Constrained Optimization with equality

machintoshlover

Hey guys got a fairly simple problem that unfortunetly my simple brain won't work out.

The question is:

Consider the following maximization problem:

max 2X + 3Y
X,Y

s.t. PxX + PyY = 40

(a.) Write down the Lagrangian:

(b.) Write out the F.O.N.C and solve for the optimal values X* and Y*

Okay so part (a.) is grand

L(X,Y) = 2X + 3Y + λ(40 - PxX -PyY)

In part (b.) the F.O.N.C (First order necessary conditions) are:

dL(X,Y) = 2 - Px λ
DX

dL(X,Y) = 3 - Py λ
DY

dL(X,Y) = 40 - PxX - PyY
dλ

It is finding the optimal values X* and Y* I am having trouble with:

If anyone could give me a hint or two to pop on it would be most appreciated!

Cheers!

machintoshlover

Sorry forgot to put what I had down for finding my optimization!

I work out that:

Px = 2
py 3 which I believe is the slope of the budget constraint but my brain stops there - thanks and sorry it is probably very easy but I had a long day!