I am trying to understand Markowitz portfolio theory with two stocks. I have set up the variance to be minimized: $\frac{1}{2}(w_1^2\sigma_2^2+w_2^2\sigma_2^2+2w_1w_2\sigma_{12})$ and the boundary constraints are: $w_1+w_2=1$ and $w_1r_1+w_2r_2=r$. So the Langrangian becomes: $L=\frac{1}{2}(w_1^2\sigma_2^2+w_2^2\sigma_2^2+2w_1w_2\sigma_{12})-\lambda_1(w_1+w_2-1)-\lambda_2(w_1r_1+w_2r_2-r)$
Taking partial derivatives I get:
$\frac{\partial{}L}{\partial{}w_1}=w_1\sigma_1^2+w_2\sigma_{12}-\lambda_1-\lambda_2r_1$ and $\frac{\partial{}L}{\partial{}w_2}=w_2\sigma_2^2+w_1\sigma_{12}-\lambda_1-\lambda_2r_2$
Setting the partial derivatives equal to zero and isolating $\lambda_1$ I get:
$w_1\sigma_1^2+w_2\sigma_{12}-\lambda_2r_1=w_2\sigma_2^2+w_1\sigma_{12}-\lambda_2r_2$
Making use of $w_2=1-w_1$ I get the equation:
$\lambda_2=\frac{w_1(\sigma_1^2+\sigma_2^2-2\sigma_{12})+\sigma_{12}-\sigma_2^2}{r_1-r_2}$
But I am not sure how to proceed from there. How can I find an expression for $w_1$?