lagrangian multiplier method help!

[WlND]

Hi,

I am trying to work on this problem, I have already sett up the Lagrange equation, i need to solve for x and y

[tex]L(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)+ \lambda[(x)+\theta (y)-\theta(z)-w][/tex]

[tex]\frac{\partial(L)}{\partial(x)}=x+\lambda=0 [/tex]
[tex]\frac{\partial(L)}{\partial(x)}=\alpha(y)+(\theta) (\lambda)=0 [/tex]
[tex]\frac{\partial(L)}{\partial(\lambda)}=(x)+\theta (y)-\theta(z)-w=0 [/tex]

[tex]\lambda=-x [/tex]
[tex]\alpha(y)=-(\theta)(\lambda)[/tex]
[tex]\alpha(y)=(\theta)(x)[/tex]
[tex](y)=\frac{\theta}{\alpha}(x)[/tex]

[tex](x)+\theta (y)=\theta(z)+w[/tex]
[tex](x)+\theta (\frac{\theta}{\alpha}(x))=\theta(z)+w[/tex]
>>>
[tex](x)=\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]}[/tex]

[tex](y)=\frac{\theta}{\alpha}(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})[/tex]

***
then I have to sub it in back into the original equation

[tex]f(x,y)=\frac{1}{2}(\alpha(y^2)+x^2)[/tex]
[tex]f(x,y)=\frac{1}{2}(\alpha(\frac{\theta^2}{\alpha^2 }[/tex][tex](\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)+(\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)[/tex]

[tex]f(x,y)=\frac{1}{2}[((\frac{\theta(z)+w}{[1+\theta (\frac{\theta}{\alpha})]})^2)((\frac{\theta}{\alpha})^2+1)][/tex]

but how can I simply this further, have I made any mistakes?

Any help will be greatly appreciated!

[WlND]

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