Solve for $y$ explicitly or prove that it is impossible Let's say you want to solve for $y$ explicitly (it is by itself on one side of the equation with no $y$'s on the other side) in terms of any sort of function (elementary or non-elementary) in the following equation:
$$\sin(y)+e^y-xy=0$$

Is this possible? Or, if it isn't, prove that it is impossible.

Edit: After seeing an answer, I am providing another example:
$$\sin(xy)-\frac xye^y=0$$
This is a very "over the top" example, but my question is the same.
Edit #2: After seeing jgon's answer (thank you), I was wondering:

Is there any way to prove that these are impossible to solve explicitly for $y$ without using graphs?

 A: It is impossible. We can solve for $x$, and then graph $x$ as a function of $y$. $x=\frac{\sin(y)+e^y}{y}$. (See below, $y$ is on the horizontal axis, and $x$ is on the vertical axis)
Since there are two values for $y$ that give $x=10$, $y$ is not a function of $x$, or how could we pick what value to assign $y$ when $x=10$?
Therefore we cannot solve for $y$, since solving for $y$ expresses $y$ as a function of $x$.

Edit
To answer your second question, Wolfram Alpha can graph implicit functions like this. The picture we get this time is the following:

This is a really interesting curve, but this time, it is neither a function of $x$ nor $y$ for the same reasons as above, so we cannot solve the equation for either $x$ or $y$.
Edit 2
In general to show that a curve defined implicitly in terms of $x$ and $y$ cannot be solved for one of the variables say $y$, what you need to do is find two points on the curve where the value of $x$ is the same, but the points differ at their $y$ coordinate. See the first part of my answer, there were two points on the curve with $x=10$, and different $y$ coordinates. Then if you had $y=f(x)$, then how would you know what $f(10)$ is?
A: 
Is there any way to prove that these are impossible to solve explicitly for $y$ without using graphs?

The implicit function theorem gives conditions for the existence of a (in general local defined) function $y = f(x)$ which satisfies $0 = F(x, f(x))$. If they do not hold, it is not possible.
The problem of expressing a solution with elementary functions starts already with the problem what an elementary function is (link). 
A famous theorem about non-expressiveness in terms of elementary functions is Liouville's theorem about antiderivatives (link, link).
A: 1.)
By looking for algebraic values of terms in the given equation, one finds at first e.g. the following solutions.
$\sin(y)+e^y-xy=0:$ $x=\frac{e^{k\pi}}{k\pi}$, $y=k\pi$, where $k\in \mathbb{Z}$
$\sin(y)+e^y-xy=0:$ $x\in \{-\frac{k\pi}{2W(-\frac{1}{2}\sqrt\frac{k\pi}{sin(k\pi)})},-\frac{k\pi}{2W(\frac{1}{2}\sqrt\frac{k\pi}{sin(k\pi)})}\}$, $y=\frac{k\pi}{x}$, where $k\in \mathbb{Z}$, $W$: Lambert W function
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2.)
Interpret your zeroing equation as $H(x,y)=0$, $F(x)=0$ and $G(y)=0$. You have to try to find the global or local inverses of the functions $H$, $F$ or $G$. If it is not possible to find an inverse by only simplifying the left-hand sides of this equations, it is not possible to solve the equation in closed form on this way.
2a)
The possibility of an elementary inverse of an elementary function is treated in Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90.
Your equations don't have a structure for which a non-constant elementary function as solution exist.
2b)
There are no commonly used Special functions like Lambert W function as inverses of $H$, $F$ or $G$.
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3.)
The posiibilitiy of solvability of equations by elementary functions or by Liouvillian functions is treated in:
Liouville, J.: Mémoire sur la classification des transcendantes, et sur l’impossibilité d’exprimer les racines de certaines équations en fonction finie explicite des coefficients. Journ. Math. pures et appl. 2 (1837) 56-105
Liouville, J.: Suite du Mémoire sur la classification des transcendantes, et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients. Journal de mathématiques pures et appliquées 3 (1838) 523-547
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4.)
The question of solvability of equations by functions from a differential field (e.g. by elementary functions or by Liouvillian functions) is treated in Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22.
An explanation of such function classes is given e.g. in section 1 of Davenport, J. H.: What Might "Understand a Function" Mean. In: Kauers, M.; Kerber, M., Miner, R.;  Windsteiger, W.: Towards Mechanized Mathematical Assistants. Springer, Berlin/Heidelberg, 2007, page 55-65.
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See Wikipedia for the corresponding mathematical technical terms.
