A consumer purchases food $X$ and clothing $Y$. Her utility function is given by: $U(X,Y) = XY +10Y$, income is $\$100$ the price of food is $\$1$ and the price of clothing is $P_y$.
Derive the equation for the consumer’s demand function for clothing.
I found the first order conditions for $X$ and $Y$ and then solved for $Y$ which gave me $Y = X/P_y -10$ I then combined this with the budget constraint to get $2X - 10P_y = 100$ Please would it be possible to advise me whether whether my answers are correct as this is my first attempt at deriving demand functions. Also, is the utility quasilinear? I know that an equation of the form $U(X,Y) = f(X) + Y$ is quasilinear but I don't know if $U(X,Y) = f(x,Y) + Y$ would fit the category.