x$.
So I should be able to write it as any normal function where $y = f(x)$ and its derivative with respect to $x$ should be expressed with only $x$'s instead of $x$'s and $y$'s - its derivative is : $\frac{-2(3x+y)}{3y^2+2x}$.
Please illuminate me, i'm really clueless.
$\endgroup$ 31 Answer
$\begingroup$According to wolfram alpha, there does exist a real explicit form:
