[my intial work on the problem](https://i.stack.imgur.com/mn0M5.jpg) I appoligize, I do not quite know how to use math formatting for this post.

I've been toying with this problem for a few days. I must have attempted to solve it in 4 different ways. Essentially we have a constrained optimization problem given by:

f(x,y)=x^2+2y^2 , constrained by x^2+16y^2=16 = g(x,y)

Please see the attached picture for some of my intial thoughts/workings.

I'm trying to disect my main problem when it comes to solving for λ. I can set up my system of three equations with the last equation being my g(x,y). Every video/notes I see online in which they attempt to solve for λ works out with the gradient of f(x,y) being a function of x, and grad(g(x,y)) either being a constant, or a function of (y,z) for example. I understand that case.

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> If L(x,y,λ) = grad(f)-λ(grad(g)), then grad(f)=λ(grad(g)). In this particular case I can see my first equation is 2x=λ(2x)
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I hope this plus the picture was specfic enough to get any advice on this problem! I'm here to answer any further questions!

tldr: how do i solve for λ given the system

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> 2x=λ(2x) , 3y=λ(32y) , 16=x^2+16y^2
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