How to solve these simultaneous equations using numerical methods? How to solve these simultaneous equations for $\alpha$ and $\lambda$ using numerical methods?
$\lambda * [(\frac{3}{4})^\frac{-1}{\alpha} - 1] = 11$
$\lambda * [(\frac{1}{4})^\frac{-1}{\alpha} - 1] = 85$ 
I am new to this. I tried resolving this using log. But, I could not solve it algebraically. 
 A: We wish to solve the following system for $\lambda$ and $\alpha$
$$\lambda\left(\left({3 \over 4}\right)^{{-1 \over \alpha}} - 1\right) = 11$$
$$\lambda\left(\left({1 \over 4}\right)^{{-1 \over \alpha}} - 1\right) = 85.$$
We will then define
$$\textbf{f}(\lambda,\alpha) = \begin{pmatrix} \lambda\left(\left({3 \over 4}\right)^{{-1 \over \alpha}} - 1\right) - 11 \\ \lambda\left(\left({1 \over 4}\right)^{{-1 \over \alpha}} - 1\right) - 85\end{pmatrix}.$$
In order to solve the system by Newton's method, we follow the following algorithm


*

*Choose an initial guess $\textbf{x}^{(1)}=(\lambda^{(1)},\alpha^{(1)})^T.$

*Solve $J_{\textbf{f}}(\textbf{x}^{(k)})\textbf{d}^{(k)} = -\textbf{f}(\textbf{x}^{(k)})$, for $k=1,2,...$ 

*Define $\textbf{x}^{(k+1)}= \textbf{d}^{(k)} + \textbf{x}^{(k)}$.

*Repeat, until you have an acceptable solution.


Note that $J_{\textbf{f}}(\textbf{x}^{(k)})$ is the jacobian of $\textbf{f}$ evaluated at $\textbf{x}^{(k)}$.
A: At some point, you will have to take log since one of the unknowns is in the power.   
An approach can be by dividing the two equations and cross-multiplying.
