I have to solve for x,y,z,λ
First, note that L(ambda) <> 0 since that would the 4th equation impossible.
I'll get you started and see if you can finish. Re-arrange each equation:
1) yz = L(y+z)
2) xz = L(x+z)
3) xy = L(x+y)
Now take (1) divided by (2) and see if you can get a relationship between x and y. If you can, that will tell you how to get a relationship between x and z. The rest will tumble down. Let me know how you make out.
Well first I forgot to tell you that in my initial problem x,y,z>0
Anyway what you said actually worked thank you very much