How does one calculate an optimal consumption bundle if the MRS only contains one variable?
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.