How does one calculate an optimal consumption bundle if the MRS only contains one variable? For example, say the MRS has simplified to the form $MRS(x,y) = -y$, given a utility function 
\begin{equation*}
U(x, y) = e^{(x + ln(y))^{1/3}}
\end{equation*}

With a budget line of, say, $2x + y = 10$ which would yield a budget line slope of -2, equating the MRS to the BLS would simply yield $y = 2$, but there is no x to solve for in the equating of the two slopes.

Is the last step, in cases like this, to plug back into the budget constraint?  In which case, one would get $x = 4$, yielding an optimal consumption bundle of $(4, 2)$?
I wanted to add this as a reply to denesp's comment, but I do not have enough reps. 

MRS and a binding BC gives a system of two equations from which we can solve the optimum bundle. In case of income  = 10, these two equations have positive solutions, in case of income = 1, these two equations do not have positive solutions. See this:



Income = 1 just makes the choice set small enough that there is no MRS = price ratio, hence the corner solution. 

I completely agree that income =1 is really the more fun question.