Now that I have the alpha I will find x,y subject to the budgetary constraint of $10. I plan to use Lagrange multiplier method.

The price of apples is $1/3 as before, but the price of oranges is $1/6 (half of the price of apples).

Let budget function

Then I need to find a common tangent point of u(x,y) and g(x,y) subject to the constraint g(x,y)=10. Let be the Langrange multiplier:

After raising into 3rd power, I have and I am not sure how to proceed next...